bertrand russell_s the analysis of matter its historical context and contemporary interest - william...

8/12/2019 Bertrand Russell_s the Analysis of Matter Its Historical Context and Contemporary Interest - William Demopoulos and Michael Friedman

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Bertrand Russell's The Analysis of Matter: Its Historical Context and Contemporary InterestAuthor(s): William Demopoulos and Michael Friedman

Source: Philosophy of Science, Vol. 52, No. 4 (Dec., 1985), pp. 621-639Published by: The University of Chicago Presson behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/187446.

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CRITICAL NOTICE:BERTRAND RUSSELL'S THE ANALYSIS OF MATTER:ITS HISTORICAL CONTEXT ANDCONTEMPORARY INTEREST*t

WILLIAM DEMOPOULOSDepartment of PhilosophyUniversity of Western Ontario

MICHAEL FRIEDMANDepartment of Philosophy

University of Illinois at ChicagoThe Analysis of Matter is perhaps best known for marking Russell'srejectionof phenomenalism in both its classical and methodologicalforms)and his development of a variety of Lockean representationalism-Rus-sell's causal theory of perception. This occupies Part 2 of the work. Part1, which is certainly less well known, contains many observations ontwentieth-centuryphysics. Unfortunately, Russell's discussion of relativ-ity and the foundations of physical geometry is carried out in apparent

ignorance of Reichenbach's and Carnap's investigations in the same pe-riod. The issue of conventionalism in its then contemporaryform is sim-ply not discussed. The only writers of the period who appear to have hadany influence on Russell's conception of the philosophical issues raisedby relativity were Whitehead and Eddington. Even the work of A. A.Robb fails to receive any extended discussion;1 although Robb's causaltheoryis certainlyrelevant to many of Russell's concerns, especially thosevoiced in Part 3, regarding the construction of points and the topologyof space-time. In the case of quantum mechanics, the idiosyncrasy ofRussell's selection of topics is more understandable,since the Heisenbergand Schr6dinger heories were only just discovered. Nevertheless, it seemsbizarre to a contemporary reader that Russell should have given suchemphasis2 o G. N. Lewis's suggestion that an atom emits light only when*Received May 1985.tAn earlier version of this paper was presented at the Grover Maxwell Memorial Con-ference on Bertrand Russell's Philosophy of Science, June, 1982.'See Winnie (1977), pp. 156 ff. for a discussion of Russell's remarks on Minkowski

space-time, and their relation to Robb's theory.2See especially pp. 125 ff.Philosophy of Science, 52 (1985)pp. 621-639.Copyright C 1985 by the Philosophy of Science Association.

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622 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANthere is anotheratom to receive it-a suggestion reminiscent of Leibniz,and one to which Russell frequently returns. In short, the philosophicalproblems of modernphysics with which Russell deals seem remote fromthe perspective of post-positivist philosophy of physics.But if the observations on philosophy of physics seem dated, this isnot true of the theory of theories which the book develops. As GroverMaxwell emphasized,' it is possible to extract from the book a theory oftheories that anticipates in several respects the Ramsey-sentence recon-structionof physical theories articulatedby Carnapand others many de-cades later.

IThe heartof the theory of The Analysis of Matter is the claim that ourknowledge of the external world is purely structural. For Russell, thisthesis was based on the contention that we are not "directly acquainted"with physical objects. On a more neutral reading, the basis for the thesisis the belief that the reference of the class of theoretical predicates hasan explanation in terms of the reference of anotherfamily of predicates.Such an explanation, if it could be given, would be highly non-trivialsince no definitionalor reductive relation between the two classes of pred-icates is claimed, althoughthere is for Russell something like a reductiverelation between our knowledge of theoretical properties and our knowl-edge of perceptual properties. In this respect the theory of The Analysisof Matter stands in marked constrast to the phenomenalism of Russell's1914 external world program.4The point of the latter, of course, is toassume only perceptual objects (sensibilia or actual and possible sense-data), perceptual relations (especially perceptualsimilarity relations), andto explicitly define all other objects and relations from these. In The Anal-ysis of Matter, Russell wishes to exploit the notion of logical form orstructure o introducescientific objects and relationsby means of so-calledaxiomatic or implicit definitions. Thus, if we representa scientific theoryby

H 01,.- . . on; TIs . .gTm)s 1say, where 01, . ., 0O are observational or perceptual terms and T1,

I., Tn are theoretical terms, then Russell in 1927 is preparedto acceptthe Ramsey-sentence3T,rI. I3T"m0(0J... 10,; Trl,. .,Tm) (2)

as the properstatementof our scientific knowledge. And (2) is legitimate3See especially his 1968 and 1971.4As articulated in his 1914a and 1914b.



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THE ANALYSISOF MATTER 623whether or not the theoretical terms in (1) are explicitly definable fromthe observational terms; indeed, the whole tenor of Russell's discussionin The Analysis of Matter is that theoretical terms will not be explicitlydefinable in purely observational terms.'It is not entirely clear why Russell abandonedhis earlier phenomenal-ism. His explicit discussion of the issue in chapter20 of The Analysis ofMatter is ratherinconclusive. Postulating non-perceptible events allowsus to maintain a very desirable continuity in stating the laws of nature,(1927, p. 216f.) but it appears that "ideal" perceptible events (the sen-sibilia of 1914) would serve equally well (1927, pp. 210, 213). A perhapsmore interesting reason for Russell's shift emerges from the discussionof recent physics in Part 1. Briefly, the point is that a phenomenalistreduction in the style of the 1914 program does not do justice to the novel"abstractness"of modem physics. Compared with the knowledge ex-pressedin classical physics or commonsense, for example, the knowledgeconveyed by twentieth-century physics appears to be on a higher level ofabstraction:t is "structural" r "mathematical"n an importantnew sense.For Russell, this leads to a partial skepticism regarding our knowledgeof the physical world:

