mathematical empiricism. a methodological proposal

Mathematical EmpiricismA Methodological Proposal

Hannes Leitgeb

LMU Munich

June 2016

I will make a proposal for how to do philosophy: mathematical empiricism.

The idea will be to take what was good about Carnapian logical empiricism,and to extend and transform it into a more tolerant and conceptually enrichedlogical-mathematical empiricism 2.0 that also embraces metaphysics.

Plan:

1 Philosophy as Rational Reconstruction

2 Mathematical Empiricism: A Method of Rational Reconstruction

3 A Toy Framework

4 Conclusions: Metaphysics and the Future

Philosophy as Rational Reconstruction

Philosophy as the discipline of rational reconstruction.

I take the term from Rudolf Carnap. (Carnap 1950 spoke of ‘explication’ later.)

The idea:

‘Re-’ means that something is already there upon which philosophersreflect; they look over someone’s shoulder (maybe one’s own), proceed onthe “metalevel”.

‘-construction’ means that they are taking apart whatever they are findingthere: studying, amending, and reassembling it.

‘rational’ means that, when doing so, philosophers are taking a particularnormative stance: they aim to make transparent, systematize, evaluate,guide, correct, and improve whatever(i) is supposedly rational or which ought be so,(ii) for the (ir)rationality of which we can take some responsibility, and(iii) the rationality of which we should therefore investigate and advance.

In a nutshell: to rationally reconstruct means to take something that is more orless rational, study its rational features, and use this to make it more rational.

Philosophy is thus a certain kind of “rational therapy”—Wittgenstein!The “medicine” for the “therapy” involves language.

But, contrary to the later Wittgenstein:

the goal of the “therapy” need not be linguistic: one may also rationallyreconstruct concepts, reasoning, decisions, methods, ideals, institutions,how they all relate to each other,. . .

the “therapy” often also involves: the definition of concepts, the defense ofphilosophical theses, the construction of philosophical theories andmodels, the development of new conceptual frameworks, applying all thatto concrete circumstances, facilitated by logical-mathematical methods.

Mathematical empiricism will be a proposal how all that can be done better(hopefully) by invoking mathematical-empirical conceptual frameworks.

Rational reconstruction is not plain “analysis” in the traditional sense ofordinary language philosophy either.

Example: What do mean by mean by the predicate ‘true’ (as in ‘true sentence’,not in ‘true friend’. . .)?

Famously, Tarski (1935) suggested an answer, and he did so by rationallyreconstructing truth:

Tarski starts by looking at examples and the history of the subject; hedetects patterns: all instances of the truth scheme seem assertable andacceptable, and they capture in some sense the vague “truth ascorrespondence” idea; he points to potential contradictions (Liar!).And then he develops a way of doing better than that.

He shows how we can define truth for a great variety of formalizedfragments of natural or scientific language in a precise framework ofsyntax and higher-order logic, such that the definition is materiallyadequate but no paradoxical claims follow. The standard laws of truth canbe derived, the theory is provably consistent, and it turned out to beenormously fruitful in logic, philosophy of language, and linguistics.




Tarski starts by looking at examples and the history of the subject; hedetects patterns: all instances of the truth scheme seem assertable andacceptable, and they capture in some sense the vague “truth ascorrespondence” idea; he points to potential contradictions (Liar!).

And then he develops a way of doing better than that.





Tarski starts by looking at examples and the history of the subject; hedetects patterns: all instances of the truth scheme seem assertable andacceptable, and they capture in some sense the vague “truth ascorrespondence” idea; he points to potential contradictions (Liar!).And then he develops a way of doing better than that.


Strawson (1963): “typical philosophical problems about the concepts used innon-scientific discourse cannot be solved by laying down the rules of use ofexact and fruitful concepts in science. To do this last is not to solve the typicalphilosophical problem, but to change the subject.”

No. Tarski still deals with truth.

The output of Tarski’s rational reconstruction is sufficiently similar to its input.They are not identical, of course—but that is the point: truth after Tarski isclearer, more exact, free from contradiction, more informative, and more fruitful.

