Embed Size (px)
Citation preview
Neo-Fregeanism
I Last week, we considered recent attempts to revive Fregeanlogicism.
I Analytic logicists try to show that HP is an analytic truth.
I But HP is only satisfiable in infinite domains. Can an analytictruth be committing to infinitely many things?
I Neo-Fregean logicism tries to stipulate HP as an implicitdefinition of number.
I But stipulation must be sentential or subsentential. Theformer fails to fix a reference for the term-forming operator.The latter fails to guarantee that HP is true.
Implicit definition
I Over the past few weeks, we’ve been discussing implicitdefinition: a functional characterisation of some notion interms of its relation to other notions.
I A functionalist about the mental may define ‘pain’ in thefollowing way:
I Pain is caused by stubbing your toe and pain makes you winceand pain is unpleasant and ...
I x is caused by stubbing your toe and x makes you wince and xis unpleasant and ...
I ‘Pain’ is then anything that satisfies this open sentence.
I But it may be multiply realizable, e.g. the physical state thatsatisfies the open sentence may be c-fibres in humans butd-fibres in akarpis.
Geometry
I We may take a similar approach in the philosophy ofmathematics:
I Given any two points, exactly one line can be drawn whichpasses through them.
I Any line can be indefinitely extended.I Given any two points, exactly one line can be drawn which
passes through them AND Any line can be indefinitelyextended AND ...
I Given any two points, exactly one x can be drawn which passesthrough them AND Any x can be indefinitely extended AND ...
I We may take this open sentence to implicitly define ‘line’. Aswith the mental case, there could be many entities that satisfythe open sentence.
Hilbert
I Every theory is only a scaffolding or schema ofconcepts together with their necessary relations toone another, and the basic elements can be thoughtof in any way one likes. If in speaking of my points,I think of some system of things, e.g. the systemlove, law, chimney-sweep ... and then assume all myaxioms as relations between these things, then mypropositions, e.g., Pythagoras’ theorem, are alsovalid for these things ... any theory can always beapplied to infinitely many systems of basic elements.(see Frege, Philosophical and mathematicalcorrespondence)
I Many systems of things may satisfy mathematical axioms.
I So mathematics isn’t really about any of those things inparticular, but the structure that those things instantiate.
Structuralism
I Further evidence for this structuralist line of thought is theexistence of relative consistency proofs.
I A theory Θ1 is proved consistent relative to Θ2 by interpretingthe nonlogical primitives of the former in the language of thelatter and showing that the results are theorems of the latter.
I Hilbert initiated this method of consistency proof, which isnow common practice.
I In particular, it is well known that virtually all of mathematicscan be modelled in set theory.
I This all suggests a form of structuralism.
What numbers could not be
I Many philosophers of mathematics have identified the naturalnumbers with set-theoretic objects.
I Even Frege, who wrote before the development of set theory,identified numbers with the extensions of second-levelconcepts, which are set-like objects.
I But, as Paul Benacerraf points out in his famous 1965 paper‘What Numbers Could Not Be’, this identification can bemade in infinitely many ways.
I Without loss of generality, let’s think about 2 ways that wemay go about making this identification.
I We’ll consider 2 ways of identifying natural numbers with puresets (sets with no urelements).
Two identifications
I 0 = ∅1 = {∅}2 = {{∅}}3 = {{{∅}}}...
II 0 = ∅1 = {∅}2 = {∅, {∅}}3 = {∅, {∅}, {∅.{∅}}}...
Identification problem
I Which of I and II is to be preferred?
I It seems that there is nothing to choose between them. Wecan define successor, addition and multiplication on I and II tomake them isomorphic, so they make precisely the samesentences true.
I And yet they cannot both be true collections of identityclaims. By I, 2 = {{∅}}. By II, 2 = {∅, {∅}}. So, by thetransitivity of identity, {{∅}} = {∅, {∅}}. But that’s false.
I The correct conclusion seems to be that neither I nor II iscorrect.
I Rather, the numbers are to be identified structurally: ‘2’ picksout a certain position in a structure, which can beinstantiated by {{∅}} and can be instantiated by {∅, {∅}}.