Whatever we infer from perceptions it is only structure that we canvalidly infer; and structure s what can be expressed by mathematicallogic. (1927, p. 254)The only legitimate attitude about the physical world seems to be oneof complete agnosticism as regards all but its mathematical proper-ties. (1927, p. 270)6

The view expressed in these passages is very strongly anticipated inRussell's Introduction to Mathematical Philosophy (1919), where itsKantianmotivation is quite evident:There has been a great deal of speculation in traditionalphilosophywhich might have been avoided if the importance of structure, andthe difficulty of getting behind it, had been realized. For example,

5Notice that whether the terms are implicitly or explicitly definable depends on the log-ical framework within which the definitions are to be constructed. Is (1) a first-order theorywith identity? An extension of first-order set theory? A formulation in some higher-orderlogic? These differences matter because, if one is working in a sufficiently strong back-ground language, one can often transform axiomatic or implicit definitions into explicitdefinitions (see Quine 1964 for one such strategy). Of course, this way of expressing theissue is anachronistic since these distinctions were not available to Russell. Consequently,the distinction between implicit and explicit definition is not so clearcut for Russell. Thus,for example, Russell appears willing to allow implicit definitions even in 1914 when, insection 9 of his 1914b he defines physical things as "those series of appearances whosematter obeys the laws of physics."6See also chap. 14, especially the discussion of Eddington, pp. 136 ff.



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624 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANit is often said that space and time are subjective, but they have ob-jective counterparts;or that phenomena are subjective, but are causedby things in themselves, which must have differences inter se cor-responding with the differences in the phenomena to which they giverise. Where such hypotheses are made, it is generally supposed thatwe can know very little about the objective counterparts. In actualfact, however, if the hypotheses as stated were correct, the objectivecounterparts would form a world having the same structure as thephenomenalworld, and allowing us to infer from phenomenathe truthof all propositions that can be stated in abstract terms and are knownto be true of phenomena. If the phenomenal world has three dimen-sions, so must the world behind phenomena; if the phenomenal worldis Euclidean, so must the other be; and so on. In short, every prop-osition having a communicablesignificancemust be true of both worldsor of neither: the only difference must lie in just that essence of in-dividuality which always eludes words and baffles description, butwhich, for that very reason, is irrelevant to science. Now the onlypurposethatphilosophers have in view in condemning phenomena isin order to persuadethemselves and others that the real world is verydifferent from the world of appearance. We can all sympathize withtheir wish to prove such a very desirable proposition, but we cannotcongratulate them on their success. It is true that many of them donot assert objective counterpartso phenomena, and these escape fromthe above argument.Those who do assert counterparts are, as a rule,very reticent on the subject, probably because they feel instinctivelythat, if pursued, it will bring about too much of a rapprochementbetween the real and the phenomenal world. If they were to pursuethe topic, they could hardly avoid the conclusions which we havebeen suggesting. In such ways, as well as in many others, the notionof structure. . . is important. (1919, pp. 61f.)

To sum up, on Russell's "structuralism"or "structuralrealism," of"percepts"we know both theirquality and structure(where Russell's useof the term "quality" ncludesrelations),while of external events we knowonly their structure. The distinction is, in the first instance, between prop-erties and relations of individuals and properties and relations of prop-erties and relations. Structuralpropertiesare thus a particularsubclass ofpropertiesof properties and relations: they are marked by the fact thatthey are expressible in the language of higher-orderlogic. Unlike Locke'sdistinctionbetween primary and secondary qualities, the structure/qualitydistinctiondoes not mark a differencein ontological status: externaleventshave both structureand qualities-indeed when we speak of the structureof external events, this is elliptical for the structuralproperties of their



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THE ANALYSISOF MA7TER 625qualities. Neither is it the case that one is "more fundamental" than theother or that qualities are "occurrent" while structure is a power. Whatis claimed is a deficiency in our knowledge: of external events we knowonly the structuralproperties of their properties and relations, but we donot know the properties and relations themselves. Of course, physicalknowledge falls within the scope of this claim, so that the theory of per-ception immediately yields the consequence that physical theories giveknowledge of structureand only knowledge of structure.Russell's emphasis on structure in The Analysis of Matter has closeaffinities with much other work of the period, especially with the twoclassics of early (pre-1930) positivism: Schlick's General Theory ofKnowledge (1918, second edition 1925) and Carnap's Aufbau (1928). In-deed, the "criticalrealism" of General Theory of Knowledge is identicalin almost every respect to Russell's structuralrealism of 1927. Schlickargues that modern physics deals with real unobservable entities (atoms,electrons, the electromagnetic field), which cannot be understood as log-ical constructions out of sense-data in the manner of Mach's Analysis ofSensations (1897) or Russell's 1914 external-worldprogram (1918, ?26).Such entities are not experienceable, intuitable, or even picturable; ac-cordingly, Schlick goes so far as to call them "transcendent"entities and"things-in-themselves"(1918, ?25). Nevertheless, this "transcendence"presents no obstacle to our knowledge or cognition, for knowledge relatesalways to purely formal or structural properties-not to intuitive quali-ties or content. Thus, while we cannot experience or intuit the entities ofmodern physics, we can grasp their formal or structural eaturesby meansof axiomatic or implicit definitions, in the style of Hilbert's Foundationsof Geometry(1899)-and this is all that knowledge or cognition requires(1918, ??5-7). The similarities with Russell's view of 1927 are patent.However, there is one significant difference: Schlick draws a sharp con-trast between knowledge (erkennen) and acquaintance (kennen). On hisaccount, knowledge by acquaintanceis a contradiction in terms, for onlystructural eatures are ever knowable (1918, ? 12). We areacquaintedwithor experience (erleben) some qualities (content), but we have knowledgeor cognition of none. As we have seen, for Russell we know both theform and the content of percepts.Camap's Aujbau does not embrace structural realism. All concepts ofscience are to be explicitly defined within a single "constructional sys-tem" whose only non-logical primitive is a "phenomenalistic"relationRsof recollected similarity. Yet the form/content distinction and the notionof logical structure are equally important. To be sure, all concepts ofscience are to be constructed from a basis in the given or "my experi-ence," but the objectivityof science is capturedin a restriction to purelystructuralstatements about this given basis. Although the "matter" or