When a predicate or concept or thesis or theory or rule or method is not clearenough for a certain purpose, or not exact enough for some purpose, or both,then we should clarify and precisify them for the sake of these purpose(s).

Often, the purposes of everyday life do not demand particularly high standardsof clarity or exactness, and that is fair enough. In such cases rationalreconstruction is normally not called for.

But for philosophical and scientific purposes the standards are often higher andshould be so. Greater clarity means that one is able to understand better, andto be understood better. Greater exactness means that one’s claims, rules ofuse, and standards of assessment have become more informative, moresystematic, with less being left implicit, such that logical and mathematicalmethods become more easily and widely applicable.

Aiming at clarity and exactness in the long run is part of the project of rationalreconstruction. (Clarity and distinctness in Descartes, Leibniz, Kant.)

At the same time, clarity and exactness constitute just minimal rationalityrequirements on discourse in academic contexts.

E.g., an inferential pattern, or the reasons for an action, or the presuppositionsof a question, might be perfectly clear and exact but wrong, in which case theyought to be rectified.

Or, e.g., a theory of rational belief might be clear, exact, and right, but it maynot be useful enough, too restricted in scope, isolated from nearby theories, orpresented in a way that may be overly complicated or lacking elegance:all of which may constitute reasons for reconstruction.

But the revisions should ultimately be clear and exact again.

Rational reconstruction vs empirical science:

The primary aim of empirical science is an epistemic one: to describe andexplain the empirical world by true (informative) declarative sentences.(It also aims to improve theories/methods in service of the primary aim.)

The primary aim of rational reconstruction is a practical one: makinglanguage, concepts, reasoning, decisions, methods, norms, institutions,. . .more rational. (Whenever assertions are made, one should also aim at thetruth, but that is in service of the primary aim.)

Rational reconstruction vs engineering and computer science:

Engineering and computer science also have practical aims. But insteadof erecting bridges or running computer programs, rational reconstructionhandles rational structures. Instead of physical engineering or softwareengineering, rational reconstruction uses conceptual engineering.

Rational reconstruction vs humanities:

Rational reconstruction is concerned with cultural products and may stillbe seen as a special form of “hermeneutical interpretation”: but one that isconstrained methodically and based upon theoretical and practical reason.

What can be rationally reconstructed?

Argumentation and proof (logic). Meaning, truth, and communication(philosophy of language). Belief, justification, theoretical rationality, andknowledge (epistemology). Scientific concepts, theories, methods, andconfirmation (philosophy of science). Moral action, moral attitudes, andmoral maxims (ethics). Preference, decision-making, and practicalrationality (decision theory). Historical philosophical concepts andpositions (history of philosophy). . . .

Qualifications:

‘Rational’ is to be understood broadly.

Rational Reconstruction vs Endorsement:

One does not need to endorse x when one reconstructs x .

Rational Reconstruction and Bootstrapping:

Sometimes one needs to use x to reconstruct x .

The Non-Uniqueness (and Open-Endedness) of Rational Reconstruction:

Usually there is more than just one rational reconstruction of x .

What cannot be rationally reconstructed:

A piece of pop music. A mountain. A law of nature as a worldly fact.

This leads to a worry concerning metaphysics (of the physical world):

If philosophy is rational reconstruction, and metaphysics is part of philosophy,metaphysics could not be the rational reconstruction of physical entities, or ofthe sum of all physical entities, or the like, at least not in any literal sense.

Metaphysics might well be engaged in rationally reconstructing ourpre-theoretic conception of the physical world, or the conception of the physicalworld according to some scientific theory, or the presuppositions of suchconceptions, or our “models” of physical reality, or general concepts, such asexistence or identity or necessity or counterfactuality. . .

But how does metaphysics differ from logic and philosophy of science then?

On the other hand, if metaphysics were in fact concerned with the physicalentities themselves, and hence would not be rational reconstruction after all,the question would be:

How does it differ from science (whether folk or proper), other than beingpursued on a distinctively high level of generality?

Upshot: the status of metaphysics vis-a-vis rational reconstruction is unclear.

