1

Counterfactuals in the Real World

Jim Woodward and Mark WilsonPreliminary draft

0 Introduction Current philosophical research into the descriptive utilities of counterfactual data within science have taken a decidedly odd turn in recent years at least in our opinion Rather than attempting to directly categorize the ways in which the direct exploitation of counterfactual data assist scientific endeavor philosophical focus has largely turned to the alleged manner in which valid counterfactuals require a ldquometaphysical groundingrdquo (or allied support) within the ldquofundamental laws of naturerdquo The feasibility of this secondary project rests upon a bed rock of philosophy of science assumptions that we personally find dubious but we will not attempt to litigate these large scale issues here The purpose of this note is rather to remark that somewhere along the metaphysical quest for ldquogroundingrdquo the intriguing manners in which sturdy appeals to counterfactual considerations serve as vehicles for injecting reliable upper scale data into real life scientific modeling have become neglected in a topsy-turvy manner that obscures the important utilities that well-selected counterfactuals offer science In practical life the canny exploitation of easy-to-assess counterfactual data frequently serves as a vital safeguard against the modeling hazards of excessive lower scale speculation In this spirit the remarkable ldquoavoiding hypotheses of an excessively grounded naturerdquo characteristics of standard Lagrangian methodologies within classical physics were warmly appreciated by all of the great philosopherscientists of the late nineteenth century (Maxwell Duhem Hertz etc) But the methodological advantages that so impressed these venerable thinkers scarcely appear within the modern narratives supplied by the metaphysicians of grounding who often dismiss ldquoreliabilityrdquo as an ldquoepistemological characteristic of no metaphysical interest whatsoeverrdquo But surely codifying the remarkable utilities that direct forms of counterfactual appeal offer science represents an important methodological concern even if the metaphysicians choose to look elsewhere

It is worth noting that Nelson Goodmanrsquos original interest in ldquothe problem of counterfactual conditionalsrdquo lay in the problem of explaining why seemingly fictive considerations have any sort of useful role to play within productive science Such concerns trace to the late nineteenth century when the injection of modal considerations of any kind into science became viewed with suspicion1 In many circumstances these anti-intensionalist concerns were well motivated but in banning all contrary-to-fact claims altogether the early methodologists clearly overshot their legitimate brief In our estimation Goodman and his many contemporary followers overreach in a reciprocal manner they seek acceptable ldquosemanticrdquo underpinnings for a very wide collection of contrary-to-fact claims even when the latter possess no evident scientific

1 The considerations that Carnap and Quine later codified into ldquothe thesis of extensionalityrdquo were framed in this eramdashsee WS for more on this

2

utility whatsoever In this way they may fail to address the original concerns about scientific practice ably at all Analogy even if we find ourselves able to supply suitable truth-conditions for the claim ldquoEvents in Europe would have proved less catastrophic if Hitlerrsquos application to Arts College had been acceptedrdquo our parsing will not explain why this claim seems out of place within a physics journal But much sharper answers can be supplied than this In particular in this essay we will explicate several concrete manners in which specific collections of counterfactual data serve as the natural vehicles for codifying important aspects of nature in a manner that directly addresses our original ldquohow can seemingly fictive concerns play a useful role within productive sciencerdquo query We make no claims of grander scope than this However we doubt that projects pursued in the wider vein of Goodman Stalnaker or Lewis will contribute helpfully to the original concerns from which such projects sprung

On this same score the extremely broad choice of targets adopted by these authors practically ensures that the answers they supply will prove reductive in some manner or other viz arising as attempts to ldquogroundrdquo the counterfactual claims upon some bed rock of more ldquoacceptable scientific ingredientsrdquo such as ldquolaws of naturerdquo But our own methodological ambitions are not reductive in this manner at all we only wish to explicate the concrete manners in which certain restricted collections of counterfactual assertion that directly capture aspects of nature critical to describing its wider behaviors in a profitable and reliable manner The factors we will highlight are mathematical and inferential in character certain collections of trusted counterfactual data provide a solid platform from which further patterns of scientific reasoning can be carried forward effectively

Perhaps the following example provides a better feel for the answers we seek Any text in celestial mechanics will traffic extensively in so-called Trojan counterfactuals that start ldquoIf a small mass L4 were located at the precise apex of an equilateral triangle framed between larger masses E and M as illustrated then helliprdquo Such weirdly exacting circumstances are never realized in nature so why does our textbook waste so much time with them An uninformative philosopherrsquos answer respondsrdquo Oh science always idealizes you find that in all of the textbooksrdquo But the same primers rarely pose comparable counterfactual questions with respect to isoscelean planetary arrangements Why not A truly informative answer observes that the special stable equilibrium features of the Trojan placement serve as a vital jumping off point for a raft of important reasoning patterns utilizing perturbative techniques

In this general manner the answers we will supply to our original ldquohow can seemingly fictive concerns play a useful role within productive sciencerdquo worries adopt a similar form certain collections of counterfactual data serve as the natural jumping points for a host of important reasoning patterns within science One of us (Woodward) has long attempted to identify some of the central counterfactual components that implicitly lie within common forms

3

of causal reasoning within science but his proposals have often been dismissed by the Goodman-Stalnaker-Lewis school as not explicating the ldquometaphysical groundingrdquo of his counterfactuals sufficiently In our view these demands even if fulfilled will not address Woodwardrsquos original methodological concerns very effectively and run the danger of directing philosophical attention along unrewarding pathways

As noted above we shall not attempt to diagnose all of the tacit presumptions that have made such reductive projects attractive to their advocates We will rest content with the baseline observation that such proposals donrsquot appear to address the original questions of the utilities of scientific counterfactuals very effectively Unfortunately the ambitious schemes of these grander ldquometaphysical groundingrdquo projects frequently appeal to notions like ldquoauxiliary conditionrdquo and ldquolaw of naturerdquo in a cavalier manner that discourages adequate attention to the scientific details we wish to highlight From time to time in the argument ahead we will pause to discuss some of the unhappy ways in which the grander reductive visions have mutilated well-established diagnostic terminologies in an unhelpful manner2

1 ldquoGroundingrdquo and manipulation counterfactuals

According to popular philosophical opinion counterfactuals require ldquosupportrdquo or ldquogroundingrdquo by laws of naturemdashindeed it often assumed that that the modal ingredients conveyed by counterfactuals are entirely supplied by underlying ldquolawsrdquo with the other ingredients (eg initial and boundary conditions) entering into their assessment in a modally inert manner Proposals along these lines have taken several different forms In ldquometalinguisticrdquo accounts of counterfactuals beginning with Goodman (1955) for a counterfactual to be true its consequent must be derivable from its antecedent in conjunction with other premises with ldquolawsrdquo being among the other premises A similar conception is defended by Maudlin (2007) and by Paul and Hall (2013) although here the grounding laws are assumed to take a rather specific form ndash they are taken to be laws of temporal evolution and the systems to which they apply are taken to admit of a well-posed initial value problem an assumption we shall discuss critically below Lewisrsquo well-known account of counterfactuals (1973) is often regarded as an alternative to such meta-linguistic theories but the similarity relations among possible worlds that determine the truth-values of counterfactuals also assign a central role to laws of nature again supporting the general claim that laws play some sort of grounding role with respect to counterfactuals3

2 PA3 Through this essay we do not employ ldquogroundrdquo in its restricted modern usage as an updated surrogate for ldquosupervenience baserdquo (as in eg Correia and Schneider 2012) but simply as a demand for some obligatory conceptual linkage between counterfactuals and ingredients that are counterfactual-free Our criticisms apply to most proposals that invoke ldquogroundingrdquo in a more technical sense but we are not limiting our attention to these

4

When we ask why such grounding in laws is required4 for counterfactuals a number of different motivations are offered Some writers (eg Psillos 2004 2007) have epistemic concernsmdashthey hold that counterfactuals cannot be reliably assessed for truth or falsity without information about grounding laws Other writers (eg Maudlin 2007 but also Psillos 2004) offer broadly semantic concernsmdashthey claim that stand-alone counterfactuals without grounding laws are (commonly or always) vague context-dependent and unclear in a way that makes them unsuitable for use in science Providing backing laws is required to repair these deficiencies Still other writers have more straightforward metaphysical concerns They claim that counterfactuals cannot be ldquobarely truerdquomdashthat counterfactuals (and modal claims more generally) require grounding in what is ldquoactual rdquo and that laws are prime (and perhaps the only) candidates for such ldquoactualrdquo truth makers Often although not inevitably the underlying goal is a reduction of some kind First true counterfactuals are understood in terms of claims about laws and non-modal initial and boundary conditions Laws in turn are understood as regularities meeting further conditions as in the Best Systems Analysis so that a complete Humean reduction is achieved

Many philosophers cling to various claims about ldquogroundingrdquo with unshakeable conviction and find it astonishing that anyone should harbor any doubts about them But from whence do these firm assurances stem As we shall see they are not based upon the attentive study of scientific practice which instead reveals more complex patterns in which notions of ldquolawrdquo and ldquocounterfactualrdquo comingle in mutually supportive yet patently non-reductive ways Nor do these supposed grounding requirements reflect any obvious semantic or conceptual connection between laws and counterfactuals For one thing children master counterfactual constructions long before they grasp the somewhat recondite contours of ldquolawrdquo (they understand that ldquoif I snitch that cookie Irsquoll be punishedrdquo long before they display any evident awareness of a ldquolaw of naturerdquo5) Nor is any epistemological motivation for maintaining a ldquogroundingrdquo thesis immediately plausible Many counterfactuals that describe the outcomes of potential manipulations along the lines of ldquoif variable A were tweaked to a new value α effect B would followrdquo can be firmly established as correct solely on the basis of a well-designed experiment in the face of total ignorance of any underlying ldquolawrdquo Firm counterfactual knowledge about what would happen to a childrsquos headache if she were to take an aspirin have been familiar to canny parents for generations long before the chemical underpinnings of these dependencies became

4 We emphasize that the grounding claims that are our target donrsquot just contend that as a matter of empirical fact it will turn out that for every true counterfactual there will be an underlying law Rather it is contended that this must be true and that the ingredients that underlie the counterfactual (laws initial and boundary conditions etc) can all be understood in non-modal ldquoactualistrdquo terms 5 See Harris (2000) Weisberg and Gopnik (2013) and Woodward 2011 for empirical support for such claims about childrenrsquos knowledge of counterfactuals

5

clear at all (these represent a recent discovery6) Even so an accurate modern description of these mechanisms doesnrsquot contain anything that looks much like the ldquolawsrdquo of traditional philosophical expectation (as wersquoll witness in simpler circumstances below)

Our concern with these claims about grounding arises in the following way Recent work on causation (eg Woodward 2003) has connected causal claims to counterfactuals concerning the outcomes of possible experimental manipulations Roughly speaking the guiding idea is that (11) ldquoC causes Erdquo should be elucidated in terms of (12) the claim that E would change in an appropriate experimental manipulation were to be performed on C (We call these manipulation counterfactuals in what follows they are instances of the ldquotweaking a variable to a new valuerdquo claims of the prior paragraph) This proposal has been frequently criticized from the perspective of the grounding claims described above counterfactuals of form (12) need to be further supported by laws that do not themselves presuppose counterfactuals or modally committed notions Hence the proposed elucidation of (11) in terms of (12) is (at best) incomplete and needs to be fleshed out by an additional account of the reductive relationships between the counterfactuals (12) and the laws that ground them When these details are provided treatments like Woodwardrsquos treatment turn into nomic (viz law-based) accounts of causal claims (and in turn perhaps into regularity accounts if the underlying laws are reduced to regularities) and the manipulationistcounterfactual elements all fade away as unnecessary

Hiddleston (2005) is representative of these criticisms

While Woodward relies heavily on counterfactuals he says surprisingly little about their truth conditions (He criticizes Lewis for using nonintervention counterfactuals but does not say what makes any of them true) This raises puzzles because standard theories appeal directly to natural laws ldquoA 1048687rarr Crdquo is true iff A background facts and actual laws jointly imply C (547)

He adds

Woodwardrsquos arguments support only the weaker contention that there issome close connection between counterfactuals and causal explanation Forexample it remains open to say that this connection is that the truth conditions of counterfactuals immediately involve laws and that their causal and explanatory force derives from that fact (547)

How do these writers explain the fact that practicing scientists interested in diagnosing

causal patterns display little evident concern with respect to the ldquogroundings in lawrdquo that philosophers demand Some of Woodwardrsquos ldquoactualistrdquo critics treat this as merely a matter of 6 In contrast the headache relieving properties of the substance we call aspirin was known to the ancient Greeks

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

2

utility whatsoever In this way they may fail to address the original concerns about scientific practice ably at all Analogy even if we find ourselves able to supply suitable truth-conditions for the claim ldquoEvents in Europe would have proved less catastrophic if Hitlerrsquos application to Arts College had been acceptedrdquo our parsing will not explain why this claim seems out of place within a physics journal But much sharper answers can be supplied than this In particular in this essay we will explicate several concrete manners in which specific collections of counterfactual data serve as the natural vehicles for codifying important aspects of nature in a manner that directly addresses our original ldquohow can seemingly fictive concerns play a useful role within productive sciencerdquo query We make no claims of grander scope than this However we doubt that projects pursued in the wider vein of Goodman Stalnaker or Lewis will contribute helpfully to the original concerns from which such projects sprung