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626 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANcontent of Carnap's constructional system is indeed subjective or "au-topsychological" and therefore private and inexpressible-what is reallyimportant s the logical form or structure of the system. For it is logicalform and logical form alone that makes objective knowledge and com-municationpossible:7

Science wants to speak about what is objective, and whatever doesnot belong to the structurebut to the material (i.e., anything that canbe pointed out in a concrete ostensive definition) is, in the final anal-ysis, subjective. One can easily see that physics is almost altogetherdesubjectivized, since almost all physical concepts have been trans-formed into purely structuralconcepts. . . . From the point of viewof construction theory, this state of affairs is to be described in thefollowing way. The series of experiences is different for each subject.If we want to achieve, in spite of this, agreement in the names forthe entities which are constructed on the basis of these experiences,then this cannot be done by reference to the completely divergentcontent, but only through the formal description of the structure ofthese entities. (1928, ?16)8

Characteristically,Carnapdoes not rest content with a simple referenceto logical form or structure,he turns his "objectivity requirement" intoa definite technical program:the program of defining all scientific con-cepts in terms of what he callspurely structuraldefinite descriptions(1928,??1 1-15). Such a definition explains a particularempirical object as theunique entity satisfying certain purely formal or logical conditions: thevisualfield, for example, is defined as the unique five-dimensional "senseclass" (??86-91). The point is that such purely structuraldescriptionscontain no non-logical vocabulary; ultimately, we will need only vari-ables and the logical machinery of Principia Mathematica (or set the-ory).9 Thus, while Russell in TheAnalysis of Matter would formulate our

7That s, one finds the form/content distinction drawn in such a way that form is ob-jective andcommunicable, while content is neither. Schlick came to view the form/contentdistinction in much the same way in his later writings, while Russell came to place muchless stress on this (somewhat romantic) view of the distinction. (For Schlick's view, seehis 1979.) Schlick explicitly acknowledges the influence of the Aufbau and of Wittgen-stein's Tractatus.8In he explanatory references to this section, Carnaprefers to the passage from Russell's1919, quoted earlier. Immediately before this section, Carnap refers to Schlick's doctrineof implicit definition in his [1918] (1925).9AsCarnaphimself emphasizes, this kind of programfor individuatingconcepts by meansof purely formal properties only begins to make sense in the context of modern, polyadiclogic. Monadic concepts correspond to unstructured sets whose only formal property istheir cardinality. Polyadic concepts have such diverse formal properties as transitivity,reflexivity, connectedness, dimensionality, and so on. In Russell's terminology from his1919 and The Analysis of Matter, only polyadic concepts have "relation-numbers."



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THE ANALYSISOF MATTER 627physical knowledge in the mannerof (2) (that is, by turningall theoreticalterms into variables), Carnapin the Aufbau goes much further:all termswhatsoever are to be replaced by variables. (The reader might very wellwonder how Carnap'sone non-logical primitive, RE, s to be itself elim-inated in favor of variables. We shall returnto this below.)

IIWe believe there are insurmountabledifficulties with the theory of the-oretical knowledge just outlined. So far as we are aware these difficultieswere first raised by M. H. A. Newman in an article published in Mind

in 1928. Newman's paper-the only philosophical paper he ever pub-lished-is not as well known as it deserves to be. (The prose is quitedelightful, and for that reason alone, it deserves a wider readership.) Thepapercontains much that is of interest today, and so, we propose to dis-cuss the structuralismof Russell's The Analysis of Matter from the per-spective of Newman's paper.Newman begins his discussion with the observation that on any theoryof scientific knowledge, the question of the truth of at least some prop-ositions of physics should turn out to be non-trivial. Consider, for ex-ample, the question whether or not matter is atomic. Newman observesthat on any account "this is a real question to be answered by consid-eration of the evidence, not a matter of definition" (p. 143). In fact,whether or not matter is atomic is a question that concerns the holdingof various "structuralproperties," althoughthey are of a fairly high levelof abstraction.(It entails, for example, thatphysical objects have isolableconstituents, and that these constituents have a certain theoretically char-acterizableautonomy.) Newman's point is that whether or not the worldexhibits such properties is a matterto be discovered, not stipulated, andwe may demand of a theory of theories that it preserve this fact. The gistof his criticism of Russell is that (with one exception) Russell's theorydoes not satisfy this constraint. Let us see how the argument goes.The difficulty is with the claim that only structure is known. On thisview "the world consists of objects, forming an aggregatewhose structurewith regardto a certain relationR is known, say [it has structure] W;butof . . . R nothing is known . . . but its existence; . . . all we can say isThere is a relation R such that the structureof the external world withreference to R is W" (Newman 1928, p. 144). But "any collection ofthings can be organized so as to have the structureW, provided there arethe right number of them" (p. 144, italics added). Thus, on this view,only cardinality questions are open to discovery Every other claim aboutthe world that can be known at all can be known a priori as a logicalconsequence of the existence of a set of ct-many objects. For, given a set