In the following, I want to propose a particular way of doing philosophy asrational reconstruction.

And yet there will be room for metaphysics as a philosophical area of its own.

Mathematical Empiricism: A Method of RationalReconstruction

Mathematical empiricism is a mathematized form of empiricism; and theempiricism is an “enlightened” one.

Mathematical empiricism is mathematical by using mathematical methods:rational reconstruction pushes towards clarity and exactness, andmathematical methods are the canonically clear and exact ones.

Mathematical methods (including logical/computational ones) became availablein philosophy, when mathematics turned into a general science of structure: thelogical structure of language, the algebraic structure of propositions/concepts,the recursive structure of algorithms/rules, the set-theoretic structure ofcollections, the category-theoretic structure of mappings.

Rationality itself, in all its different forms and presuppositions, turned out toinstantiate non-trivial kinds of mathematical structure.

Mathematical empiricism is empiricist by aiming, in the course of rationalreconstruction,

(ontic)

to refer solely to empirical objects (or mathematical ones): objectspostulated to exist by successful empirical sciences or everydayexperience, when necessary after some rational reconstruction;

(epistemic)

to refer solely to sources of epistemic justification used in mathematics,the empirical sciences, personal or social experience (if compatible withempirical science), or conceptual understanding.

The reasons for these “self-restraints” are: rational reconstruction does notrequire anything else, and it should not risk empty reference.

(On the conceptual side, mathematical empiricism will be rather liberal.)

On the ontic side:

Empirical objects in that sense include: past and future events and persons;groups and conflicts; supplies and markets; utterances and texts; feelings andconscious states of self-awareness; neurotransmitters and patterns ofactivation; functions and traits; water and bonding of molecules; forces andelectrons; tables and phone calls. (And mathematical constructions on them.)

“Small” possible worlds will be fine, since they can be constructed frommathematical and empirical objects. But “maximally large or fine-grained”possible worlds, e.g., primitive concrete Lewisian possible worlds, are out.

No ‘observable vs non-observable’ distinction will play any role.

The boundaries of ‘empirical’ will be regarded as just as vaguely determinedand changing in time as the meaning of the term ‘empirical science’ itself.

On the epistemic side:

Mathematical empiricism lacks trust in intuition in the sense of quick orimmediate or commonsensical or seems-so-very-much-to-be-true verdicts.

Lewis: “One comes to philosophy already endowed with a stock of opinions.It is not the business of philosophy either to undermine or to justify thesepreexisting opinions, to any great extent, but only to try to discover ways ofexpanding them into an orderly system.” No.

Mathematical empiricism is happy to invoke thought experiments (Williamson)but recommends their rational reconstruction (e.g., of counterfactuals).

Mathematical empiricism is not committed to philosophy being a priori oranalytic. (But frameworks may reconstruct what is “relatively a priori”.)

The mathematical and the empirical aspect of mathematical empiricism cometogether in conceptual frameworks.

(≈ Carnap’s “construction systems” in the Aufbau, “languages” in LogicalSyntax, “linguistic frameworks” in Empiricism, Semantics and Ontology,. . .)

The mathematical empiricist credo:

You want to rationally reconstruct something? Build a framework.(A mathematical-empirical conceptual framework, that is).

Mathematical structures:

ss

s s��3

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n2

n1

n3n4

Mathematical structures with empirical labels:

ss

s s��3

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n2: Carloman

B FF

n1: Charles

n3: Pepinn4: Charles Martel“Actual”:

ss

s s��3

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n3: Pepin

B FF

n1: Charles

n2: Carlomann4: Charles Martel“Logically possible”

(“not metaphysically possible”):

Mathematical empiricism uses such empirically labeled mathematicalstructures as “mathematical-empirical possibilities” in frameworks:

!

n4!n3!

n2!

n1!n4!

n2!

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n1!

n1!n3!

n2!

n2!

n4!n1!

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n4!

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...!...! ...!

Charles!

Carloman!

Pepin! Charles!!M.!Father'

Father'Father'

Brother'

≅!ACTUAL!