On this same score the extremely broad choice of targets adopted by these authors practically ensures that the answers they supply will prove reductive in some manner or other viz arising as attempts to ldquogroundrdquo the counterfactual claims upon some bed rock of more ldquoacceptable scientific ingredientsrdquo such as ldquolaws of naturerdquo But our own methodological ambitions are not reductive in this manner at all we only wish to explicate the concrete manners in which certain restricted collections of counterfactual assertion that directly capture aspects of nature critical to describing its wider behaviors in a profitable and reliable manner The factors we will highlight are mathematical and inferential in character certain collections of trusted counterfactual data provide a solid platform from which further patterns of scientific reasoning can be carried forward effectively

Perhaps the following example provides a better feel for the answers we seek Any text in celestial mechanics will traffic extensively in so-called Trojan counterfactuals that start ldquoIf a small mass L4 were located at the precise apex of an equilateral triangle framed between larger masses E and M as illustrated then helliprdquo Such weirdly exacting circumstances are never realized in nature so why does our textbook waste so much time with them An uninformative philosopherrsquos answer respondsrdquo Oh science always idealizes you find that in all of the textbooksrdquo But the same primers rarely pose comparable counterfactual questions with respect to isoscelean planetary arrangements Why not A truly informative answer observes that the special stable equilibrium features of the Trojan placement serve as a vital jumping off point for a raft of important reasoning patterns utilizing perturbative techniques

In this general manner the answers we will supply to our original ldquohow can seemingly fictive concerns play a useful role within productive sciencerdquo worries adopt a similar form certain collections of counterfactual data serve as the natural jumping points for a host of important reasoning patterns within science One of us (Woodward) has long attempted to identify some of the central counterfactual components that implicitly lie within common forms

3

of causal reasoning within science but his proposals have often been dismissed by the Goodman-Stalnaker-Lewis school as not explicating the ldquometaphysical groundingrdquo of his counterfactuals sufficiently In our view these demands even if fulfilled will not address Woodwardrsquos original methodological concerns very effectively and run the danger of directing philosophical attention along unrewarding pathways

As noted above we shall not attempt to diagnose all of the tacit presumptions that have made such reductive projects attractive to their advocates We will rest content with the baseline observation that such proposals donrsquot appear to address the original questions of the utilities of scientific counterfactuals very effectively Unfortunately the ambitious schemes of these grander ldquometaphysical groundingrdquo projects frequently appeal to notions like ldquoauxiliary conditionrdquo and ldquolaw of naturerdquo in a cavalier manner that discourages adequate attention to the scientific details we wish to highlight From time to time in the argument ahead we will pause to discuss some of the unhappy ways in which the grander reductive visions have mutilated well-established diagnostic terminologies in an unhelpful manner2

1 ldquoGroundingrdquo and manipulation counterfactuals

According to popular philosophical opinion counterfactuals require ldquosupportrdquo or ldquogroundingrdquo by laws of naturemdashindeed it often assumed that that the modal ingredients conveyed by counterfactuals are entirely supplied by underlying ldquolawsrdquo with the other ingredients (eg initial and boundary conditions) entering into their assessment in a modally inert manner Proposals along these lines have taken several different forms In ldquometalinguisticrdquo accounts of counterfactuals beginning with Goodman (1955) for a counterfactual to be true its consequent must be derivable from its antecedent in conjunction with other premises with ldquolawsrdquo being among the other premises A similar conception is defended by Maudlin (2007) and by Paul and Hall (2013) although here the grounding laws are assumed to take a rather specific form ndash they are taken to be laws of temporal evolution and the systems to which they apply are taken to admit of a well-posed initial value problem an assumption we shall discuss critically below Lewisrsquo well-known account of counterfactuals (1973) is often regarded as an alternative to such meta-linguistic theories but the similarity relations among possible worlds that determine the truth-values of counterfactuals also assign a central role to laws of nature again supporting the general claim that laws play some sort of grounding role with respect to counterfactuals3

2 PA3 Through this essay we do not employ ldquogroundrdquo in its restricted modern usage as an updated surrogate for ldquosupervenience baserdquo (as in eg Correia and Schneider 2012) but simply as a demand for some obligatory conceptual linkage between counterfactuals and ingredients that are counterfactual-free Our criticisms apply to most proposals that invoke ldquogroundingrdquo in a more technical sense but we are not limiting our attention to these

4

When we ask why such grounding in laws is required4 for counterfactuals a number of different motivations are offered Some writers (eg Psillos 2004 2007) have epistemic concernsmdashthey hold that counterfactuals cannot be reliably assessed for truth or falsity without information about grounding laws Other writers (eg Maudlin 2007 but also Psillos 2004) offer broadly semantic concernsmdashthey claim that stand-alone counterfactuals without grounding laws are (commonly or always) vague context-dependent and unclear in a way that makes them unsuitable for use in science Providing backing laws is required to repair these deficiencies Still other writers have more straightforward metaphysical concerns They claim that counterfactuals cannot be ldquobarely truerdquomdashthat counterfactuals (and modal claims more generally) require grounding in what is ldquoactual rdquo and that laws are prime (and perhaps the only) candidates for such ldquoactualrdquo truth makers Often although not inevitably the underlying goal is a reduction of some kind First true counterfactuals are understood in terms of claims about laws and non-modal initial and boundary conditions Laws in turn are understood as regularities meeting further conditions as in the Best Systems Analysis so that a complete Humean reduction is achieved

Many philosophers cling to various claims about ldquogroundingrdquo with unshakeable conviction and find it astonishing that anyone should harbor any doubts about them But from whence do these firm assurances stem As we shall see they are not based upon the attentive study of scientific practice which instead reveals more complex patterns in which notions of ldquolawrdquo and ldquocounterfactualrdquo comingle in mutually supportive yet patently non-reductive ways Nor do these supposed grounding requirements reflect any obvious semantic or conceptual connection between laws and counterfactuals For one thing children master counterfactual constructions long before they grasp the somewhat recondite contours of ldquolawrdquo (they understand that ldquoif I snitch that cookie Irsquoll be punishedrdquo long before they display any evident awareness of a ldquolaw of naturerdquo5) Nor is any epistemological motivation for maintaining a ldquogroundingrdquo thesis immediately plausible Many counterfactuals that describe the outcomes of potential manipulations along the lines of ldquoif variable A were tweaked to a new value α effect B would followrdquo can be firmly established as correct solely on the basis of a well-designed experiment in the face of total ignorance of any underlying ldquolawrdquo Firm counterfactual knowledge about what would happen to a childrsquos headache if she were to take an aspirin have been familiar to canny parents for generations long before the chemical underpinnings of these dependencies became

4 We emphasize that the grounding claims that are our target donrsquot just contend that as a matter of empirical fact it will turn out that for every true counterfactual there will be an underlying law Rather it is contended that this must be true and that the ingredients that underlie the counterfactual (laws initial and boundary conditions etc) can all be understood in non-modal ldquoactualistrdquo terms 5 See Harris (2000) Weisberg and Gopnik (2013) and Woodward 2011 for empirical support for such claims about childrenrsquos knowledge of counterfactuals

5

clear at all (these represent a recent discovery6) Even so an accurate modern description of these mechanisms doesnrsquot contain anything that looks much like the ldquolawsrdquo of traditional philosophical expectation (as wersquoll witness in simpler circumstances below)

Our concern with these claims about grounding arises in the following way Recent work on causation (eg Woodward 2003) has connected causal claims to counterfactuals concerning the outcomes of possible experimental manipulations Roughly speaking the guiding idea is that (11) ldquoC causes Erdquo should be elucidated in terms of (12) the claim that E would change in an appropriate experimental manipulation were to be performed on C (We call these manipulation counterfactuals in what follows they are instances of the ldquotweaking a variable to a new valuerdquo claims of the prior paragraph) This proposal has been frequently criticized from the perspective of the grounding claims described above counterfactuals of form (12) need to be further supported by laws that do not themselves presuppose counterfactuals or modally committed notions Hence the proposed elucidation of (11) in terms of (12) is (at best) incomplete and needs to be fleshed out by an additional account of the reductive relationships between the counterfactuals (12) and the laws that ground them When these details are provided treatments like Woodwardrsquos treatment turn into nomic (viz law-based) accounts of causal claims (and in turn perhaps into regularity accounts if the underlying laws are reduced to regularities) and the manipulationistcounterfactual elements all fade away as unnecessary

Hiddleston (2005) is representative of these criticisms

While Woodward relies heavily on counterfactuals he says surprisingly little about their truth conditions (He criticizes Lewis for using nonintervention counterfactuals but does not say what makes any of them true) This raises puzzles because standard theories appeal directly to natural laws ldquoA 1048687rarr Crdquo is true iff A background facts and actual laws jointly imply C (547)

He adds

Woodwardrsquos arguments support only the weaker contention that there issome close connection between counterfactuals and causal explanation Forexample it remains open to say that this connection is that the truth conditions of counterfactuals immediately involve laws and that their causal and explanatory force derives from that fact (547)

How do these writers explain the fact that practicing scientists interested in diagnosing

causal patterns display little evident concern with respect to the ldquogroundings in lawrdquo that philosophers demand Some of Woodwardrsquos ldquoactualistrdquo critics treat this as merely a matter of 6 In contrast the headache relieving properties of the substance we call aspirin was known to the ancient Greeks

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

3

of causal reasoning within science but his proposals have often been dismissed by the Goodman-Stalnaker-Lewis school as not explicating the ldquometaphysical groundingrdquo of his counterfactuals sufficiently In our view these demands even if fulfilled will not address Woodwardrsquos original methodological concerns very effectively and run the danger of directing philosophical attention along unrewarding pathways

As noted above we shall not attempt to diagnose all of the tacit presumptions that have made such reductive projects attractive to their advocates We will rest content with the baseline observation that such proposals donrsquot appear to address the original questions of the utilities of scientific counterfactuals very effectively Unfortunately the ambitious schemes of these grander ldquometaphysical groundingrdquo projects frequently appeal to notions like ldquoauxiliary conditionrdquo and ldquolaw of naturerdquo in a cavalier manner that discourages adequate attention to the scientific details we wish to highlight From time to time in the argument ahead we will pause to discuss some of the unhappy ways in which the grander reductive visions have mutilated well-established diagnostic terminologies in an unhelpful manner2

1 ldquoGroundingrdquo and manipulation counterfactuals

According to popular philosophical opinion counterfactuals require ldquosupportrdquo or ldquogroundingrdquo by laws of naturemdashindeed it often assumed that that the modal ingredients conveyed by counterfactuals are entirely supplied by underlying ldquolawsrdquo with the other ingredients (eg initial and boundary conditions) entering into their assessment in a modally inert manner Proposals along these lines have taken several different forms In ldquometalinguisticrdquo accounts of counterfactuals beginning with Goodman (1955) for a counterfactual to be true its consequent must be derivable from its antecedent in conjunction with other premises with ldquolawsrdquo being among the other premises A similar conception is defended by Maudlin (2007) and by Paul and Hall (2013) although here the grounding laws are assumed to take a rather specific form ndash they are taken to be laws of temporal evolution and the systems to which they apply are taken to admit of a well-posed initial value problem an assumption we shall discuss critically below Lewisrsquo well-known account of counterfactuals (1973) is often regarded as an alternative to such meta-linguistic theories but the similarity relations among possible worlds that determine the truth-values of counterfactuals also assign a central role to laws of nature again supporting the general claim that laws play some sort of grounding role with respect to counterfactuals3

2 PA3 Through this essay we do not employ ldquogroundrdquo in its restricted modern usage as an updated surrogate for ldquosupervenience baserdquo (as in eg Correia and Schneider 2012) but simply as a demand for some obligatory conceptual linkage between counterfactuals and ingredients that are counterfactual-free Our criticisms apply to most proposals that invoke ldquogroundingrdquo in a more technical sense but we are not limiting our attention to these

4

When we ask why such grounding in laws is required4 for counterfactuals a number of different motivations are offered Some writers (eg Psillos 2004 2007) have epistemic concernsmdashthey hold that counterfactuals cannot be reliably assessed for truth or falsity without information about grounding laws Other writers (eg Maudlin 2007 but also Psillos 2004) offer broadly semantic concernsmdashthey claim that stand-alone counterfactuals without grounding laws are (commonly or always) vague context-dependent and unclear in a way that makes them unsuitable for use in science Providing backing laws is required to repair these deficiencies Still other writers have more straightforward metaphysical concerns They claim that counterfactuals cannot be ldquobarely truerdquomdashthat counterfactuals (and modal claims more generally) require grounding in what is ldquoactual rdquo and that laws are prime (and perhaps the only) candidates for such ldquoactualrdquo truth makers Often although not inevitably the underlying goal is a reduction of some kind First true counterfactuals are understood in terms of claims about laws and non-modal initial and boundary conditions Laws in turn are understood as regularities meeting further conditions as in the Best Systems Analysis so that a complete Humean reduction is achieved