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628 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMAN

A of cardinality a, we can with a minimal amountof set theory or second-orderlogic, establish the existence of a relation having the structure W,provided that W is compatible with the cardinality constraint that JAI=cx. (The relevant theorem from set theory or second-order logic is thepropositionthatevery set A determines a full structure,that is, one whichcontainsevery relation [in extension] of every arity on A; such a structureforms the basis for a [standard]model for the language of second- [orhigher-] order logic.)It is importantto be clear on the natureof the difficulty Newman hasuncovered. The problem is not a failure of the theory to specify the do-main of objects on which a model of our theories of the world is to bedefined. The difficulty is not the one of Pythagoreanismraised by Quine(1964, pp. 214ff.) and John Winnie (1967, pp. 227ff.); that is to say, itis not thatRussell cannot rule out abstractmodels. Indeed Russell himselfraises this problem in the Introductionto The Analysis of Matter:

It frequentlyhappens that we have a deductive mathematicalsystem,startingfrom hypotheses concerning undefined objects, and that wehave reason to believe that there are objects fulfilling these hy-potheses, although, initially, we are unable to point out any suchobjects with certainty. Usually, in such cases, although many dif-ferent sets of objects are abstractly available as fulfilling the hy-potheses, there is one such set which is much more important thanthe others. . . . The substitutionof such a set for the undefined ob-jects is "interpretation."This process is essential in discovering thephilosophical import of physics.

The difference between an importantand an unimportantinterpre-tation may be made clear by the case of geometry. Any geometry,Euclidean or non-Euclidean, in which every point has coordinateswhich are real numbers, can be interpretedas applying to a systemof sets of real numbers-i.e. a point can be taken to be the series ofits coordinates. This interpretationis legitimate, and is convenientwhen we are studying geometry as a branch of pure mathematics.But it is not the importantinterpretation.Geometry is important,un-like arithmeticand analysis, because it can be interpretedso as to bepartof applied mathematics-in fact, so as to be part of physics. Itis this interpretationwhich is the really interestingone, and we cannotthereforerest content with the interpretationwhich makes geometrypartof the study of real numbers, and so, ultimately, partof the studyof finite integers. (pp. 4f.)

In this case we have a simple criterion for separating important fromunimportantinterpretations:the important interpretationsare connectedwith our observations, the unimportantones are not. Newman's problem



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THE ANALYSISOF MATTER 629arises after the domain has been fixed: In the case of abstract versusphysical geometry we have to distinguish different relations on differentdomains. The problem is made easier by the fact that we can exclude oneof the domains and therefore one class of relations. When the domain ofthe model is given, we must "distinguish between systems of relationsthathold among the members of a given aggregate" (Newman, p. 147).This is a difficulty because there is always a relation with the structureW. Russell cannot avoid trivialization by claiming that the relation withstructure W which exists as a matter of logic is not necessarily the im-portant relation with structure W (as the interpretation in R3 is not theimportant interpretationof geometry). That is to say, one cannot avoidtrivializationin this way without some means of distinguishing importantfrom unimportantrelations on a given domain. But the notion of impor-tance which must be appealed to is one for which Russell can give noexplanation;and in fact, Russell's own theory precludes giving any ex-planation of the notion.Newman's summary is well worth quoting in full:

In the present case [i.e., in the case where we must choose amongrelations on a given domain rather than choose among different re-lations on different domains] we should have to compare the impor-tance of relations of which nothing is known save their incidence (thesame for all of them) in a certain aggregate. For this comparisonthereis no possible criterion, so that "importance"would have to be reck-oned among the prime unanalysable qualities of the constituents ofthe world, which is, I think, absurd. The statement that there is animportant relation which sets up the structureW among the unper-ceived events of the world cannot, then, be accepted as a true inter-pretationof our beliefs about these events, and it seems necessary togive up the "structure/quality" division of knowledge in its strictform. (p. 147)

Recall, giving up the structure/quality division in its strict form meansgiving up the idea that we do not know the qualities of unperceived ob-jects. This of course would rob Russell's theory of its distinctive char-acter, and, so far as we can tell, Russell remainedequivocal on this issuefor the remainderof his philosophical career.It might be thought that Newman's objection depends on too exten-sional an interpretationof relations, and that once this is given up, a wayout of the difficulty is available. The idea is that logic (or logic plus settheory) only guaranteeswhat Newman callsfictitious relations-relationswhose only propertyis that they hold between specified pairs of individ-uals-e.g., binary relations whose associated propositional functions areof the form 'x is a and y is b, or x is a and y is c, etc.', where a, b, and