These frameworks rationally reconstruct presuppositions of what is to berationally reconstructed within them:

?

-

-?

presuppositions of

presupposition of

rationally reconstructed by rationally reconstructed by

Conceptual framework

Categories, propositions,. . .

Rational features

Rational/irrational features

Presuppositions of all kinds and senses (e.g. Friedman 2001, Stalnaker 1984)!

Different parts of formal philosophy serve as role models here:

The Carnapian and Montagovian intensional semantics of formal andnatural language.

The Kripkean (and Hintikkean and. . .) possible worlds semantics ofnon-classical logic and of modalities.

The Stalnakerian-Lewisian pragmatics of formal and natural language.

The Non-Statement view of scientific theories (cf. Suppes, and more).

Measurement theory (Krantz, Luce, Suppes, Tversky,. . .).

Probability theory and its applications (e.g. Bayesian epistemology,decision theory).

Game theory and its applications (e.g. Lewis’ in philosophy of language,Skyrms’ in social philosophy).

In my terminology, all of these approaches carry out rational reconstructionssuccessfully, based upon mathematical-empirical conceptual frameworks.

A Toy Framework

Let us construct a framework. The situation is this: we study an inclined plane.The ball bl is located either at pl

1 or pl3; it rolls down and ends up either at pl

2 orat pl

4. This is done thrice. It is left open in the framework where the ball islocated at times t l

1, t l3, t l

5, and where it ends up at times t l2, t l

4, t l6.

pl1

pl3

pl2

pl4

The presuppositions to be rationally reconstructed are these “rules of thegame” and the existence of “worldly” laws governing the ball’s rolling.

Let the mathematical domain D of the framework be the set {1, . . . ,11} ofnatural numbers b = 1, t1 = 2,. . ., t6 = 7, p1 = 8,. . ., p4 = 11.

Let the set W of worlds of the framework be the set of all sextuples

w = 〈D, Indiv ,Time,Space,s,Law〉such that:

– Indiv = {b}, Time = {t1, . . . , t6}, Space = {p1, . . . ,p4}.

– s maps pairs of members of D to members of D, such that:

for all m in Indiv , for all t in Time, s(m, t) ∈ Space.s(m, t1),s(m, t3),s(m, t5) ∈ {p1,p3}.s(m, t2),s(m, t4),s(m, t6) ∈ {p2,p4}.

– Law maps triples of members of D to members of D, such that:

for all m in Indiv , for all t in {t1, t3, t5}, for all p in {p1,p3}, it holds thatLaw(m, t,p) ∈ Space, and

s(b, t +1) = Law(b, t,s(b, t)).

(So the trajectory s of ball b at world w must be in line with Law at w .)

The empirical labels of the framework are given by the (unique) quintuple

label = 〈Dl , Indiv l ,Timel ,Spacel ,sl〉

where:

– Dl = Indiv l ∪Timel ∪Spacel , with Indiv l = {bl}, Timel is the set of the sixpoints of time t l

1,. . ., t l6, Spacel is the set of four positions pl

1, . . . ,pl4.

– sl is the empirical position function for the ball bl at times t l ∈ Timel .

– There is no empirical label for the “worldly” Law components!

DefinitionF = 〈W , label〉 is “the rolling ball framework”.

(For some purposes, a group of transformations needs to be added.And those are just the bare essentials.)

Actuality of worlds can be rationally reconstructed as:

DefinitionFor all worlds w = 〈D, Indiv ,Time,Space,s,Law〉 over F :

w is actual in F if and only if

there is a “labeling function” l : D→ Dl over F that is a structure-preserving mapbetween w and label = 〈Dl , Indiv l ,Timel ,Spacel ,sl〉 in the following sense:

l(b) = bl ;

for each i in {1, . . . ,6}: l(ti) = t li ;

for each i in {1, . . . ,4}: l(pi) = pli ;

for all m ∈ Indiv , for all t ∈ Time: l(s(m, t)) = sl(l(m), l(t)).