Many philosophers cling to various claims about ldquogroundingrdquo with unshakeable conviction and find it astonishing that anyone should harbor any doubts about them But from whence do these firm assurances stem As we shall see they are not based upon the attentive study of scientific practice which instead reveals more complex patterns in which notions of ldquolawrdquo and ldquocounterfactualrdquo comingle in mutually supportive yet patently non-reductive ways Nor do these supposed grounding requirements reflect any obvious semantic or conceptual connection between laws and counterfactuals For one thing children master counterfactual constructions long before they grasp the somewhat recondite contours of ldquolawrdquo (they understand that ldquoif I snitch that cookie Irsquoll be punishedrdquo long before they display any evident awareness of a ldquolaw of naturerdquo5) Nor is any epistemological motivation for maintaining a ldquogroundingrdquo thesis immediately plausible Many counterfactuals that describe the outcomes of potential manipulations along the lines of ldquoif variable A were tweaked to a new value α effect B would followrdquo can be firmly established as correct solely on the basis of a well-designed experiment in the face of total ignorance of any underlying ldquolawrdquo Firm counterfactual knowledge about what would happen to a childrsquos headache if she were to take an aspirin have been familiar to canny parents for generations long before the chemical underpinnings of these dependencies became

4 We emphasize that the grounding claims that are our target donrsquot just contend that as a matter of empirical fact it will turn out that for every true counterfactual there will be an underlying law Rather it is contended that this must be true and that the ingredients that underlie the counterfactual (laws initial and boundary conditions etc) can all be understood in non-modal ldquoactualistrdquo terms 5 See Harris (2000) Weisberg and Gopnik (2013) and Woodward 2011 for empirical support for such claims about childrenrsquos knowledge of counterfactuals

5

clear at all (these represent a recent discovery6) Even so an accurate modern description of these mechanisms doesnrsquot contain anything that looks much like the ldquolawsrdquo of traditional philosophical expectation (as wersquoll witness in simpler circumstances below)

Our concern with these claims about grounding arises in the following way Recent work on causation (eg Woodward 2003) has connected causal claims to counterfactuals concerning the outcomes of possible experimental manipulations Roughly speaking the guiding idea is that (11) ldquoC causes Erdquo should be elucidated in terms of (12) the claim that E would change in an appropriate experimental manipulation were to be performed on C (We call these manipulation counterfactuals in what follows they are instances of the ldquotweaking a variable to a new valuerdquo claims of the prior paragraph) This proposal has been frequently criticized from the perspective of the grounding claims described above counterfactuals of form (12) need to be further supported by laws that do not themselves presuppose counterfactuals or modally committed notions Hence the proposed elucidation of (11) in terms of (12) is (at best) incomplete and needs to be fleshed out by an additional account of the reductive relationships between the counterfactuals (12) and the laws that ground them When these details are provided treatments like Woodwardrsquos treatment turn into nomic (viz law-based) accounts of causal claims (and in turn perhaps into regularity accounts if the underlying laws are reduced to regularities) and the manipulationistcounterfactual elements all fade away as unnecessary

Hiddleston (2005) is representative of these criticisms

While Woodward relies heavily on counterfactuals he says surprisingly little about their truth conditions (He criticizes Lewis for using nonintervention counterfactuals but does not say what makes any of them true) This raises puzzles because standard theories appeal directly to natural laws ldquoA 1048687rarr Crdquo is true iff A background facts and actual laws jointly imply C (547)

He adds

Woodwardrsquos arguments support only the weaker contention that there issome close connection between counterfactuals and causal explanation Forexample it remains open to say that this connection is that the truth conditions of counterfactuals immediately involve laws and that their causal and explanatory force derives from that fact (547)

How do these writers explain the fact that practicing scientists interested in diagnosing

causal patterns display little evident concern with respect to the ldquogroundings in lawrdquo that philosophers demand Some of Woodwardrsquos ldquoactualistrdquo critics treat this as merely a matter of 6 In contrast the headache relieving properties of the substance we call aspirin was known to the ancient Greeks

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

4

When we ask why such grounding in laws is required4 for counterfactuals a number of different motivations are offered Some writers (eg Psillos 2004 2007) have epistemic concernsmdashthey hold that counterfactuals cannot be reliably assessed for truth or falsity without information about grounding laws Other writers (eg Maudlin 2007 but also Psillos 2004) offer broadly semantic concernsmdashthey claim that stand-alone counterfactuals without grounding laws are (commonly or always) vague context-dependent and unclear in a way that makes them unsuitable for use in science Providing backing laws is required to repair these deficiencies Still other writers have more straightforward metaphysical concerns They claim that counterfactuals cannot be ldquobarely truerdquomdashthat counterfactuals (and modal claims more generally) require grounding in what is ldquoactual rdquo and that laws are prime (and perhaps the only) candidates for such ldquoactualrdquo truth makers Often although not inevitably the underlying goal is a reduction of some kind First true counterfactuals are understood in terms of claims about laws and non-modal initial and boundary conditions Laws in turn are understood as regularities meeting further conditions as in the Best Systems Analysis so that a complete Humean reduction is achieved

Many philosophers cling to various claims about ldquogroundingrdquo with unshakeable conviction and find it astonishing that anyone should harbor any doubts about them But from whence do these firm assurances stem As we shall see they are not based upon the attentive study of scientific practice which instead reveals more complex patterns in which notions of ldquolawrdquo and ldquocounterfactualrdquo comingle in mutually supportive yet patently non-reductive ways Nor do these supposed grounding requirements reflect any obvious semantic or conceptual connection between laws and counterfactuals For one thing children master counterfactual constructions long before they grasp the somewhat recondite contours of ldquolawrdquo (they understand that ldquoif I snitch that cookie Irsquoll be punishedrdquo long before they display any evident awareness of a ldquolaw of naturerdquo5) Nor is any epistemological motivation for maintaining a ldquogroundingrdquo thesis immediately plausible Many counterfactuals that describe the outcomes of potential manipulations along the lines of ldquoif variable A were tweaked to a new value α effect B would followrdquo can be firmly established as correct solely on the basis of a well-designed experiment in the face of total ignorance of any underlying ldquolawrdquo Firm counterfactual knowledge about what would happen to a childrsquos headache if she were to take an aspirin have been familiar to canny parents for generations long before the chemical underpinnings of these dependencies became

4 We emphasize that the grounding claims that are our target donrsquot just contend that as a matter of empirical fact it will turn out that for every true counterfactual there will be an underlying law Rather it is contended that this must be true and that the ingredients that underlie the counterfactual (laws initial and boundary conditions etc) can all be understood in non-modal ldquoactualistrdquo terms 5 See Harris (2000) Weisberg and Gopnik (2013) and Woodward 2011 for empirical support for such claims about childrenrsquos knowledge of counterfactuals

5

clear at all (these represent a recent discovery6) Even so an accurate modern description of these mechanisms doesnrsquot contain anything that looks much like the ldquolawsrdquo of traditional philosophical expectation (as wersquoll witness in simpler circumstances below)

Our concern with these claims about grounding arises in the following way Recent work on causation (eg Woodward 2003) has connected causal claims to counterfactuals concerning the outcomes of possible experimental manipulations Roughly speaking the guiding idea is that (11) ldquoC causes Erdquo should be elucidated in terms of (12) the claim that E would change in an appropriate experimental manipulation were to be performed on C (We call these manipulation counterfactuals in what follows they are instances of the ldquotweaking a variable to a new valuerdquo claims of the prior paragraph) This proposal has been frequently criticized from the perspective of the grounding claims described above counterfactuals of form (12) need to be further supported by laws that do not themselves presuppose counterfactuals or modally committed notions Hence the proposed elucidation of (11) in terms of (12) is (at best) incomplete and needs to be fleshed out by an additional account of the reductive relationships between the counterfactuals (12) and the laws that ground them When these details are provided treatments like Woodwardrsquos treatment turn into nomic (viz law-based) accounts of causal claims (and in turn perhaps into regularity accounts if the underlying laws are reduced to regularities) and the manipulationistcounterfactual elements all fade away as unnecessary

Hiddleston (2005) is representative of these criticisms

While Woodward relies heavily on counterfactuals he says surprisingly little about their truth conditions (He criticizes Lewis for using nonintervention counterfactuals but does not say what makes any of them true) This raises puzzles because standard theories appeal directly to natural laws ldquoA 1048687rarr Crdquo is true iff A background facts and actual laws jointly imply C (547)

He adds

Woodwardrsquos arguments support only the weaker contention that there issome close connection between counterfactuals and causal explanation Forexample it remains open to say that this connection is that the truth conditions of counterfactuals immediately involve laws and that their causal and explanatory force derives from that fact (547)

How do these writers explain the fact that practicing scientists interested in diagnosing

causal patterns display little evident concern with respect to the ldquogroundings in lawrdquo that philosophers demand Some of Woodwardrsquos ldquoactualistrdquo critics treat this as merely a matter of 6 In contrast the headache relieving properties of the substance we call aspirin was known to the ancient Greeks

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

5

clear at all (these represent a recent discovery6) Even so an accurate modern description of these mechanisms doesnrsquot contain anything that looks much like the ldquolawsrdquo of traditional philosophical expectation (as wersquoll witness in simpler circumstances below)

Our concern with these claims about grounding arises in the following way Recent work on causation (eg Woodward 2003) has connected causal claims to counterfactuals concerning the outcomes of possible experimental manipulations Roughly speaking the guiding idea is that (11) ldquoC causes Erdquo should be elucidated in terms of (12) the claim that E would change in an appropriate experimental manipulation were to be performed on C (We call these manipulation counterfactuals in what follows they are instances of the ldquotweaking a variable to a new valuerdquo claims of the prior paragraph) This proposal has been frequently criticized from the perspective of the grounding claims described above counterfactuals of form (12) need to be further supported by laws that do not themselves presuppose counterfactuals or modally committed notions Hence the proposed elucidation of (11) in terms of (12) is (at best) incomplete and needs to be fleshed out by an additional account of the reductive relationships between the counterfactuals (12) and the laws that ground them When these details are provided treatments like Woodwardrsquos treatment turn into nomic (viz law-based) accounts of causal claims (and in turn perhaps into regularity accounts if the underlying laws are reduced to regularities) and the manipulationistcounterfactual elements all fade away as unnecessary

Hiddleston (2005) is representative of these criticisms

While Woodward relies heavily on counterfactuals he says surprisingly little about their truth conditions (He criticizes Lewis for using nonintervention counterfactuals but does not say what makes any of them true) This raises puzzles because standard theories appeal directly to natural laws ldquoA 1048687rarr Crdquo is true iff A background facts and actual laws jointly imply C (547)

He adds

Woodwardrsquos arguments support only the weaker contention that there issome close connection between counterfactuals and causal explanation Forexample it remains open to say that this connection is that the truth conditions of counterfactuals immediately involve laws and that their causal and explanatory force derives from that fact (547)

How do these writers explain the fact that practicing scientists interested in diagnosing

causal patterns display little evident concern with respect to the ldquogroundings in lawrdquo that philosophers demand Some of Woodwardrsquos ldquoactualistrdquo critics treat this as merely a matter of 6 In contrast the headache relieving properties of the substance we call aspirin was known to the ancient Greeks

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

6

short term focus perhaps on some distant occasion scientists will return to these undischarged counterfactuals and supply them with proper law-based foundations Or perhaps there is a division of labor with the job of the metaphysician being to show that such foundations are always ldquopossible in principlerdquo even if scientists donrsquot seem concerned to provide them On this basis they view Woodwardrsquos linkages between causal claims and manipulation counterfactuals as ldquomerely of epistemological significancerdquo and of no relevance to the underlying ldquometaphysicsrdquo of causation

This claim strikes us as deeply mistaken It is true that in Woodward 2003 one of the roles assigned to manipulationalist counterfactuals is that of bringing out or making apparent causal structures or relationships that cannot be read off just from correlational information If we are merely given evidence for a correlation between X and Y this by itself does not tell us whether the underlying causal structure is one in which (i) X causes Y (ii) Y causes X (iii) there is some common cause C of both X and Y or whether (iv) there is some more complex story about why the correlation arisesmdashas far as the correlational information goes the underlying causal structure remains hidden unless we willing to make further assumptions some of which may be very speculative Performing appropriate experimental manipulations (ldquointerventionsrdquo) on X and Y is one important (and often very reliable) way of distinguishing among the possibilities just described and bringing out the correct underlying structure However as we will show this point does not just apply to counterfactuals directly associated with experiments and is not just a matter of the epistemology of causal discovery instead counterfactuals that record manipulation information play an important (and not ldquopurely epistemologicalrsquo) role in many areas of science that are far removed from direct experimental evidence For example every modern work in applied mathematics ensures that its studies are conducted within the framework of what are called ldquowell-set problemsrdquo which often require considerable complexities in their technical articulation Looked at closely these requirements codify crucial information about how the target system will behave under a wide range of carefully delineated counterfactual conditions These manipulationist counterfactuals have a role very similar to the role played by direct experimentation ndash both have the role of bringing forward or making apparent structural relationships that would otherwise be hidden If one objects to the way in which Woodward uses undischarged counterfactuals one must also object on similar grounds to the extensive use of such counterfactuals in modeling practice