Systems
I Following the Hilbert quote, a system is a collection of objects– concrete or abstract – with relations between them.
I We could think of a family as a system of objects with certainrelations holding between them.
I A mathematical number system is a countably infinitecollection of objects with a designated initial object and asuccessor relation that satisifes the usual axioms.
I The Arabic numerals are such a system, as are an infinitesequence of temporal points.
I A Euclidean system is three collections of objects, one called‘points’, one ‘lines’ and one called ‘planes’, with relationsbetween them that ensure they satisfy the Euclidean axioms.
Structures
I A structure is the abstract form of a system.
I It abstracts away from all properties of the system other thanthe structural relations that hold between the objects.
I The Arabic numerals and the infinite sequence of temporalpoints share the natural number structure.
Varieties of structuralism
I All structuralists contend that mathematics is the science ofstructure, but they differ as to the nature of these structures.
I The main distinction is between platonist and nominaliststructuralists.
I Platonist structuralists, most notably Stewart Shapiro andMichael Resnik, hold that the structures exist.
I Nominalist structuralists, most notably Geoffrey Hellman andCharles Chihara, hold that the structures can be eliminated.
I Let’s take them in turn.
Platonic and aristotelian universals
I In the literature on universals, authors disagree about thespatio-temporal location of universals.
I Platonic universals are not spatio-temporally located: they areabstract entities.
I Aristotelian universals are spatio-temporally located: they arelocated in their instances.
I The defender of aristotelian universals may have ametaphysical advantage.
I They need not accept anything abstract over and above theconcrete instances.
I But they also have to say that one universal can be whollypresent at different places at the same time, and that twouniversals can occupy the same place at the same time.
Ante rem and in re structuralists
I Platonist structuralists generally come in two sorts: ante remand in re.
I The former contend that structures are platonic universals,the latter that they are aristotelian universals.
I In re structuralists have to say that the structures areontologically dependent on their systems: destroy the systemsand you destroy the structures.
I But while this may have some appeal in the non-mathematicalcase, it seems problematic here: if mathematical truths arenecessary, then we want to say that they hold in all worlds,even those without any instantiating systems.
I We’ll proceed with ante rem structuralism as our paradigmexample of Platonist structuralism.
Advantages of Platonist structuralism
I The Platonist structuralist can read mathematical sentencesat face value.
I The sentence ‘there are at least 2 primes less than 7’ appearsto have the same form as ‘there are at least 2 cities smallerthan Paris’.
I They can also accommodate the Benacerraf’s insights, andhave a straightforward explanation for e.g. relative consistencyproofs.
Epistemology
I Like most Platonist positions, Platonist structuralism hasepistemological problems
I If the objects are abstract, how do we manage to access them?
I Shapiro has the most sophisticated epistemological story forPlatonist structuralism.
I Let’s look at his explanation of how we gain access to anterem structures.
I Notice the Kantian echoes in Shapiro’s story.
Shapiro’s epistemology
Abstraction We are able to discern small, instantiated patterns.E.g. we can recognise the 2-pattern in all systemsconsisting of just 2 objects, and so on. By attendingto such tokens, we can apprehend the types.
Projection We can gain knowledge of larger structures, e.g. the10,000-pattern, by recognising the order of smallerabstractions, and projecting further.
Description Projection will only get us as far as denumerableinfinities, however. To deal with larger structures, wecan use our powers of description. If we can knowthe coherence of such descriptions, we can gainknowledge of the structures described.
Problems for projection
I It seems plausible that we are capable of something likeabstraction.
I But Fraser MacBride (1997) casts doubt on whetherprojection and description can deliver knowledge of structures.
I It seems plausible that we do in fact project our abstractionsabout smaller patterns to larger cases, e.g. we discern the 2-,3- and 4-patterns and believe that this progression willcontinue indefinitely.
I But such belief does not suffice for knowledge without (atleast) some sort of warrant or justification.
Problems for projection
I We need more than the descriptive claim that we do project.