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630 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANc are elements of A. But in fact this is not so. Given some means ofidentifying the individuals of the domain A, we can always find an iso-morphic structureW' which holds of a non-fictitious relation R' on do-main A'. We may now define a non-fictitious relation S whose field isincluded in A: S is just the image of R' under the inverse of the iso-morphism which correlates A and A'. Newman's example is a simplecombinatorialone. Consider a graphF(R) of a relationR on A x A, whereA= J{a,03,P,'y}

a-1

F(R)Figure 1

Define a new relation S on A be letting S hold of (x,y) if f(x) andf(y)belong to different alphabets where 'a' f(a), 'of = f(lo), '' ,' =f(y). Then S is non-fictitious. It is clear that this strategyis perfectlygeneral. Following Newman, call a non-fictitious relation real. Then anyextensionally specified relation may be regarded as the image of a real(non-fictitious) relation. And this real relation exists if the fictitious onedoes. Thus a claim of the form 'there exists a real relation R such that.' will be true given a claim of the form, 'there exists a relation Rsuch that . . .'. Of course S may nonetheless be trivial or unimportant.But this just shows that eliminating extensionalism does not solve ourproblem.We might try to add furtherconstraints on the notion 'R is a real re-lation'. But then we move away from the relatively clear intuition whichmotivated its original characterization,namely, that a relation with onlya purely extensional characterization s fictitious. (Notice that certain ob-vious candidates such as "projectibility"cannot be appealed to since itis precisely the class of relations and propertieswhich enter into laws thatwe are attemptingto delimit.)The conclusion Newman draws from this analysis is, we think, theright one: since it is indisputably true that our knowledge of structureisnon-trivial-we clearly do not stipulatethe holding of the structuralprop-erties our theories postulate-it cannot be the case that our knowledgeof the unperceivedpartsof the world is purely structural.The observable/unobservabledichotomy is not explicable in termsof the structure/qualitydivision of knowledge without giving up the idea that our knowledge ofthe unobservableparts of the world is discovered ratherthan stipulated.



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THE ANALYSISOF MATTER 631Of course, it is also possible to give up the naive intuition on whichNewman's conclusion from his analysis depends. In this case one wouldaccept the stipulative character of the theoretical component of ourknowledge; and, indeed, much of the philosophy of science that imme-diately preceded and subsequently followed The Analysis of Matter didjust this. Russell is unique in wanting to preserve a non-conventionalistview of the world's structure, while retaining a structure/qualityor form/content division of knowledge which is intended to more or less corre-spond to the division between theoretical and observational knowledge.By the way, Newman's objection also explains the intuition that, de-spite its intention, Russell's structuralismcollapses into phenomenalism:The difficulty is that every assertion about unperceived events is triviallytrue:that is, true as a matterof logic plus an empirical assumption con-cerning cardinality. Now, the phenomenalist claims that statementsaboutthe external world are "reducible"to statements about perception. What-ever else this may mean, it requires that, given the appropriatereductivedefinitions, statementsabout the physical world areentailed by statementsabout perception. But if Russell's view is accepted, this characteristicconsequence of phenomenalism is almost guaranteed: If statements aboutthe physical world are true as a matterof logic, then they are implied byevery proposition, and, in particular, they are implied by the statementsof perception. Thus, modulo the single non-logical assumption concern-ing the cardinality of the external world, Russell's structuralism guar-antees the truth of this phenomenalist thesisThe only reference to Newman's paper of which we are aware is inthe second volume of Russell's autobiography. Russell very graciouslyacknowledges Newman's criticism in a letter which he reprints withoutcomment. The letter is worth quoting at length. Dated April 24th, 1928,it is identified simply as "To Max Newman, the distinguished mathe-matician."

Dear Newman,Many thanks for sending me the off-print of your article about mein Mind. I read it with great interest and some dismay. You make itentirelyobvious thatmy statements o the effect that nothingis knownabout the physical world except its structureare either false or trivial,and I am somewhat ashamed at not having noticed the point for my-self.It is of course obvious, as you point out, that the only effectiveassertion about the physical world involved in saying that it is sus-ceptible to such and such a structure s an assertion about its cardinalnumber. (This by the way is not quite so trivial an assertion as itwould seem to be, if, as is not improbable, the cardinal number in-



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632 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANvolved is finite. This, however, is not a point upon which I wish tolay stress.) It was quite clear to me, as I read your article, that I hadnot really intendedto say what in fact I did say, that nothing is knownabout the physical world except its structure. I had always assumedspacio-temporalcontinuity with the world of percepts, that is to say,I had assumed that there might be co-punctuality between perceptsand non-percepts, and even that one could pass by a finite numberof steps from one event to another compresent with it, from one endof the universe to the other. And co-punctuality I regarded as a re-lation which might exist among percepts and is itself perceptible.(1968, p. 176)

To our knowledge, Russell never discusses the puzzle in any of hislater work. He seems to give up the idea that our knowledge of the phys-ical world is purely structural, but there is no account of how, on histheory of knowledge (e.g., the theory developed in Human Knowledge:Its Scope and Limits [1948]), such non-structural knowledge can arise.Yet all the elements of the earlier and later theories are the same-theonly difference is in the conclusion drawn. Thus either the original claim(that we are restricted to purely structuralknowledge) was theoreticallyunmotivated or the argumentof the later theory contains a lacuna. Thedifficulty in adjudicating between these alternatives is that the theoreticaldevelopment is made to depend on what we regard as falling within "ac-quaintance." And this makes the resolution quite artificial: in the earliertheory we could not assume acquaintance with a cross category notionsuch as spacio-temporalcontiguity or causality; but in light of the diffi-culties of that theory we now find that we can assume this 10We are notsaying that one cannot resolve the issue in this way. But it seems quiteclear that without a considerable advance in the theoretical articulationof this rather elusive Russellian concept, no such resolution of the dif-ficulty can be very compelling.