In an allied vein many of the most highly prized advances in descriptive sciencemdashLagrangersquos techniques in his Analytical Mechanics for examplemdashhave traded precisely upon their ability to enhance the reliability of a modeling through the clever exploitation of information recorded in manipulationist counterfactuals Most physicists would be dismayed to learn that many philosophers are apparently committed to regarding such techniques as problematic because they lack proper metaphysical grounding

Most of this essay will be devoted to explaining in non-technical terms how these

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

7

fruitful blends of ldquocomplementary but coequal7rdquo types of information having to do with laws and manipulationist counterfactuals strategically operate within some simple applied mathematics circumstances We will not attempt to critique in any detail the background metaphysical and semantic concerns that have led to the present impasse We do maintain however that the underlying argumentative burden has become unaccountably shifted within the prevailing debates so that the ldquogrounding in lawrdquo presumptions have assumed the status of a null (or default) hypothesis within many philosophical circles despite the fact that such advocates generally employ ldquolaw of naturerdquo in such a loose and elastic way that one is unable to ascertain what their ldquonull hypothesisrdquo actually demands We believe that these inadequacies become readily apparent as soon as we scrutinize a typical ldquowell-set problemrdquo in any detail and attempt to locate the purported ldquogroundingsrdquo of the metaphysicians

Accordingly this essay will present a range of non-epistemic examples in which scientists exploit counterfactuals with a manipulationist interpretation to strategic purposes much like Woodwardrsquos We shall argue that such descriptive techniques comprise important aspects of how effective science assembles reliable accounts of naturersquos workings But our metaphysical andor linguistic opponents appear to insist upon philosophical strictures that blur over or eliminate the distinctive blends of ingredients that prima facie advance scientific understanding in a fruitful manner So we set the following challenge to these critics Working with real life scientific examples can you supply a more finely wrought explication of why you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner

2 Well-set Modeling

This section introduces the notion of well-set modeling Every consideration we discuss will already be familiar to applied mathematicians who thanks to the pioneering work of Jacques Hadamard (1952) in the early twentieth century recognize the motivational underpinnings behind the ingredients demanded within a well-set problem in a focused manner That enlightenment on our questions should be found within the narrow corridors of mathematized physics should not seem altogether surprising in light of the fact that the logical empiricist forebears of modern philosophy of science borrowed most of their standard classifications (ldquoinitial conditionsrdquo ldquoboundary conditionsrdquo ldquosystem lawsrdquo etc) from the mathematicians and applied them more loosely and broadly in ways that have inflicted considerable diagnostic harms upon philosophy as a whole Hadamard and his coworkers had originally developed these terminological discriminations in order to address some contemporaneous methodological confusions that are closely allied in underlying theme to the

7 In speaking of laws and information recorded in counterfactuals as ldquocomplimentary and co-equalrdquo we mean that each requires the other and there is no clear sense in which one is ldquopriorrdquo to the other

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

8

puzzles about the utilities of counterfactuals in science that will concern us A major theme of this section will be the importance of restoring ldquoinitial conditionsrdquo et al to their originally intended discriminations

Working from this vantage point we will find that the well-set models of the applied mathematician (1) directly represent large collections of counterfactual assertion that (2) further demand that more basic ranges of manipulationist counterfactuals be credited with reliably established truth-values as well These counterfactuals are reflected in choices about initial and boundary conditions and these although often containing modal elements are not grounded in ldquolawsrdquo in any clear sense of that word Furthermore the mathematicians impose these requirements for very much the same reasons that Woodward emphasizes in his account of the connections between causation and experimentation viz a well-set modeling can sever the internal behavioral properties of a target system S from the greater complexities of its real world environmental entanglements E In the sequel we shall call these techniques of descriptive disentanglement policies of S versus E effacement and will find that such stratagems can be carried out in a distinct variety of mathematical ways depending largely upon our abilities to manipulate the target system S freely

To get a preliminary sense of what a well-set problem looks like let us adapt an old example of Arthur Eddingtonrsquos (Eddington 1928) A huge elephant walks across a tightrope over Niagara Falls For the purposes of understanding how the rope internally responds to this duress a modeler only needs to boil down the elaborate details of this complete loading into an upper surface distribution of downward force8 The elephant and scenery vanish from view replaced by a simple schedule of downward arrows upon the string (let us suppose that the elephant remain immobile over the time interval we are interested in) We also presume that the two far endpoints of the rope remain completely immobile This fixed end boundary specification allows us to efface our modeling from the greater complexities of the external environmentmdashwe shall return to the importance of this isolation later on Once our problem has become effectively well-set in this effaced manner we gain the wherewithal to address a wide range of counterfactual questions crisply What would happen if we had introduced a small perturbation into the wire somewhere

8 Technical remark due to the collapsed one-dimensionality of this reduced modeling the elephant loading isnrsquot always categorized as a proper ldquoboundary conditionrdquo until we begin modeling our string as a two or three-dimensional solid But wersquoll ignore this terminological qualification here For most regular solids and liquids exterior bounding surfaces supply suitable opportunities upon which a worthy policy of internal behavior effacement can reasonably build

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

9

say with a brisk hammer blow What sorts of waves would travel to our elephantrsquos feet Will they disturb the animalrsquos position very much And so forth If our problem is properly well-set we will be able to address all of these problems in a reliable manner based upon our modeling

Why did we write that a well-set modeling of this character ldquorepresents a large collection of counterfactual assertionsrdquo Consider the boundary condition that textbooks normally assign to a problem such this They invariably first presume that the ropersquos two endpoints will remain completely immobile whatever else happens within the ropersquos interior over the interval of time under investigation (such a stipulation is called a Dirichlet boundary condition) But we know that the

endpoints of such ropes never satisfy such a requirement in the real world any pachydermal loading will surely cause them to wobble Despite these lapses from descriptive realism applied mathematicians feel that they havenrsquot modeled a target system S in an adequate way unless their proposal satisfies standard formal criteria for qualifying as ldquowell-setrdquo Why A full answer will consume most of this essay but the short version is because an improperly set modeling will often overlook many of Srsquos most important internal characteristics

What are the basic structural ingredients within our modeling that allow us to address our counterfactuals in a firm manner The mathematicians tell us (1) The boundary conditions that reflect in a suitably effaced way how the environment E impinges upon the ropersquos interior along its outer surface viz that (a) the rope remains entirely immobile at its two far endpoints and (b) we are supplied with the locations and magnitudes of the gravitational loads that the elephant permanently places upon the string (2) The initial condition of the string at some starting time t0 here assumed to be the exact moment at which we apply our little hammer (3) The set of ldquosystem lawsrdquo that capture the physics involved in how the interior of the rope responds to local conditions In our simple circumstances only one ldquolawrdquo of this character is neededmdashthe familiar differential wave equation (party2partt2 = c2party2partx2 mdashthat encodes the fact that the string generates a local restoring force ρparty2partt2 proportional to the degree to which the string is curved (party2partx2) at that same location But in more general circumstances a number of differential equations will be required working together in what we will later describe as proper equational closure

Note that initial conditions specify data with respect to a time slice cutting through our diagram at time t0 whereas boundary conditions supply partial data pertinent to future states of the system Hadamard stresses the fact that the formal character of the data differs between these two circumstances viz normally two pieces of data (position and velocity of the string at

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

10

t0) are needed for proper initial conditions but only a single (partial) specification is permissible for a coordinate boundary condition Furthermore the character of the counterfactual variability required of these stipulations differs considerably as well9 One of our tasks in this essay is to tie these distinct demands to the basic methodological motivations behind the articulation of well-set modelings in the first place

Herersquos a small bit of useful terminology we shall consistently employ If we wish to group initial and boundary condition requirements together under a brisker heading we shall employ the mathematiciansrsquo term side conditions10 But we shall studiously avoid the philosopherrsquos analogous phrase ldquoauxiliary conditionsrdquo for this phrase carelessly blurs together modeling ingredients that we will need to distinguish sharply

Before we move ahead let us note another significant consideration that Hadamard brought to our attention Nature actually suggests a variety of structurally distinct well-set problems to us each implementing somewhat different strategies for how a satisfactory modeling effacement should be achieved Consider the task of determining how a childrsquos soap film distributes itself across the interior of a twisted wire frame possibly of several pieces Mathematicians call this a ldquopure boundary value problemrdquo and do not assign any initial conditions in the proper ldquotime slice at t0rdquo sense to the problem at all11 Why Because we are tacitly seek the configurations that the soap might reach once it stops jiggling about and reaches a settled equilibrium state To address this kind of modeling task we donrsquot need to know the initial positions and velocities of the film when it is applied to the rim at time t0 we merely need to fix

9 The usual requirements for an initialboundary value being well-set are (1) solutions exist locally for a freely varied collection of initial conditions (2) these solutions prove unique and (3) are well-posed in the sense that solutions that are close to one another in initial conditions remain close in behavior for short periods of time Other forms of structural stability (say with respect to boundary condition variation) are often imposed as well depending upon circumstances Generally when one or more of these requirements fail for a target system S modelers believe that that theyrsquove not captured the behaviors of S in a revealing way The most salient aspect of these formal requirements for the present essay is that (1-3) demand an assessment of S under a wide range of counterfactually varied circumstances if the projection of unwarranted external characteristics into S itself is to be avoided10 Other forms of attached side condition include driving forces interfacial joins constraints and much more We shall only consider the ramifications of the latter briefly here11 The official classification for our wiggling string problem is as ldquoan initialboundary problem of a hyperbolic evolutionary characterrdquo of which our vibrating string problem often serves as a canonical textbook illustration Our soap case qualifies as a ldquopure boundary problem of an elliptic characterrdquo

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

11

the fixed shape that the wire maintains over time So only boundary conditions enter our descriptive task as required side conditions initial conditions per se are completely unwanted in such a setting

What accounts for these discrepancies in side condition requirement between the string and the soap bubble They stem from the fact that the equilibria forthcoming in the soap case substantially alter our capacities to control the target system through exterior modifications The well-set modelings that mathematicians apply to soap film-like cases codify the interior information about S that we can extract from these altered experimental determinations

These differing data requirements usually carry other changes in modeling expectation in their wake Because we anticipate that our film may be able to establish satisfactory equilibria in a variety of manners we no longer expect that our problem will possess unique answers in the same manner as our vibrating violin string where we typically demand that its evolving states must flow uniquely from its initial conditions if the problem is to be considered well-set In the jargon of the mathematicians the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the violin string)

Here then is one of our central theses in a nutshell A number of the considerations that philosophers have regarded as demands that ldquocounterfactual assertions must be grounded in lawsrdquo represent a distortion (as in a funhouse mirror) of the methodological objectives that lead applied mathematicians to divide their modelings into various classes of well-set problem As indicated before these objectives revolve around a desire to isolate (or efface) the internal responses of the target system S as effectively as possible from the greater complications of its real world environmental setting E These isolate-internal-behaviors-cleanly objectives require that system laws and side conditions must work together in cooperative integration but they do not demand that direct appeals to ungrounded manipulationist counterfactuals be absent within the motivational background of the proposed modeling As wersquoll see quite the contrary is true Philosophical presumptions otherwise stem from a simple misunderstanding of the requirements upon a ldquosystem lawrdquo that emerge within the context of a well-set problem (which we shall detail in later sections) To not recognize the reliability-enhancing contributions of these unacknowledged forms of counterfactual appeal is ipso facto to misunderstand many of the methodological safeguards that allow us to trust our scientific models as much as we do To the critics who argue that ldquostand-alone counterfactuals without grounding laws are vague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo we reply ldquoNo science owes much of its vaunted reliability to the effective exploitation of certain special classes of manipulationist counterfactuals which donrsquot require backing by lawsrdquo Their autonomous capacities for truth-functional reinforcement will emerge into the bright daylight of accurate philosophical analysis as soon as the characteristic ingredients within a well-set problem are scrutinized with the same attention to methodological detail that an applied mathematician brings to the task

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

12

Analytic metaphysicians are fond of declaring that they aim at ldquocarving nature at the jointsrdquo Insofar as these ambitions can be correlated with sound scientific practice the locales of effective S versus E effacement represent the ldquojointsrdquo that a mathematical modeler attempts to capture within a specimen well-set problem To be sure these opportunistic ldquocutsrdquo may not be quite as absolutist as the ldquojointsrdquo that the metaphysicians seek but they represent objective facts about nature of central importance to descriptive science So we reject the presumption that the considerations we shall advance are ldquomerely of epistemological significancerdquo