Let us assume the empirical trajectory sl of the ball bl is this:it is dropped at pl

1 and rolls to pl2, then it is dropped at pl

3 and rolls to pl4, and

finally it is dropped at pl1 and rolls to pl

2 again.

Thus, the actual worlds in F are of the form

〈. . . ,{〈b, t1,p1〉,〈b, t2,p2〉,〈b, t3,p3〉,〈b, t4,p4〉,〈b, t5,p1〉,〈b, t6,p2〉}︸︷︷︸s

,Law〉

with varying Law components (but all of which are in line with s).

There is more than one actual world here, because we built non-representinglaw components into worlds.

Now we can rationally reconstruct propositions, truth, and concepts overF = 〈W , label〉:

DefinitionX is a proposition over F if and only if X is a subset of W .

X is true at w over F if and only if w is a member of X .

X is true simpliciter over F if and only if there is a w , such that w is actual in Fand X is true at w .

C a binary propositional operator concept over F if and only if C maps pairs ofpropositions over F to propositions over F .

For each w = 〈D, Indiv ,Time,Space,s,Law〉 in W , let <w be the following(uniquely determined) binary relation on W :

For all w ′ in W : w <w w ′ if and only if w ′ , w .

For all w ′ = 〈. . . ,s′,Law ′〉,w ′′ = 〈. . . ,s′′,Law ′′〉 in W : if w ′,w ′′ , w , then

w ′ <w w ′′ if and only if Law ′ = Law and Law ′′ , Law .

Let� be the function that maps propositions X and Y to the proposition

X � Y

= {w ∈W |X = ∅,or there is w ′ in X ∩Y s.t. for all w ′′ ∈ X ∩¬Y : w ′ <w w ′′}

In words: X � Y is the set of worlds w , such that the X -worlds closest to ware Y -worlds.

� is a binary propositional operator concept over F . Following Lewis (1975)we may regard� as rationally reconstructing the counterfactual if-then:

if X had been the case, then Y would have been the case.

Now consider the propositions

A = {w = 〈D, Indiv ,Time,Space,s,Law〉 ∈W | 〈b, t1,p3〉 ∈ s}

and

B = {w = 〈D, Indiv ,Time,Space,s,Law〉 ∈W | 〈b, t2,p4〉 ∈ s}.

A says bl is at position pl3 at time t l

1, B says bl is at position pl4 at time t l

2.

Neither A nor B includes any actual worlds as members, as the ball follows adifferent trajectory (by assumption).

One can show that A� B includes an actual world over F (“intuitively right”),as does A� ¬B (“intuitively wrong”).

Let us now construct a probability measure Pr that rationally reconstructs someagent’s degrees of belief. Say, the agent is unaware as yet of the law of gravityand also where the ball will be dropped, and she assigns greater degrees ofbelief to worlds with “simple” laws than to those with “complex” ones:

“Simple” (time-invariant) laws:– Law1: Law1(b, t1,p1) = p2, Law1(b, t1,p3) = p4, Law1(b, t3,p1) = p2,

Law1(b, t3,p3) = p4, Law1(b, t5,p1) = p2, Law1(b, t5,p3) = p4.– Law2: Law2(b, t1,p1) = p4, Law2(b, t1,p3) = p2, Law2(b, t3,p1) = p4,



Law4(b, t3,p3) = p4, Law4(b, t5,p1) = p4, Law4(b, t5,p3) = p4.

“Complex” (non-time-invariant) laws:– Law5: Law5(b, t1,p1) = p2, Law5(b, t1,p3) = p4, Law5(b, t3,p1) = p2,

Law5(b, t3,p3) = p4, Law5(b, t5,p1) = p2, (!!!) Law5(b, t5,p3) = p2 (!!!)....

If w has a simple Law , let Pr({w}) = 10800 ; if not, let Pr({w}) = 1

800 .

Assume the agent acquires evidence about the first two rolling events,

E = {w = 〈. . . ,s,Law〉 |s(b, t1) = p1,s(b, t2) = p2,s(b, t3) = p3,s(b, t4) = p4},

and she considers law hypotheses such as

L1 = {w = 〈. . . ,s,Law〉 |Law = Law1}, . . . ,

L5 = {w = 〈. . . ,s,Law〉 |Law = Law5}, . . .