I We need the normative claim that we are right to project.
I But how can the normative claim be supported?
I It can’t be that the projection is supported by bestmathematical theories, on pain of circularity.
I And if we can’t appeal to that, we face rule-following worries:how do we know that we should not be following some lessstandard rule?
Problems for description
I Here, Shapiro echoes Hilbert: if we establish the consistencyof an axiomatization, we can establish the existence of thestructure described.
I Shapiro doesn’t use Hilbert’s notion of consistency but thewider notion of coherence.
I ‘Coherence’ gets explicated in terms of set-theoreticsatisfiability.
Problems for description
I If we know that an axiomatization is coherent, and we knowthat it is categorical, then we know that the structurecorresponds to the descriprion.
I But how do we come to know that a description is coherentand categorical?
I These notions get spelt out in set-theoretic terms, and thequestion of their justification gets pushed back tomathematical theory.
I Again, these are the very things we’re out to justify.
Nominalist structuralism
I Given the problems faced by Platonist structuralists, somehave endorsed a form of nominalist structuralism.
I We’ll focus on Hellman’s modal structuralism.
I The central idea is that we can enjoy the benefits of Platoniststructuralism but without the ontological commitment.
I Rather than thinking about actual structures, we need only beconcerned with possible structures.
Modal structuralism
I Consider some arithmetic sentence S . Hellman wouldparaphrase S :
�∀X (X is an ω-sequence → S holds in X )
I In words: necessarily, if there is an ω-sequence, then S is truein that sequence.
I Hellman calls this the hypothetical component of his view.
Vacuity
I This paraphrase faces an immediate vacuity problem.
I The nominalist structuralist does not want to be committedto mathematical objects, since they are abstract and sopresent e.g. access problems.
I These are the very things that caused issues for Shapiro.
I But, for all we know, the universe may not contain enoughphysical objects for all of mathematics.
I E.g. if the universe contains only 10100,000 objects, then therewon’t be any ω-sequences.
Vacuity
I Even if we are convinced that the universe contains enoughobjects for arithmetic, there may be some limit to the numberof objects it contains, and there will be branches ofmathematics that require more.
I If there are not enough objects, then the antecedent of thehypothetical component will be false and so the wholeconditional will be vacuously true.
I For this reason, Hellman introduces the categoricalcomponent:
♦∃X (X is an ω-sequence)
I In words: it is possible that an ω-sequence exists.
The categorical component
I The categorical component guarantees that, if thehypothetical component is true, it is non-vacuously true.
I There’s one wrinkle. Both components are expressed insecond-order S5. But it had better be S5 without the BarcanFormula:
BF ♦∃xFx → ∃x♦Fx
I With BF, the categorical component entails:
∃X♦(X is an ω-sequence)
which Hellman obviously doesn’t want.
Modality
I Overall, then, the thought is that mathematical sentences areelliptical for longer sentences in second-order S5 (without BF).
I They have a hypothetical component to achieve this, and acategorical component to avoid vacuity.
I The account therefore avoids Shapiro’s metaphysical burden,but does so at the cost of respecting mathematical sentencesat face value.
Modality
I Further, the nature of the invoked modality must also beexplained.
I Is it metaphysical possibility, or logical, or mathematical?
I Hellman says logical, but now there’s a problem.
I Logical modality usually gets explicated in set-theoretic terms,but that is to let abstract objects back in.
I Instead, he refuses to explicate the logical modality at all, butleave it primitive.
Ontology vs ideology
I Quine says that a theory carries an ontology and an ideology
I The ontology consists of the entities which the theory saysexist.
I The ideology consists of the ideas expressed within the theoryusing predicates, operators, etc.
I Ontology is measured by the number of entities postulated bya theory.
I Ideology is measured by the number of primitives.
I It is often thought that ideological economy hasepistemological benefits. A theory with fewer primitives islikely to be more unified, which may aid understanding.
Ontology vs ideology
I So Hellman has increased ideology in order to reduce ontology.
I Is this satisfactory?
I We’ll discuss this further next week in light of Hartry Field’snominalism.