Although we will not appeal primarily to epistemological concerns here we believe that strong affinities link the arguments advanced in this essay to the experimental and quasi-experimental considerations to which Woodward has appealed elsewhere (Woodward 2015) In particular the modeling opportunities where nature allows us to establish ldquocutsrdquo of viable descriptive effacement are motivationally closely allied with the experimental considerations that dictate how a well-planned experiment can evince trustworthy conclusions with respect to Srsquos causal operations But our primary purpose is to dispel the philosophical misapprehension that trustworthy counterfactuals require grounding within more basic forms of scientific law In our estimation these presumptions arise from a failure to appreciate the integrated and coequal manner in which notions of ldquoside conditionrdquo and ldquosystem lawrdquo must fit together within a well-set modeling if it is to achieve its desired S versus E objectives These misconceptions in turn trace to the unfortunate manner in which generations of philosophers of science have systematically misapplied the carefully wrought discriminations of the applied mathematician

3 S versus E Effacement

As just suggested most of the methodological insights advanced here have been familiar to the applied mathematician since the early twentieth century albeit couched in the somewhat rebarbative vocabulary of differential equations As often happens sharp messages become muddled after they become transmitted across many generations of ldquopurely philosophicalrdquo discourse never revisiting the practical concerns that prompted the original discriminations We believe that common presumptions with respect to the alleged grounding of counterfactuals have arisen in this manner As we noted standard philosophy of science primers of the last century misappropriated the crisp terminologies (ldquoinitial and boundary conditionsrdquo) that Hadamard and his coworkers developed for explicating the effacement rationales inherent in standard forms of physical modeling and reapplied them in a misty manner that obscures the very methodological lessons for which the mathematicians had originally devised those distinctions As ldquoside conditionsrdquo in the proper sense have morphed into the ill-diagnosed vagaries of ldquoauxiliary conditionrdquo the coordinate notion of ldquosystem lawsrdquo has suffered as well engendering all of the misapprehensions about grounding dependence that we criticize here

As a case in point consider Hiddlestonrsquos reference to ldquobackground factsrdquo in the criticism of Woodward quoted earlier

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

13

[Woodwardrsquos proposals raise] puzzles because standard theories [of counterfactuals] appeal directly to natural laws lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

Now in the context of a well-set modeling one can supply a robust reading for the structural ingredients (A background facts laws C) that Hiddleston invokes but none of them support his implicit presumption that they are all intrinsically ldquocounterfactual freerdquo Letrsquos examine more carefully our string case in its simplest setting where we omit the elephant loading and only consider a stringrsquos unencumbered wave behaviors after an initial plucking at time t0 Here the relevant system law is the familiar wave equation whose exact status as a ldquolawrdquo in the philosopherrsquos conception we will probe more closely later In terms of the mathematiciansrsquo side condition assignments Hiddlestonrsquos ldquobackground factsrdquo plainly align with the standard Dirichlet boundary condition proviso that the endpoints of our string remain completely immobile over a relevant interval of time Likewise his A corresponds to whatever initial condition plucking ltP(x t0) V(x t0)gt we choose to apply to our string at time t0

But are Hiddlestonrsquos altered Arsquos and his ldquobackground factsrdquo truly modality free No and it is exactly here that Hadamardrsquos penetrating methodological insights come into play We must examine our problems over properly extended counterfactual families if we hope to extract the correct internalist lessons from our investigations In particular if a modeling doesnrsquot supply correct counterfactual answers over a suitable range of varied conditions it is likely to mistake some extraneous aspect of Srsquos environmental placement for a genuine internalist characteristic Herersquos a typical if somewhat artificial Hadamard-like example12 Suppose that we only consider special initial conditions ltP(x t0) V(x t0)gt in which P(x t0) = 1radicc intL

x V(x t0) dx Call this restriction R Within these delimited R circumstances the induced waves will travel in a resolutely right-handed direction until they collide with the nut at the far end Within the range of R variation a simpler form of system law becomes viable the one-directional wave equation (partypartt = cpartypartx) Whatrsquos wrong with this modeling Physically a proper registration of the restorative processes active within the interior of a string S should reveal that a pure disturbance in initial position P(x t0) will normally split into two left and right heading waves as a natural outcome of Srsquos attempts to straighten itself out By only considering the range of Rndashobedient variations we hide this even-handed natural response within a special requirement on initial velocity that suppresses the normal left-heading reaction Expressed in another manner Srsquos apparent propensity in favor right-heading waves does not

12 His chief illustration in (1952) is quite substantive for he shows how a limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling In our toy substitute the restriction on initial conditions turns off the leftward heading component within drsquoAlembertrsquos general solution for the wave equation A(x ndash at) + B(x + at)

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

14

reflect a genuine internal characteristic of S but merely represents an ersatz projection into Srsquos interior of features that properly characterize the restricted manner in which we have initially manipulated the string at time t0 The proper corrective Hadamard advises is to examine our modeling of S under a wider range of potential starting conditions capable of erasing externalist biases of a Rndashprojected character Avoiding faulty internalist projections of this type supplies the primary reason why mathematicians build freely assignable initial conditions into the formal requirements of a well-set problem (if the modeling calls for initial conditions at all)

Herersquos an allied example Suppose that the region in which our string is located contains an ambient 60 cycle hum Its coupling with our string may induce wave patterns to appear within the latter that once again are not characteristic of the stringrsquos intrinsic propensities and so we once again should examine our core modeling under a wider range of variations that can filter away these unrepresentative behaviors Situations of this ilk are standardly labeled as ldquoforcing conditionrdquo problems and their contributions to a systemrsquos behavior are generally investigated in a manner that more closely approximates the policies that modelers adopt in considering the behavioral consequences of a systemrsquos boundary conditions For reasons wersquoll discuss in a moment boundary region variability is usually approached in a somewhat different manner than the free assignability policies the modelers apply to initial conditions largely on the basis of backgrounded considerations of manipulative control that differ from those operative with respect to initial conditions

Anticipated ranges of possible initial state variation affect our choice of suitable system laws in other ways as well Letrsquos consider the case of the Distressed Steam Shovel At time t0 let us subject the devicersquos parts to various initial conditions such as a big bump when the mechanism hits a rock We are interested in determining how much of the kinetic energy entering the gizmo through the jolt on its caterpillars is likely to transfer to the boom and dipper stick causing them to swing around dangerously In such circumstances we would not want to assign our shovel the full range of permissible position and velocity values that we commonly employ in other modeling circumstances because most of those initial choices will deposit its parts in locations where they are no longer attached to one another Studying how those generic detached part scenarios unfold wonrsquot reveal the vital internal-to-an-intact-steam-shovel behaviors we are interested in unraveling we want to know how the device internally redistributes disturbances amongst its parts under the presumption that they remain attached to one another How do we rectify this problem We should readjust our notion of ldquopossible starting conditionrdquo to variables (labeled as xyαβγδ in the diagram) that directly reflect the freedoms of movement open to an intact steam shovel (a physicist calls these new variables ldquogeneralized coordinatesrdquo)

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

15

But how do we know that these xyαβγδ capture the proper range of initial conditions we should study Typically by simple induction from manipulative experiment ldquoWiggle the sundry parts of the mechanism yoursquoll discover that you can freely choose the angles xyαβγδ without tearing the damned thing apartrdquo 13 But once we determine a suitable range of possible initial states we will want to examine all of them in an even-handed manner to avoid ersatz externalist projections of a Hadamard-like character

When a system exhibits locked-together behaviors amongst its parts in the manner of our mechanism mathematicians say that its movements have become subject to constraints In the case before us these constraints can be captured in entirely geometrical terms the boom can move relative to the cab only by turning through the angle β and so forth By employing descriptive variables (such as xyαβγδ) that naturally reflect these geometric limitations scientists find that they can significantly simplify the models they supply for their target systemrsquos behaviors They exploit their easy-to-obtain knowledge of the steam shovelrsquos movement constraints to ldquocut offrdquo a huge amount of unwanted lower scale complexity they can ignore all of the detailed physical processes that keep the parts intact This represents a second policy of descriptive effacement that we shall investigate later in the paper For the moment we merely note that the de facto ldquofree variability of initial conditionsrdquo we attribute to a target system tacitly incorporates a large amount of raw counterfactual data with respect to the possible positions into which we can place our steam shovel without destroying its integrity In other words both the assumption that initial conditions are free to vary independently of one another (to the extent that this is true) and the presence of constraints reflect counterfactual claims And although one can assert that these in turn must be grounded in laws (understood non-modally) and other non-modal information nothing in the practices we have been describing provides support for the claim that this must be true14

When we consider the ranges of variability to be correlated with the boundary conditions of a problem we find that we should implement policies of a strikingly different character again reflecting the interior facts we hope to learn about our target system (once again Jacques Hadamard is chiefly responsible this important methodological aperccedilu) When we speak of effacing the behaviors of a target system S from the complexities of its environmental entanglements E we rarely wish to study S as a wholly isolated system (eg a string floating freely through interstellar space) but rather within special simplified environments E in which the central influences of E upon S can be studied in simple controllable terms For example

13 Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay we donrsquot need to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws 14 We remind the reader that the issue is whether the use undischarged non-lawbacked counterfactuals indicates a deficiency of some kind Merely postulating that the underlying laws exist does not identify such a deficiency

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

16

when we model a target system S (an iron bar say) as maintained at a constant temperature T by a surrounding ldquoheat bathrdquo we do not attempt to sever S from its externalist connections entirely but we restrict our attention to special choices of E that most clearly reveal Srsquos basic modes of response to simple and statically conceived forms of environmental embedding In other words ldquoheat bathrdquo boundary conditions insert a useful descriptive S versus E cut that allows us to isolate the basic manners in which S reaches adjustment with its environment more generally15

These requirements of informative effacement directly explain why the standard boundary conditions studied within standard forms of physical modeling carry counterfactual implications for they have replaced actual facts about the complicated physical region of the cut with artificially simplified ingredients such as immobile endpoints constant loads heat baths or static gravitational fields

Invocation of allied S versus E cuts is nearly ubiquitous within mathematical physics The rapid movements of the planets and asteroids within the inner solar system are intimately entangled with the larger planets and sun but the latter move sluggishly in comparison with the former so common practice models the situation to first order as a circumstance in which the larger bodies are descriptively ldquofrozenrdquo in place and we study how the smaller bodies whisk about within the simplified E of a constant gravitational field created by the large bodies Physicists speak of having inserted a descriptive ldquocutrdquo between S and E in such circumstances (indeed we have borrowed our talk of ldquoeffacementrdquo from the jargon of celestial mechanics as well)

Finally when an introductory textbook models the endpoints of a tightrope or violin string as completely immobile it likewise enforces a rather severe and somewhat unrealistic effacement between string and environment that is absolutely critical to understanding string behavior more generally (wersquoll explain why in a moment) Boundary region requirements of this simplified (Dirichlet boundary condition) character represent the stuff of which well-set problems are standardly composed and they collectively provide the materials with which textbooks and journal articles in mathematical physics come amply stuffed Stipulations of this purified character are important because they mathematically implement the S versus E cuts that help us understand the internal responses of strings within much wider forms of environmental embedding ` This is because properly selected (and often rather idealized) S versus E cuts can showcase the characterizations of internal response of exactly the sort that modelers seek In our string stationary end conditions crisply evince the tonal qualities that we generally regard as the most characteristic trait of a violin string namely its harmonic spectrum Mathematically these

15 An important lesson from both thermodynamics and quantum theory by carefully controlling a selected system variable such as temperature conjugate quantities (such as the total energy) frequently become unregulated We shall not deal with these complications here

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

17

ldquostanding waverdquo sine wave patterns emerge as proper solutions to our wave equation only if the bridge and nut remain completely stationary and do not tolerate any end point energy leakage In those purist circumstances the traveling wave energy arising from a time t0 excitation will reflect back into the string (upside down) and eventually coalesce in the stationary vibrational patterns of familiar acoustic experience Those repeated reflections supply the critical mechanism that eventually locks distantly separated portions in the string together into coherently organized patterns (the standing waves) that span the entire span of the string and which store energy in discrete bundles These are the patterns that underwrite the spectral analysis that supply us with our greatest insights into the interior dynamics of the string (and many other physical systems as well) But if we hadnrsquot attached the Dirichlet boundary conditions to our problem our modeling would not have brought these vital interior characteristics forward clearly16