IIIThere is anotherconsequence thatNewman draws, although it is muchless explicit in the article. Russell's structuralism can be viewed as atheoryof how the reference of the theoretical vocabulary is fixed. On this

view the reference of a T-predicate is fixed by two things: (1) its con-'?GroverMaxwell opted for this solution. (See the papers previously cited.) Unlike Rus-sell, Maxwell was quite explicit about the nature of the problem.



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THE ANALYSISOF MA ITER 633nection with observation, and (2) its structural properties. As we haveseen the connection with observation fixes the domain (or at least, es-tablishes that very many sets are not the domain of interpretation). Struc-ture is supposed to complete the task while preserving the non-stipulativecharacterof the truth of our theoretical knowledge. As we saw, Newmanobserves that within Russell's theory, there is no analogue for relationsof the important/unimportant criterion for domains. From a contempo-rary, model-theoretic standpoint, this is just the problem of intended ver-sus unintended interpretations:Newman shows that there is always somerelation, R, (on the intended domain) with structure W. But if the onlyconstraints on something's being the intended referent of 'R' are obser-vational and structuralconstraints, no such criterion for distinguishing theintended referent of 'R' can be given, so that the notion of an intendedinterpretation s, in Quine's phrase, provided by our background theory,and hence, cannot be a formal or structural notion in Russell's sense.Now, in fact, something strikingly similar to this argument has recentlybeen rediscovered by Hilary Putnam and it has been used by him to posea puzzle for certain "realist" views of reference and truth.Putnam's formulation of the argument is model theoretic and generalratherthan informal and illustrative: Suppose we are given a theory T allof whose observational consequences are true. We assume, for the sakeof the argument, that the observational consequences of T may be char-acterizedas a subset of the sentences generated from a given "O-vocab-ulary." It is assumed that the interpretationof the language L(T) of T isspecified for its 0-vocabulary, so that T is "partially interpreted"in thesense that the interpretation unction is a partial function. Assume that Tis consistent and that M is an abstract model of the uninterpreted sen-tences of T of the same cardinality as T's intended domain. FollowingPutnam we let "THEWORLD" denote the intended domain of T. Nowextend the interpretation o the theoretical vocabulary of T by letting eachpredicate of the theoretical vocabulary of T denote the image in THEWORLD of its interpretationin M under any one-one correspondencebetween M and THE WORLD. Call this interpretation which extends thepartial interpretation)"SAT." Clearly SAT is completely arbitrary,andshould be an utterly unacceptable extension since it trivializes the ques-tion whether T is true. Any theory of knowledge and reference that isincapable of rejecting SAT, i.e., any theory which is incapable of dis-tinguishingtruth from truthunder SAT-TRUTH (SAT)-is committedto the implication, T is true if T is TRUE (SAT). But T is TRUE (SAT)as a matterof logic That is, "T is TRUE (SAT)" follows, via the com-pleteness theorem, from "Tis consistent." So any theory that cannot ex-clude SAT as an intended interpretationof L(T) cannot account for our



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634 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANnaive confidence in the belief that our theories, if true, are "significantlytrue."11

Except for its metalogical flavor, this argument parallels Newman's.Where Putnam argues from the consistency of a set of sentences, New-man argues directly from the existence of a relational structure satisfyingthe intuitive conditions of a model. (Recall that Newman was writing in1928 and in the logicist tradition of Russell and Whitehead-a traditionto which the metalogical turn in mathematicallogic was quite alien. SeeGoldfarb 1979.)The chief difference between Putnam and Newman is in the use madeof the argument. Putnam, as we have said, employs the argument againsta certain form of realism-one which is rather difficult to isolate, al-though there are some suggestions. For example Putnam says it is thatform of realism which holds that even an epistemically ideal theory mightbe false; or any view which regards truth as radically non-epistemic; orany theory of truthand reference which "treatslanguage as a 'mirrorofthe world', rather than one which supposes merely that 'speakers mirrorthe world' i.e., their environment-in the sense of constructing a sym-bolic representationof that environment" (1978, p. 123). 12 All these re-marks, though certainly suggestive, are vague and elusive.

In his recent book (1981, pp. 33f.), Putnam expresses the significanceof the result slightly differently. There he characterizes the argument assupporting Quine's observation (1969) that fixing the set of true obser-vation sentences together with the set of accepted theoretical sentenceswill not determine the reference of the theoretical vocabulary. Putnamshows (1981, Appendix) that the indeterminacy affects the reference oftheT-vocabularyn every possible world, andin thisrespectextendsQuine'sobservation. The formal connection with Newman's argument is this.Newman shows: fixing the domain and models up to isomorphism doesnot fix the intended reference of the T-vocabulary. Putnam shows: fixingthe domain and models up to elementary equivalence does not fix theintended reference of the T-vocabulary. (Thus, formally Newman's ar-gument is stronger than Putnam's, since isomorphism strictly impliesequivalence, although we do not wish to lay much stress on this fact.)It may be that Newman's argument contains an important observation

"John Winnie (1967) employs an argument of this form in the proof of his Theorem 1.Winnie's discussion focuses on formal features of the puzzle: that there is not a uniqueinterpretationof the theory and that there is always an arithmetical interpretation.Newmanand (we believe) Putnam emphasize a different point, namely, that mere consistency seemsto be sufficient for truth; that is, since T has an arithmetical interpretation, it has a trueinterpretation n the world (that is, in THE WORLD).'2Theother suggestions may also be found in this chapter, which is a reprint of Putnam's1978 A. P. A. Presidential Address.