Such law hypotheses may be said to be metaphysical propositions, since bothof them say that “worldly” laws are so-and-so (that have not been labeledempirically). They are not merely propositions concerning the trajectory sl .

Pr(E) = 0.0625.

Pr(L1) = Pr(L2) = Pr(L3) = Pr(L4) = 0.1.

Pr(L5) = 0.01.

Pr(L1|E) = 0.4. (Confirmed!) Pr(L2|E) = Pr(L3|E) = Pr(L4|E) = 0.

Pr(L5|E) = 0.04. (Confirmed!)

The evidence (E) favors one metaphysical proposition (L1) over the others!

Pr(A� B) = 0.5.

Pr(A� ¬B) = 0.5.

Pr(A� B |E) = 0.68.

Pr(A� ¬B |E) = 0.32.

Hence, once the evidence E is in, our agent regards A� B as more likelythan A� ¬B.

The next step for our agent might be to assert a hypothesis, that is,

L1,

the most likely law hypothesis. (L1 includes an actual world.)

Let us assume she does even more than that: she accepts L1 and decides tocontinue operating under that presupposition. (E.g., she might think that takingL1 for granted is going to simplify her subsequent work.)

We can rationally reconstruct that new presupposition by a new framework:

DefinitionF ′ = 〈W ∩L1, label〉 is “the rolling ball framework”, now restricted to L1.

Within the boundaries of the framework, L1 cannot be questioned anymore(but relative to a larger framework it can be).

There is exactly one actual world left in W ∩L1.

Summing up: a true metaphysical law hypothesis (L1) and a true metaphysicalcounterfactual (A� B) have become presuppositions, supported by empiricalevidence. If the agent’s beliefs had been different, some other propositionmight have been supported empirically and accepted as presupposition, andthe proposition might well have been true, too.

Thus, here, the metaphysical presuppositions one ends up with depend onone’s inductive presuppositions.

Conclusions: Metaphysics and the Future

We have rationally reconstructed presuppositions, “worldly” laws, worlds,propositions, truth, concepts, counterfactuals, degrees of belief, andinductive learning from evidence in a mathematical-empirical framework.All of mathematical empiricism’s recommendations have been satisfied.Metaphysical concepts are compatible with that: surplus structure!

The Mathematical-Empiricist conception of metaphysics:

Metaphysics is the philosophical discipline that constructs, studies, andapplies mathematical-empirical conceptual frameworks, subject to themetaphysical attitude: it endorsing a framework to be true and useful andrecommending it to be presupposed. The truth aspect of thatpresupposition consists in presupposing that the set of worlds of aframework is true simpliciter—that the empirical world conforms to one ofthe possible worlds of the framework.

Logicians and philosophers of science usually do not endorse theirframeworks. Scientists usually do not rationally (re)construct presuppositions.

When a framework includes an actual world by mathematics alone, thechoice of the framework reduces to a matter of practical rationality(Carnap 1950). If not, metaphysics comes with empirical presuppositionsthat need to answer to practical rationality and epistemic rationality (truth).

There is a great plurality of mathematical-empirical frameworks available,many of which may be useful (for some purpose) and based on a set ofworlds that is true over the framework. (Carnapian Tolerance!)

E.g.: build a Measurement framework, Morning-Star/Evening-Starframework, Newtonian framework, Intuitionistic Framework,. . .

“Deep” metaphysical problems may be dissolved by framework choices;which framework to choose may be a deep question. (E.g.: Mind-body?)

The presuppositions of rational reconstruction may themselves berationally reconstructed in a philosophical framework. Mathematicalempiricism will do so in a mathematical-empirical framework again.

There is philosophical progress without convergence:we improve our rational reconstructions in diverse frameworks.

In our philosophical work, let us neither be poets nor scientists nor politicians.

But let us share the poetic sense of possibility, the scientific aim of truth, andthe political ambition of reform.

mathematical empiricism. a methodological proposal

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