Note that our central system equation--party2partt2 = c2party2partx2--says nothing at all about the endpoint physical processes responsible for the crucial energy reflection that produces the standing wave behavior The relevant physical facts appear only in the boundary condition guise of a stipulation ldquoHowever we might internally excite our string at time t0 its endpoints will remain completely stationary over the time interval ltt0 tigtrdquo Is this a ldquolaw of naturerdquo It scarcely appears so So how do we know to first approximation that this condition is appropriate to violin strings even for initial conditions (ie manners of plucking) that wersquove never implemented Answer by extrapolation from simple experiment Wersquove manipulated a lot of violin strings over our lifetimes and those bridges and nuts have always remained nearly stationary So circumstances in which these endpoints remain entirely immobile should be regarded as feasible and their study may reveal important facets of a stringrsquos interior capacities through the purity of the responses invoked Within these basic rationales for standard forms of boundary condition attribution we witness how the crucial roles that experimentally-founded manipulationist counterfactuals perform in rendering evident the most important aspects of a 16 The entanglement of boundary conditions with the internal characteristics of S should not seem surprising in light of the fact that standard boundary stipulations register both the environmental setting E around S and Srsquos own response to that E within its outer layers of material Many of the physical factors that hold a solid material S together and allow S to be transported from a natural setting to a testing bench reside within those outer layers of material and are not reflected within the interior processes captured within Srsquos governing system equations alone

We should mention that traveling wave to stationary wave subsidence does not transpire quite so rapidly in real violin strings as our discussion suggests but we shall ignore these complications here

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

18

target systemrsquos internal behaviors In performing these descriptive duties our side condition counterfactuals act as coequal partners to Srsquos system equations and record the physical factors that pertain to Srsquos boundary region interface with its wider environment E In this manner standard boundary condition assignments cut off the usual complexities of E by reducing them to static or easily monitored forms of exterior surface condition

As we have already noted the severe environmental effacement of a Dirichlet boundary condition is rarely realized in real lifemdashwe wonrsquot hear our violin at all if some of its interior energy doesnrsquot leak from the string to the body of the instrument through the bridge and then into the surrounding air To study these more complicated entanglements modelers progressively set the locations of their cuts further out in the universe moving first to the rather difficult problem of how a string (W) interacts with a plate P (viz a bridge) within a wider environmental setting in which the W + P system is modeled as a well-set problem in which the body of violin is still modeled as rigidly fixed And so on But these hierarchies of well-set modeling must be arranged in properly conceived layersmdashto understand our violinrsquos tone we must recognize that its standing wave patterns comprise its dominant manner of environmental response which are further perturbed by energetic leakage and allied factors

As our soap film case demonstrates the formal requirements imposed within each well-set modeling assembled in this progressive manner may differ from case to case in their precise features reflecting the fact that nature itself allows us to disentangle a target S from its environment E according to various forms of effacement rationale The strategic logic inherent in the different patterns of modeling elaboration that applied mathematicians utilize makes for a fascinating study but is too elaborate to be pursued further here

We stress the fact that fruitful S versus E cuts cannot be placed just anywhere nature itself must provide the special opportunities where such divisions can be fruitfully implemented How should we locate these descriptive opportunities Our steam shovel considerations suggest an essential manipulationist clue locate the S versus E divisions where we can sever a target system S from its normal environmental entrapments without destroying Srsquos internal integrity in the process If that disentanglement can be executed we can probably relocate S to a testing lab and study its responses to simple controlled environments in a systematic way In this manner nature suggests that violin strings will retain many of their vital characteristics if we transmit them to a testing bench where their side condition environment can be carefully controlled Or at least to first order for we will subsequently want to determine how the real life wiggling at the bridge complicates the stringrsquos basic tonal spectrum17 17 The present themes relate to the issues that (Wilson 2017) characterizes as ldquothe stupidity of mathematicsrdquo--the fact that modeling equations must be probed with artificial tools to reveal

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

19

Note that the S versus E considerations we have highlighted very much resemble the factors that a technician weighs in framing a well-designed experiment In our opinion this methodological affinity explains why undischarged counterfactuals play roles within well-set problems similar to those emphasized within Woodwardrsquos own studies

One of the great fallacies that undergirds standard grounding expectation stems from a tacit assumption that vital ldquoobjective traitsrdquo of a target system S will be directly revealed within the descriptive vocabulary employed within the set of differential equations relevant to S Every applied mathematician recognizes the naivety of this assumption The standing wave tonal spectrum of a violin string represents one of its most important physical characteristics and natural selection has designed our ears to filter away the extraneous noise that surrounds its manifestation in everyday life But these traits do not directly appear as ldquokind term vocabularyrdquo within party2partt2 = c2party2partx2 indeed that formula doesnrsquot pretend to capture all of the physical factors responsible for making standing wave behavior so prominent in our musical lives These hidden features only become clearly revealed when our system equation is attached mathematically to effaced endpoint descriptions that crisply register the energetic reflections crucial to standing wave energetic storage Accordingly a mathematician probes the behavioral ramifications of party2partt2 = c2party2partx2 with the carefully wrought instrumentalities of counterfactual consideration to bring these latent but vital behaviors forward

We earlier observed that some metaphysicians have criticized methodological appeals to manipulationist counterfactuals as merely of ldquoepistemological significancerdquo whereas they pride themselves in their interest in naturersquos own inventory of objective properties But an applied mathematician must also appeal to comparable collections of manipulationist consideration to extract vital ldquoobjectiverdquo conclusions from a mathematical modeling That doesnrsquot appear to be a matter of ldquoepistemologyrdquo

These significant parallelisms have become lost to view within philosophical circles due to a pervasive propensity for invoking the word ldquolawrdquo in a loose manner allied to the misapplications of ldquoinitial and boundary conditionsrdquo of which we have already complained In particular the contents of ldquosystem lawsrdquo as we have used the term here--to capture the differential equations at the core of a well-set modeling--are rarely scrutinized with the attention required to notice that standard ldquolaw versus ungrounded counterfactualrdquo dichotomies are poorly drawn We shall turn to this task in section 5 For the moment let us merely underscore one of the formal features that Hadamard requires of a well-set mathematical modeling This is simply the demand that our modeling exhibit adequate equational closure at least over a respectable interval of space and time In the simplest circumstances (like our string) this ldquoclosurerdquo demands that our modelrsquos assembled descriptive elements side conditions and system laws must fit together in a manner that supplies the mathematician with enough equations that the systemrsquos effaced behaviors can be projected forward into the future in a unique and stable manner for an appreciable span of time In the jargon of the applied mathematician a well-set problem (of a

their deepest ldquodominant behaviorrdquo secrets

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

20

deterministic initialboundary character) must match system laws to descriptive variables in such a manner that solutions (1) will exist for a certain span of time (2) prove unique and (3) demonstrate behavioral stability under various criteria once the problemrsquos defining side conditions have been appropriately fixed Other species of well-set problem (eg for our soap film case) alter these requirements mutatis mutandis (because eg initial conditions are not required in their specification)

These demands for equational closure should be viewed as the calculus analog to the high school algebra notion of a solvable set of linear equations such as x - 2y + 3z = -6 How many equations will be needed to solve for variables x y z hellip As a rough rule and barring hidden redundancies n equations for n variables are required Smaller or larger equation sets are apt to be underspecified or excessively demanding In the same way a well-set problem involving differential equations must arrange its descriptive ingredients in a manner that will generate stable and well-defined models over useful spans of time These considerations supply the criteria of equational closure we require in successfully effacing a locally focused modeling from the target systemrsquos wider interactions with the universe beyond and below18

As suggested above equational closure captures the descriptive self-sufficiency that we expect to find when a systemrsquos internal workings can be adequately captured within a properly effaced localized modeling We recognize that these self-enclosed treatments rarely render full justice to our target systemrsquos de facto engagements with its wider physical environment but we should first isolate our systemrsquos internalist responses as best we can before we turn to more ambitious forms of descriptive project So wersquod first like to know how our system behaves when its interactions with the wider world have been either turned off or controlled in a carefully monitored way In this way an insistence upon wisely chosen counterfactual circumstances comes into view

4 Not all counterfactuals are created equal

It is commonly acknowledged that many counterfactual claims are vague and that their truth-values are difficult or impossible to assess A familiar illustration is ldquoif Caesar had been in charge of UN forces during the Korean War he would have used catapultsrdquomdashhow do we 18 We write ldquobelowrdquo because sensible effacement policies also allow us to cut off descriptive parameters relevant to detailed behaviors active at size scales below our current level of interest For example a simple characterization of temperature on a macroscopic level usually captures the randomized complexities of the systemrsquos molecular degrees of freedom in a manner adequate to everyday effaced description On occasion such presumptions fail and the lower scale details demand closer attention on the modelerrsquos part In such circumstances nature doesnrsquot support the discriminations between macroscopic and microscopic characteristic of appeals to ldquotemperaturerdquo As wersquoll see appeals to constraints on movement exhibit this same ldquoeffacement from belowrdquo character

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

21

determine whether this claim is true or whether he would have used atomic weapons instead Maudlin (2007) agrees that tamer scientific counterfactuals like those we have considered possess unproblematic truth-values and notes that many of these appear to be intimately connected with modeling circumstances of an initialboundary value character19 On this basis he argues that the laws invoked within these models must serve as the essential ontological ldquoprimitivesrdquo that ground the truth-values of these acceptable counterfactuals in a context-insensitive manner

Maudlin is correct in associating counterfactual evaluation within mathematical physics with the well-set problems it studies but he falsely assumes that the latter completely lack counterfactual ingredients themselves independently of whatever laws are involved He also presumes that all model building unfolds in the mode of what applied mathematicians call an ldquoinitialboundary value problem of a simple hyperbolic characterrdquomdashthat is the modeling adopts the basic explanatory format we have assigned to our vibrating string20 But we have noted that equally firm counterfactuals can be supported within alternative forms of effacement opportunitymdashour soap film case serves as a typical exemplarmdashthat piece together their modeling ingredients according to somewhat different patterns (eg in not citing initial conditions at all) We noted that these structural discrepancies commonly trace to the fact that manipulative capacities available in these circumstances differ significantly Accordingly we recommend that we should directly consider the overarching effacement policies that explicate why modelers approach vibrating strings as initialboundary value problems but do not apply this treatment to soap films On this approach the key to understanding the role of counterfactual data within useful science lies in appreciating the general value of S versus E effacement not of initialboundary value problems per se

As already noted our S versus E constructions do not pace Maudlin eschew counterfactual appeals altogether but instead rely quite substantially upon several restricted ranges of manipulationist counterfactuals which are often established by direct extrapolation from experiment without any evident appeal to ldquolawrdquo whatsoever We have already witnessed a simple exemplar of these dependencies in our steam shovel case above we extrapolate its full set of movement freedoms from simple experiments where we attempt to relocate its parts in different initial conditions In section 5 we will find allied manipulationist dependencies lie within most forms of ldquosystem lawsrdquo as well 19 Putting aside Maudlinrsquos claims that laws are more primitive than counterfactuals and his restriction of the grounding laws to those that have an ldquoevolutionaryrdquo character we think that his account of counterfactuals has much to recommend it In particular we agree with this suggestion that the antecedents of scientifically legitimate counterfactuals correspond to commandsmdashthis is very close to the interventionist interpretation of counterfactuals as having to do with what would happen if we were to ldquodordquo or bring about the condition specified in their antecedents 20 This is apparent from the role Maudlin assigns to what he calls ldquoFundamental Laws of Temporal Evolutionrdquo in supporting counterfactuals (2007 pp 13ff)

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

22

The moral we should extract from the Caesar example is merely that not all counterfactuals should be evaluated in the same manner not that they should be eschewed altogether in understanding how effective scientific modeling works In particular appeals to manipulationist behavior under varied (and frequently counterfactual) circumstances enjoy a high degree of reliability with respect to truth-value precisely because they have been firmly tested within both laboratory and everyday circumstance (we would amazed if someone doubted that steam shovels really exemplify their familiar freedoms of movement and thought that these presumptions merit rigorous laboratory examination) But these often obvious facts comprise the key behavioral data upon which modelers rely in deciding where effective S versus E cuts should be reasonably placed As noted previously with normal solid objects we generally attach our boundary conditions to their exterior surfaces or as in our steam shovel circumstances to the connected sets of parts that appear to move together as an integral whole

In other words the side condition data and system laws appearing within a properly constructed well-set modeling have been so selected precisely because they effectively isolate the internal behavioral characteristics of the target system S by factoring away extraneous environmental factors E as much as possible A wisely modeled well-set problem assembles the evaluative tools that ldquocarve most closely to the behavioral bonerdquo of a target system through capturing the empirical opportunities for effective effacement that nature offers us in sharp mathematical terms That is nature first informs us ldquoTo first order strings and steam shovels can be profitably understood in terms of their responses to environmental conditions of a simple and closely monitored characterrdquo and the applied mathematicians then implements these suggestions in the formal guise of a side condition + system laws montage Within these methodological confines no ambiguity whatsoever attaches to the counterfactual claims that hew closely to our target systemrsquos tightly focused S versus E behaviors eg ldquoif the boom of our steam shovel were turned through an additional angle β the other parts of the apparatus can be maintained in their original relative positionsrdquo