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THE ANALYSISOF MA7TER 635about realism in general-not merely about structural realism. The re-cent work of Putnam, to which we have drawn attention, is intended tosuggest that it does. But we have been unable to find or construct a clearstatement of the connection. Certainlythe argument poses difficulties forpartial interpretationaccounts of theories which exactly parallel the dif-ficulties confronted by Russell's theory: both accounts of theoreticalknowledge and reference fail to square with our naive beliefs regardingthe nature of the truth of theoretical claims. The conclusion we have sug-gested is a conservative one: the argumentshows that neither Russell northe neopositivist doctrine of partial interpretationhas gotten the analysisof theoretical knowledge and reference quite right. A satisfactory accountof these notions must do justice to such "obvious facts" as that the world'sstructure s discovered ratherthan stipulated. We are skeptical of our abil-ity to do this on any view of reference and truth that does not take thelanguage of physical theory as the "ultimate parameter" within whichreference is fixed.

IVThe difficulty that Newman raises for Russell's use of the notion of

structurecan be clarified and deepened if we look at an analogous dif-ficulty that arises for Carnapin the Aufbau and then take a brief glanceatWittgenstein's Tractatus. Newman's problemcan be put this way. Rus-sell wishes to turn theoretical terms into variables by Ramsification; ac-cordingly, physics becomes the assertion that there exist properties andrelations having certain logical features, satisfying certain implicit defi-nitions. The problem is thatthis proceduretrivializes physics: it threatensto turn the empirical claims of science into mere mathematical truths.More precisely, if our theory is consistent, and if all its purely obser-vational consequences are true, then the truth of the Ramsey-sentencefollows as a theorem of set theory or second-order logic, provided ourinitialdomain has the right cardinality-if it doesn't, then the consistencyof our theory again implies the existence of a domain that does. Hence,the truthof physical theory reduces to the truthof its observational con-sequences; and as we saw above, Russell's realism collapses into a ver-sion of phenomenalism or strict empiricism after all: all theories with thesame observational consequences will be equally true.

We observed above that Carnap attempts to go Russell one better intheAujbauby turningall scientific termsinto variablesby means of purelystructural efinite descriptions. So Carnap'sprogram s faced with a closelyanalogous difficulty, a difficulty he himself articulateswith characteristichonesty and precision in ??153-55. Up to this point in the Aufbau, allscientific concepts have been introducedby means of "structural"definite



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636 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANdescriptions containing the single non-logical relation R, of recollectedsimilarity: the visual field, for example, is the unique five-dimensional"sense class" based on R. However, the goal of "complete formaliza-tion" converting all definitions into purely structural definite descrip-tions-can only be achieved if the basic relation Rs is itself eliminated(? 153). How is this to be done? Well, Rs also has a logical form or struc-ture: it has a graph F(Rs). For example, Rs is asymmetrical (?108) andthere is one and only one "sense class" based on Rs that is five-dimen-sional (??117-19). To be sure, we can know these formal properties ofRsonly empirically; but once we know them, we can express them in apurely formal schema F(R), whereR is now a relation-variable. The idea,then, is to define Rs as the unique relation with this graph:

Rs= (iR)F(R) (3)However, as Carnap mmediately points out, there is a serious problemwith this procedure. Since F(R) is now a purely formal schema containingno non-logical primitives, the uniqueness claim implicit in (3) will neverbe satisfied. As long as the condition F is consistent, there will alwaysbe infinitely many distinct relations with this graph, even if we confineourselves to relations on the field of Rs:Our procedure is justified only if the new relation extensions are notarbitrary,unconnected pair lists, but if we require of them that theycorrespondto some experienceable, "natural" elations (to give a pre-liminary, vague expression).

If no such requirement s made, then there are certainly other relationextensions [besides Rs itself] for which all constructional formulascan be produced. . . . All we have to do is carry out a one-to-onetransformationof the set of basic elements into itself and determineas the new basic relations those relation extensions whose inventoryis the transformed inventory of the original basic relations. In thiscase, the new relation extensions have the same structure as the orig-inal ones (they are "isomorphic"). (? 154)The analogy with the Newman problem is evident.How does Carnap handle the problem? He introduces the notion offoundedness to characterize those relation extensions that are "experi-enceable and 'natural'," and argues that one can view this notion as aprimitive concept of logic:

[Foundedness] does not belong to any definite extralogical object do-main, as all other nonlogical objects do. Our considerations con-cerning the characterizationof the basic relations of a constructionalsystem as founded relation extensions of a certain kind hold for every



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THE ANALYSISOF MAITER 637constructional system of every domain whatever. It is perhaps per-missible, because of this generality, to envisage the concept of found-edness as a concept of logic and to introduce it, since it is undefin-able, as a basic concept of logic . . . let us introduce the class offounded relation extensions as a basic concept of logic (logistic sym-bol: found) without considering the problem as alreadysolved. (? 154)

In other words, Carnap proposes to do precisely what we saw Newmanrecoil from, above: he wants to reckon "importance" (foundedness) as"amongthe prime unanalysablequalitiesof the constituentsof the world"Once this is done, the problemof defining Rsis easily solved. We replace(3) withRs = (iR)(found(R) and F(R)) (4)

(see ? 155). It is clear, however, that we must side with Newman here:the idea that (4) is a purely logical formula is absurd.Actually, it is not entirely obvious why Carnap needs a definition like(3); it is not clear why he needs to make a uniqueness claim. He couldexpress our empirical knowledge, as Russell does, with a purely existen-tial claim3 RF(R), (5)

and (5) fulfills the goal of "complete formalization" just as well as (3):both contain no non-logical vocabulary. Of course, (5) is vulnerable tothe Newman objection in precisely its original form-it again threatensto turnempiricaltruth into mathematical truth-but let us slow down andreflect for a moment. For, when applied to our current physical theories,the assumptions needed to prove (5) turn out to be ratherstrong. In par-ticular, we will need some version of the axiom of infinity and someversion of the power set axiom applied to infinite sets (that is, we willneed the continuum). Now contemporary model theory for second-orderlogic makes both these assumptions, and that is why (5) is a theorem.Suppose, however, that we are transportedback into the 1920s. The linebetween logic and mathematics is much less clear, and it is equally un-clear whether strong existence assumptions like the axiom of infinity andthe power set axiom are to be counted as part of logic or mathematics. 3(In Principia Mathematica, for example, Russell carries along the axiomof infinity as an undischarged hypothesis.) Accordingly, the status of (5)is much less clear-cut.