Ever since Nelson Goodmanrsquos original discussion (1955) most of the literature on counterfactuals has presumed that philosophers must supply a ldquogeneral theory of counterfactualsrdquo adequate to every intelligible instance including unconstrained cases such as our Caesar-in-Korea example We see no reason why this should be the case In itself the employment of the subjunctive mood merely alerts an audience to the fact that something fictive is afoot But the byways of pretense are incredibly varied and any discussion of their focal purpose requires a delicate discussion of the specific authorial intentions behind the fictional act (if they can be pinned down at all) Assertions like ldquoif Sherlock Holmes hadnrsquot smoked a pipe he could have still deduced that a snake was involved in the Case of the Speckled Bandrdquo can sometimes be reasonably adjudicated and sometimes not But none of these wider dimensions of interpretative purpose affect the specific counterfactuals highlighted here whose truth-values are frequently established through simple induction from experiment

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

23

With respect to Goodmanrsquos classic match example the informative purpose of his counterfactual ldquoif the match were struck it would lightrdquo strikes us as very similar to the backgrounded rationales behind the effacement policies of standard mathematical modeling We employ Goodmanrsquos counterfactual to characterize interior characteristics of the match in the same evocative manner that applied mathematicians apply to their mathematized string and steam shovel models For these internalist purposes firm truth-values need to be conceded to a restricted range of manipulationist counterfactuals that report upon potential system variability in the manner of our earlier discussion (eg ldquothe temperature or applied force acting upon the outer boundary of target system S can be held constant through a certain range of possible valuesrdquo) To be sure these claims extend beyond the range of de facto experimental realization but only in a harmless extrapolated-from-simple-experiment kind of way In contrast ldquosemanticrdquo approaches to general counterfactual evaluation as exemplified within David Lewisrsquo (1973) or Robert Stalnakerrsquos (1968) well-known proposals hew a completely different course for they attempt to construct full ldquopossible worldsrdquo around the match without any effacement whatsoever These embedding-in-a-full-world propensities characteristically embroil their authors in extraneous worries about the ldquomiraclesrdquo that must be tolerated to bring these disparate elements into consistent accord In doing so they direct our attention away from the robust Hadamard-style rationales that judiciously explicate why direct appeals to manipulationist counterfactuals play vital roles within the descriptive practices of real life science In this fashion we believe that semantic misdirection of a ldquogeneral theory of counterfactualsrdquo ilk has seriously hampered our ability to understand the working rationales of real life science in a coherent and rational manner

The specific forms of manipulationist counterfactual that we have associated with the side conditions of a standard well-set modeling within mathematical physics are quite elementary in character but serve a vital role in extracting further forms of counterfactual information from our target system such as its potential capacities with respect to the formation of standing wave patterns Many of these conclusions are naturally captured within manipulationist language as well as when we rightfully claim that ldquoIf discrete packets of energy E0 E1 hellip were deposited into the vibrational modes of a violin string at t0 in the absence of dispersive coupling the energies E0 E1 hellip will remain constant in those modes over an appreciable intervalrdquo (expressed in everyday language violin strings ring out with a constant harmonic flavor) We shall label assertions of this character (temporally) extended manipulationist counterfactuals (ldquoextended counterfactualsrdquo for brevity)21 Once again we frequently credit these extended counterfactuals

21 Under this heading we primarily have in mind ldquocounterfactuals with fully specified side conditionsrdquo On this basis we often indulge in more generic forms of assertion such as ldquoif this pendulum bob were displaced from its rest position it will (usually) start movingrdquo Here the

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

24

with firm and reliable truth-values whether we have gathered this assurance from a well-established mathematical model or directly from experimental test In the first circumstances the system laws within the modeling play a significant role in underwriting the truth-values of these extended counterfactual claims but so do the simpler side condition counterfactuals as well In our diagnosis these natural entanglements of extended counterfactuals with system laws within a well-set modeling represent the original source of the various philosophical mythologies that claim that ldquocounterfactuals must be conceptually grounded in laws to achieve firm truth-valuesrdquo But that conclusion is mistaken manipulationist facts and system laws operate as coequal partners in bringing these extended forms of behavioral pattern into structural prominence Letrsquos see why

5 System laws and extended counterfactuals

A good way to recognize that stark ldquolaw versus counterfactualrdquo contrasts are poorly drawn is to observe that ldquosystem lawsrdquo as the term is used here generally incorporate significant amounts of undischarged counterfactual data within their own interior workings (in the next section wersquoll pry some of these out of our old friend party2partt2 = c2party2partx2) And real life science benefits greatly from these counterfactual incursions We believe that these methodological recognitions have become unduly obscured by the careless manner in which most philosophers of science invoke the term ldquolawrdquo So let us remind our readers of the equational closure required of the system laws appearing within a well-set modeling they must collectively possess enough inferential power that they can fill out the interiors of their modeling assignments in a proper manner In our violin stringrsquos circumstances party2partt2 = c2party2partx2 must be able to fill out the complete interior of the open-ended box bounded on its bottom and sides by the problemrsquos anointed initial and boundary conditions respectively This task demands a good deal of inferential strength from our system laws In consequence the relevant bundles of modeling typically contain many ingredients that philosophers donrsquot usually regard as ldquolawlikerdquo including a fair amount of undischarged manipulationist appeal of the same sort as we have already detected within their companion side conditions

Unfortunately philosophers rarely delineate what they expect of a ldquolawrdquo in detailed consideration preferring to gesture vaguely in the direction of ldquoOh I mean stuff like Newtonrsquos

parenthetical ldquo(usually)rdquo has been inserted to acknowledge the possibility that we might move the bob to the pendulumrsquos unstable equilibrium position

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

25

laws Maxwellrsquos equations the law of gravitation and so forthrdquo These selective appeals are not broad enough to encompass the differential equation ldquosystem lawsrdquo central within most well-set modelings On other occasions philosophers swing to the opposite extreme of tolerance and freely accept ldquoall ravens are blackrdquo and ldquoall dry matches ignite when struckrdquo as candidate ldquolawsrdquo despite the fact that neither of these assertions look like plausible ldquolaws of naturerdquo in any traditionalist sense In truth considerable caution is mandated anytime ldquolawsrdquo become invoked in this off-handed manner Historically the notion emerged within the annals of science in a wide variety of highly irregular ways often carrying along fossilized remnants of archaic conceptions of scientific method Significant confusions can arise from the simple practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a ldquolawrdquo We shall find that ldquoHookersquos lawrdquo comprises a sterling exemplar of such a pattern in the next section

These tensions can readily recognized if we attempt to apply Hiddlestonrsquos reductive schema

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

to our string case Within its standard textbook modeling our old friend party2partt2 = c2party2partx2 is the most natural candidate to serve as Hiddlestonrsquos governing ldquoactual lawrdquo But it fulfills its formal ldquosystem lawrdquo responsibilities by filling out the interior of its initialboundary condition box admirably In non-temporal circumstances such as our soap film case the formal task has to be formulated somewhat differently the pertinent system equation(s) (eg Laplacersquos formula (partz2partx2 + partz2partx2 = 0) must fill in the spatial region within the span of the two wire rings As noted before several equilibrium distributions of this character may prove physically viable

Logical empiricist tradition regards the differential equation contents of a well-set modeling as ldquolawsrdquo due to the manner in which they perform this ldquofilling out an interior regionrdquo role in delineating the complete behaviors of target system S under a suitable range of side condition stipulations This identification appears to stem from a logicist inclination to label all of the inferential agencies that fill out an explanatory grid as ldquolawsrdquo whether that labeling enjoys any scientific precedent or not This loose policy has generated a large amount of conceptual confusion including contributing to the apparent plausibility of the grounding claims under examination in this essay So let us review what we have learned so far We have granted that we will feel confident in the truth-values assigned to various extended counterfactuals if we know that an experimentally verified well-set modeling has evinced those general features of Srsquos internalist responses in an evocative way We say ldquoOh I feel pretty confident that S will do X if S were manipulated in manner Mrdquo because the applied mathematicians have shown that such responses are native to system Srdquo On this basis we know that effaced circumstance counterfactuals such as ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo possess well-supported truth-values In contrast we may remain comparatively at sea with respect to the truth of ldquoif Caesar had been in charge of UN

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

26

forces during the Korean War he would have used catapultsrdquo Why Because we recognize the effacement policies followed in the first case have targeted Srsquos internal behaviors in a focused and reliable manner whereas in the second case we lack a firm understanding of its informative intent Indeed without further clues from a speaker we will remain uncertain which target system is at issue Are you trying to say something about Caesar Or the Korean War But in the favorable cases before us we are only trying to characterize the string or match in its own right

However let us reiterate the fact that we can become equally confident in the truth of these same extended counterfactuals through other means such as simple extrapolation from properly designed experiment We donrsquot require a wave equation modeling to know that ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo is a correct counterfactual to apply to our string In our diagnosis one source of the claim that our extended counterfactuals ldquodemand laws for their truth-conditionsrdquo stems from mistaking the formal fact that well-set modelings utilize system laws to inferentially fill out their interior behaviors for an inherent conceptual (or ldquometaphysicalrsquo) necessity No well-set modelings ldquodemand system lawsrdquo only in the sense that coupling interior system laws to properly simplified and varied side conditions supplies an excellent recipe for extracting many significant but hidden structural features latent within a target system of interest But an excellent policy for extracting information from a mathematical model does not constitute a conceptual dependency or necessary truth of the sort that grounding advocates anticipate Yes a properly composed well-set modeling reveals our systemrsquos central S versus E behaviors in crisp mathematical terms But these extraction capacities donrsquot demonstrate that our confidence in the truth-values of properly focused manipulationist counterfactuals ldquoconceptually dependsrdquo upon the availability of these mathematical models To the contrary we can gain an increased confidence in the same extended claims through many routes including direct forms of experimental test A laboratory testing bench can accord the same firm truth-value to ldquoif the string were sharply plucked at time t0 two sharp cornered waves will move to the right and left thereafterrdquo as our wave equation analysis the truth of the counterfactual doesnrsquot depend conceptually upon the wave equation modeling in any evident sense The only ldquodependencerdquo at issue lies in the fact that within a well-set modeling system laws and side conditions work together to bring otherwise hidden behaviors into clear view

The presumption that the truth-conditions of counterfactuals must rest upon ldquolawsrdquo and other forms of ldquooccurrentrdquo fact appears to derive at least in part from a puzzlement as to how counterfactual considerations can reveal useful facts about a real life target system It is then presumed that some sort of reductive analysis is conceptually required to address this methodological conundrum When Hiddleston writes

lsquoA 1048687rarr Crsquo is true iff A background facts and actual laws jointly imply C

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

27

his motivation appears to be that except for the variations tolerated in the selection of A every ingredient on the right side of his biconditional represents a fact that can be understood in a non-modal counterfactual-free way22 thereby yielding a satisfactory reduction or explication of the counterfactual on the left hand side23 In the same general vein (Psillos 2007) claims

The [story] I favor ties the truth-conditions of counterfactual assertions to laws of nature It is then easy to see how the evidence-conditions (that is actual and hypothetical experiments) are connected with the truth-conditions of a counterfactual actual and hypothetical experiments are symptoms for the presence of a law

Both authors regard these connections as ldquoconceptualrdquo or ldquosemanticrdquo because they believe that the informational utilities offered by common place manipulationist counterfactuals canrsquot be explained in any other way In our view however the correct answer to the question of counterfactual informativeness in the context of well-set modelings is that given by Hadamard examination of a target system S under a range of carefully monitored potential variations can reveal important behavioral characteristics of S in a manner that minimizes the risk of mistaking extraneous environmental influences for intrinsic features of S This is sensible scientific policy but it is not purist or reductive in any evident sense And its underlying rationales apply to sagacious experimental testing just as firmly as they constrain the exertions of the mathematical

22 Again as argued in Section 2 if this were not the case and some of the right hand side ingredients cannot be understood in a counterfactual-free way then we would have a regress with additional laws required to understand those counterfactual ingredients and so on The motivation for an account of the truth conditions for counterfactuals in terms of laws would be lost 23 Philosophers frequently presume that the ldquoinitial conditionsrdquo studied within a modeling should simply represent the actualist position and velocity data that we might extract from any space-like slice running through the interior of a target system S thereby persuading themselves that no modal information is coded therein This is incorrect for the reason already mentioned the mathematician requires behavioral data from a wide variety of initial conditions to reach proper conclusions about S In addition the initial states that offer the deepest insights into S are often not of an arbitrary time slice character For example to understand heat flow in a pipe we want to see what happens when we apply a blowtorch to a restricted region But such a condition lies outside the applicable range of the standard heat equation due to the non-analytic character of the initial data But of course knowing that we can apply blowtorches to pipes at will is one of the Basic Facts of Manipulationist Life so mathematicians reframe their notion of ldquoinitial conditionsrdquo so that this eventuality is tolerated (Readers may wish to ask themselves ldquoIs this Basic Fact grounded in some law of naturerdquo) In an allied vein they tolerate the impulsive hammer blows we applied to our string despite the fact that this starting state is not of a conventional positionvelocity stripe either Generally proper standards of initial condition toleration are motivated by background considerations of experimental manipulability rather than as data appearing on actualist time slices

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

28

modelerAs already noted we doubt that any general ldquoconceptualrdquo explanation of the

informativeness offered by counterfactuals of every stripe is possible or that any uniform and informative explication of their ldquotruth conditionsrdquo is forthcoming The vagaries of fictive consideration are too varied to permit such an analysis But if we narrow our sights to the unproblematic extended counterfactuals of scientific experience all of our ldquowhat are these things good forrdquo puzzles vanish for we can cite Hadamardrsquos compelling rationales for how various easy-to-establish counterfactual claims can assist the project of evaluating the internal capacities of target systems Mathematically these claims must be counterfactually extended in order to squeeze out the desired conclusions in a precise manner24 For these probes to work important classes of manipulationist counterfactuals must work together with appropriate system laws in symbiotic cooperation But this methodological entanglement is neither semantic nor conceptual but merely represents a mathematicianrsquos implementation of a general policy for understanding a target system in S versus E terms Related extraction policies are pursued by Woodwardrsquos experimentalists as well without any conceptual link to the system laws of the mathematical modeler

6 Entanglements of system laws with counterfactuals

Perhaps the clearest route to seeing that no general policy of eschewing appeals to counterfactuals is truly involved in schemas such as Hiddlestonrsquos is to scrutinize the actual contents of any typical set of system equations closely for these commonly involve covert appeals to further forms of undischarged counterfactual fact This is the chief juncture where loose thinking about ldquolawsrdquo plays a dirty trick on us Traditionally ldquolaws of naturerdquo are held to be universal non-dependent upon manufacture or place and time These are the standards of ldquolawlikenessrdquo that get invoked when a philosopher airily appeals to ldquostuff like Newtonrsquos laws Maxwellrsquos equations the law of gravitation and so forthrdquo Indeed lofty principles of this ilk do not rely upon humble manipulationist counterfactuals in any evident sense However the ldquosystem lawsrdquo at the core of a well-set modeling generally lack this unsullied character and typically consort with undischarged counterfactuals quite promiscuously Formally the reason for this disparity is quite simple in most circumstances we donrsquot possess enough ldquogeneral laws of naturerdquo to insure the equational closure required within a properly posed mathematical modeling These confusions would have never arisen if the logical empiricists had not employed ldquolawrdquo in such an indiscriminate manner

24 Specifically important system traits are often mathematically uncovered through fixed point searches through carefully monitored spaces of constrained possibilities Generally these spaces must contain enough variations to guide this search in a proper manner These formal concerns are discussed at greater length in (Wilson 2017) under the heading of ldquoSturm-Liouville problemsrdquo

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

29

In particular letrsquos briskly survey the derivations one finds of the standard string equation in introductory textbooks These vary considerably in rigor from one source to another but the best ones rest their arguments on three central pillars (61) F = ma (62) Hookersquos law (in one-dimension along the axis of the string) and (63) the constraint presumption that each section of string will remain directly above its rest position when stretched (as illustrated) Only the first looks like a ldquogeneral law of naturerdquo in the expected sense but its content is far too feeble to provide the equational closure required to fill in the interior of our evolving behavior chart To gain this closure we must appeal to other considerations that allow us to link the stringrsquos vertical acceleration (part2ypartt2) to some particular feature of its current position y(x) such as its local curvature (part2ypartx2)25 To this purpose considerations (62) and (63) enter the scene as essential ingredients within the system laws needed within a well-set modeling of an adequately specified nature

(62) and (63) donrsquot appear to be ldquogeneral laws of naturerdquo in a traditional sense so why do we trust their input into our differential equation reasoning principles so firmly The surprising answer is that both directly rely upon various collections of undischarged manipulationist data for their reliable support Letrsquos first consider (62) the ldquoHookersquos lawrdquo assumption that that a stretched unit of string will develop a restoring force directly proportional to its displacement from its natural rest position (along the axis of the string) Although behavioral assumptions of this linear character have been traditionally labeled as ldquolawsrdquo for nearly five centuries their provisos do not have the contents that grounding enthusiasts expect under the heading of a ldquolaw of naturerdquo Certainly its scope is not ldquouniversalrdquo in any plausible sense for no realistic materials obey Hookersquos principle perfectly The carefully wrought products sold as violin strings successfully approximate to Hookean behavior only through devoted attention to fabricational detail (a rather complex and delicate internal structure within the string is required) In modern classificatory parlance within mechanics ldquoHookersquos lawrdquo operates as a contingent constitutive principle and does not qualify as a ldquogeneral law of mechanicsrdquo in the same mode as F = ma or the balance of angular momentum26 Its applicability to a target material is generally established on the basis of direct testing bench experiment rather than through any appeal to the elaborate forms of molecular modeling that grounding enthusiasts appear to anticipate in their demands for law-like underpinnings Hookersquos law itself merely represents a simple inductive extrapolation of these laboratory findings into an empirical formula linking local stresses to local strains

When experimentally based constitutive assumptions intercede in the formulation of

25 In modern usage these are often called ldquoconstitutive principlesrdquo26 Worse yet the forces posited in Hookersquos law possess a character prohibited by Newtonrsquos third law as normally construed for they presume a natural rest configuration to which the system strives to return For a discussion of the general problem of articulating ldquofundamental force lawsrdquo capable of backing up the common procedures of classical physics see (Wilson 2008)

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

30

internalist system laws in this manner they function as a second form of modeling effacement here serving to segregate the large scale macroscopic behaviors of a target system from its lower scale behavioral complexities Our constraint-on-string-movement ingredient (63) exemplifies this effacement capacity in an even more naked form for no one pretends that ldquostring particles follow vertical paths directly above their rest positionsrdquo represents a ldquolaw of naturerdquo in any evident sense To the contrary it is a grateful testament to the finesse of the Pirastro String Company that real violin strings can be manufactured to approximate these geometrical behaviors even roughly

A more blatant illustration of the complexity-effacing capacities of constraints on movement is illustrated in the limited degrees of freedom we attribute to our steam shovels on a macroscopic scale A chief virtue of variational techniques in science as perfected by Lagrange lies precisely in their capacity to drastically simplify a modeling effort by exploiting known limitations on movement often to very dramatic effect In the jargon of the discipline these techniques allow a modeler ldquoto ignore the forces that perform no workrdquo upon our steam shovel This dramatic ldquophysics avoidancerdquo is achieved by describing the machine in terms of the independent angular coordinates αβγδ rather than employing regular Cartesian spatial locations and velocities These choices of descriptive parameters wall off the upper scale behaviors of our shovel from its lower scale complexities in much the same manner that properly selected boundary condition cuts efface the interior behaviors of system S from its real world environmental complexities E (the trick is to not avoid environmental influences altogether but to study their effects in restricted and controllable ways)

Although we will not enlarge further on these topics here we believe that the sagacious effacement of a modeling from unnecessary amounts of lower scale speculation comprises an important characteristic of the scientific method Every engineer knows that models that have been strengthened with strong injections of verified manipulationist experiment are commonly more reliable in practice than models that engage in excessive amounts of lower scale speculation One of our chief complaints about claims that ldquocounterfactual conditionals demand a grounding in lawsrdquo is that such contentions seem tone-deaf to one of the chief imperatives of proper science to undergird onersquos claims with as much reliability as possible Something has gone deeply astray when the very forms of counterfactual data that supply science with some of its greatest sources of reliability-enhancing backbone have become ungratefully characterized as ldquovague context-dependent and unclear in a way that makes them unsuitable for use in sciencerdquo

7 Conclusions

Within the real life circumstances of a well-set physical modeling the formative ingredients that loosely labeled as ldquoA background facts and actual lawsrdquo do not function in a conceptually reductive manner but instead assemble collections of data that have been selected to highlight the characteristic behaviors that should be credited to Srsquos internal workings rather

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

31

than comprising projected images of more complicated forms of external contingency In searching for proper locales in which effective policies of S versus E effacement can be implemented we are guided by known facts about system variability as reported in firmly established manipulationist counterfactuals of the sort ldquoif disturbed strings will displace in vertical directionsrdquo ldquowhen disturbed steam shovels display along xyαβγδ pathwaysrdquo ldquoif stretched nylon displays a nearly linear restorative responserdquo and so on But these dependencies are not conceptual they merely reflect sagacious descriptive strategies based upon the objective opportunities offered within nature itself

At this juncture defenders of counterfactual grounding sometimes retreat to the comforting arms of near tautology ldquoIsnrsquot there some richer story to be told with respect to the elasticity of nylon strings and wonrsquot this account require laws of a more fundamental variety All we wish to maintainrdquo they protest ldquois that all natural behaviors rest upon fundamental law in some sense of lsquorest uponrsquo that we donrsquot know how to precisifyrdquo27 At this stage the original requirements for a ldquoconceptual grounding in lawrdquo appear to have devolved into some hazy expression of molecular naturalism with which we neednrsquot disagree By the same non-conceptual token X being a bachelor depends upon the fact that Xrsquos body must contain a smidgen of molybdenum without which X would quickly expire We surely wouldnrsquot want to claim that part of the truth-conditions for X being a bachelor is that Xrsquos body must contain some portion of that element But that is exactly the manner in which many defenders of grounding demands express themselves for equally inappropriate reasons

So once again we reiterate our challenge to the ldquogrounding in lawrdquo demands of the metaphysicians why do you insist upon critiquing established descriptive technique in a seemingly inaccurate and counterproductive manner 27 A popular rejoinder appeals to hypothetical sets of purist ldquofundamental lawsrdquomdashprinciples that we donrsquot presently know how to formulate but must be forthcoming someday (it is presumed) On this basis grounding advocates concede ldquoOkay you have established that various classes of undischarged counterfactuals play significant roles in underpinning the derived laws that we invoke within conventional modeling But these supplements merely represent lsquoauxiliary assumptionsrsquo S(x) that have been added to the antecedents of genuine fundamental laws in the fashion that (x)(Ax rarr Bx) is specialized to obtain (x)((Ax amp S(x)) rarr Bx)rdquo

One should be wary of arguments such as this Appeals to constraints in the standard manner of Lagrangian mechanics represent stipulations on system behaviormdashldquowhatever happens these rods and gears will remain approximately rigid over the lifetime of system Srdquo Strong assumptions of this character can effectively ldquoturn offrdquo underlying fundamental law requirements in favor of their own approximating ldquostipulationsrdquo Reconciling rigid body behavior with Newtonrsquos third law (as conventionally understood) is difficult but practitioners typically brush these obstacles aside with an airy ldquoWell on a macroscopic level those parts will stay approximately rigid and thatrsquos good enough to support the conclusions I wish to reachrdquo Unapologetic appeals to ldquosuccessful approximationrdquo can readily overpower any pertinent form of ldquounderlying physicsrdquo For more on these subtle issues see (Wilson 2017)

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

32

References

Correia Fabrice and Schneider Benjamin 2012 Metaphysical Grounding Cambridge UK Cambridge University Press

Eddington AS (1928) Nature of the Physical World Cambridge UK Cambridge University Press

Goodman Nelson (1955) Fact Fiction and Forecast Cambridge MA Harvard University Press

Hadamard Jacques (1952) Lectures on Cauchyrsquos Problem in Linear Partial Differential Equations New York Dover

Harris Paul (2000) The Work of the Imagination Hoboken NJ Wiley- Blackwell

Hiddleston Eric (2005) ldquoReview of Woodward Making Things Happenrdquo Philosophical Review 114 545-547

Lewis David (2001) Counterfactuals Oxford Wiley-Blackwell

Maudlin Tim (2007) The Metaphysics Within Physics Oxford Oxford University Press

Paul LA and Hall Ned (2013) Causation A Userrsquos Guide Oxford Oxford University Press

Lewis David (1973) Counterfactuals Cambridge MA Harvard University Press

Psillos S (2004) ldquoA Glimpse of the Secret Connection Harmonizing Mechanisms with Counterfactualsrdquo Perspectives on Science 12 (3)

Psillos S (2007) ldquoCausal Explanation and Manipulationrdquo in Johannes Persson amp Petri Ylikoski (eds) Rethinking Explanation New York Springer

Stalnaker Robert (1968) ldquoA Theory of Conditionalsrdquo in N Rescher ed Studies in Logical Theory Oxford Blackwell

Weisberg D S and Gopnik Alison (2013) ldquoPretense Counterfactuals and Bayesian Causal Models Why What is not Real Really Mattersrdquo Cognitive Science 37 1368ndash1381

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599

33

Wilson Mark (2009) Determinism The Mystery of the Missing Physics British Journal for the Philosophy of Science 60 173-193

------ (2017) Physics Avoidance and other Essays Oxford Oxford University Press

Woodward James (2003) Making Things Happen New York Oxford University Press

----- (2011) ldquoPsychological Studies of Causal and Counterfactual Reasoningrdquo in C Hoerl T McCormack amp S R Beck (Eds) Understanding Counterfactuals Understanding Causation (pp 16ndash53) Oxford UK Oxford University Press

---- (2015) ldquoMethodology Ontology and Interventionismrdquo Synthese 192 3577ndash3599