'3Carnap,for one, seems remarkably sanguine about this. He is apparently willing tocount both Axinf and PowerSet as logical. In ?125, for example, he introduces an n-di-mensional real numberspace with a completely innocent air. Apparently, then, he is will-ing to count virtually all of set theory as logic (see also ? 107). This is why (5) could notpossibly serve as an expression of our empirical knowledge for Carnap.



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638 WILLIAM DEMOPOULOS AND MICHAEL FRIEDMANTo see some of the issues here, consider the strictest and most rigorousversion of logicism devised in this period, namely, Wittgenstein's Trac-

tatus (1921). The Tractatus is clear and emphatic about the axiom ofinfinity and set theory: neither is part of logic (5.535, 6.031). In fact,logic makes no cardinality claims whatsoever: there are no privilegednumbers (5.453). By the same token, however, such cardinality claimsarenot partof mathematics either, for mathematics is a "logical method"involving only the simplest forms of combinatorialequations (6.2-6.241).Although Wittgenstein is not very explicit about the precise scope ofmathematicsin the Tractatus, it is apparently exhausted by a rather smallfragment of elementary arithmetic. On this kind of conception, then, for-mulae like (5) will not be theorems of logic or mathematics; if true at allthey can only be empirically (synthetically) true. Hence, on a Tractarianconception of logic and mathematics, we could perhaps make sense ofCarnap's programof "complete formalization" after all. And, for all weknow, Wittgensteincould have had something very much like this in mind.(See Tractatus5.526-5.5262, for example, where Wittgenstein says thatwe can describe the world completely by means offully generalized prop-ositions. See also 6.343-6.3432.) In any case, whether or not these con-siderations have any connection with Wittgenstein's intentions in theTractatus, they do illustrate some of the intellectual tensions producedby logicism's attempt to account simultaneously for both pure mathe-matics and applied mathematics (mathematical physics). In general, itappearsthat we can account for the distinctive characterof one only atthe expense of the other. This is perhapsthe final lesson of the Newmanproblem.

REFERENCESCarnap, R. (1929), Der logisches Atf4au der Welt. Berlin: Schlachtensee Weltkreis-Ver-lag.Goldfarb, W. (1979), 'Logic in the Twenties: The Nature of the Quantifier", Journal ofSymbolic Logic 44: 351-68.Mach, E. [1897] (1959), Analysis of Sensations and the Relation of the Physical to thePsychological: New York: Dover.Maxwell, G. (1968), "Scientific Methodology and the Causal Theory of Perception", inProblems in the Philosophy of Science, I. Lakatos and A. Musgrave (eds.). Am-sterdam: North-Holland.. (1971), "StructuralRealism and the Meaning of Theoretical Terms", in Analysesof Theories and Methods of Physics and Psychology, Michael Radner and StephenWinokur(eds.). Minnesota Studies in the Philosophy of Science, vol. 8. Minneapolis:University of Minnesota Press.Newman, M. H. A. (1928), "Mr. Russell's Causal Theory of Perception", Mind n.s. 37:137-48.Putnam,H. (1978), Meaning and the Moral Sciences. Boston: Routledge and Kegan Paul.. (1981), Reason, Truthand History. Cambridge: Cambridge University Press.Quine, W. V. 0. (1964), "Implicit Definition Sustained", Journal of Philosophy 61: 71-73.



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THE ANALYSISOF MA7TER 639.(1969), "Ontological Relativity", in Ontological Relativity and Other Essays. NewYork: Columbia University Press.Russell, B. (1914a), Our Knowledge of the External World as a Fieldfor Scientific Methodin Philosophy. London: George Allen & Unwin.. (1914b), "The Relation of Sense-Data to Physics", Scientia.. (1919), Introduction to Mathematical Philosophy. London: George Allen & Un-

win.. (1927), The Analysis of Matter. London: Allen & Unwin.. (1948), Human Knowledge: Its Scope and Limits. London: Allen & Unwin.. (1968), The Autobiography of Bertrand Russell, vol. 2. London: Allen & Unwin.Schlick, M. [1918] (1925), General Theory of Kniowledge.Translated by A. E. Blumbergand H. Feigl. New York: Springer-Verlag.. (1979), "Experience, Cognition and Metaphysics", (1926), and "Lectureson Formand Content", (1932), in Collected Papers, vol. 2. Dordrecht: D. Reidel.Winnie, J. (1967), "The Implicit Definition of Theoretical Terms", British Journalfor thePhilosophy of Science 18: 223-29.(1977), "The Causal Theory of Space-Time", in Foundations of Space-Time The-ories, John Earman et al. (eds.). Minnesota Studies in the Philosophy of Science,vol. 4. Minneapolis: University of Minnesota Press.Wittgenstein, L. (1922), Tractatus Logico-Philosophicus. London: Routledge and KeganPaul.

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