Philosophy of Mathematics Structuralism Owen Griffiths oeg21@cam.ac.uk St John s College, Cambridge 17/11/15
Neo-Fregeanism Last week, we considered recent attempts to revive Fregean logicism. Analytic logicists try to show that HP is an analytic truth. But HP is only satisfiable in infinite domains. Can an analytic truth be committing to infinitely many things? Neo-Fregean logicism tries to stipulate HP as an implicit definition of number. But stipulation must be sentential or subsentential. The former fails to fix a reference for the term-forming operator. The latter fails to guarantee that HP is true.
Implicit definition Over the past few weeks, we ve been discussing implicit definition: a functional characterisation of some notion in terms of its relation to other notions. A functionalist about the mental may define pain in the following way: Pain is caused by stubbing your toe and pain makes you wince and pain is unpleasant and... x is caused by stubbing your toe and x makes you wince and x is unpleasant and... Pain is then anything that satisfies this open sentence. But it may be multiply realizable, e.g. the physical state that satisfies the open sentence may be c-fibres in humans but d-fibres in akarpis.
Geometry We may take a similar approach in the philosophy of mathematics: Given any two points, exactly one line can be drawn which passes through them. Any line can be indefinitely extended. Given any two points, exactly one line can be drawn which passes through them AND Any line can be indefinitely extended AND... Given any two points, exactly one x can be drawn which passes through them AND Any x can be indefinitely extended AND... We may take this open sentence to implicitly define line. As with the mental case, there could be many entities that satisfy the open sentence.
Hilbert Every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and the basic elements can be thought of in any way one likes. If in speaking of my points, I think of some system of things, e.g. the system love, law, chimney-sweep... and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras theorem, are also valid for these things... any theory can always be applied to infinitely many systems of basic elements. (see Frege, Philosophical and mathematical correspondence) Many systems of things may satisfy mathematical axioms. So mathematics isn t really about any of those things in particular, but the structure that those things instantiate.
Structuralism Further evidence for this structuralist line of thought is the existence of relative consistency proofs. A theory Θ 1 is proved consistent relative to Θ 2 by interpreting the nonlogical primitives of the former in the language of the latter and showing that the results are theorems of the latter. Hilbert initiated this method of consistency proof, which is now common practice. In particular, it is well known that virtually all of mathematics can be modelled in set theory. This all suggests a form of structuralism.
What numbers could not be Many philosophers of mathematics have identified the natural numbers with set-theoretic objects. Even Frege, who wrote before the development of set theory, identified numbers with the extensions of second-level concepts, which are set-like objects. But, as Paul Benacerraf points out in his famous 1965 paper What Numbers Could Not Be, this identification can be made in infinitely many ways. Without loss of generality, let s think about 2 ways that we may go about making this identification. We ll consider 2 ways of identifying natural numbers with pure sets (sets with no urelements).
Two identifications I II 0 = 1 = { } 2 = {{ }} 3 = {{{ }}}. 0 = 1 = { } 2 = {, { }} 3 = {, { }, {.{ }}}.
Identification problem Which of I and II is to be preferred? It seems that there is nothing to choose between them. We can define successor, addition and multiplication on I and II to make them isomorphic, so they make precisely the same sentences true. And yet they cannot both be true collections of identity claims. By I, 2 = {{ }}. By II, 2 = {, { }}. So, by the transitivity of identity, {{ }} = {, { }}. But that s false. The correct conclusion seems to be that neither I nor II is correct. Rather, the numbers are to be identified structurally: 2 picks out a certain position in a structure, which can be instantiated by {{ }} and can be instantiated by {, { }}.
Systems Following the Hilbert quote, a system is a collection of objects concrete or abstract with relations between them. We could think of a family as a system of objects with certain relations holding between them. A mathematical number system is a countably infinite collection of objects with a designated initial object and a successor relation that satisifes the usual axioms. The Arabic numerals are such a system, as are an infinite sequence of temporal points. A Euclidean system is three collections of objects, one called points, one lines and one called planes, with relations between them that ensure they satisfy the Euclidean axioms.
Structures A structure is the abstract form of a system. It abstracts away from all properties of the system other than the structural relations that hold between the objects. The Arabic numerals and the infinite sequence of temporal points share the natural number structure.
Varieties of structuralism All structuralists contend that mathematics is the science of structure, but they differ as to the nature of these structures. The main distinction is between platonist and nominalist structuralists. Platonist structuralists, most notably Stewart Shapiro and Michael Resnik, hold that the structures exist. Nominalist structuralists, most notably Geoffrey Hellman and Charles Chihara, hold that the structures can be eliminated. Let s take them in turn.
Platonic and aristotelian universals In the literature on universals, authors disagree about the spatio-temporal location of universals. Platonic universals are not spatio-temporally located: they are abstract entities. Aristotelian universals are spatio-temporally located: they are located in their instances. The defender of aristotelian universals may have a metaphysical advantage. They need not accept anything abstract over and above the concrete instances. But they also have to say that one universal can be wholly present at different places at the same time, and that two universals can occupy the same place at the same time.
Ante rem and in re structuralists Platonist structuralists generally come in two sorts: ante rem and in re. The former contend that structures are platonic universals, the latter that they are aristotelian universals. In re structuralists have to say that the structures are ontologically dependent on their systems: destroy the systems and you destroy the structures. But while this may have some appeal in the non-mathematical case, it seems problematic here: if mathematical truths are necessary, then we want to say that they hold in all worlds, even those without any instantiating systems. We ll proceed with ante rem structuralism as our paradigm example of Platonist structuralism.
Advantages of Platonist structuralism The Platonist structuralist can read mathematical sentences at face value. The sentence there are at least 2 primes less than 7 appears to have the same form as there are at least 2 cities smaller than Paris. They can also accommodate the Benacerraf s insights, and have a straightforward explanation for e.g. relative consistency proofs.
Epistemology Like most Platonist positions, Platonist structuralism has epistemological problems If the objects are abstract, how do we manage to access them? Shapiro has the most sophisticated epistemological story for Platonist structuralism. Let s look at his explanation of how we gain access to ante rem structures. 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 systems consisting of just 2 objects, and so on. By attending to such tokens, we can apprehend the types. Projection We can gain knowledge of larger structures, e.g. the 10,000-pattern, by recognising the order of smaller abstractions, and projecting further. Description Projection will only get us as far as denumerable infinities, however. To deal with larger structures, we can use our powers of description. If we can know the coherence of such descriptions, we can gain knowledge of the structures described.
Problems for projection It seems plausible that we are capable of something like abstraction. But Fraser MacBride (1997) casts doubt on whether projection and description can deliver knowledge of structures. It seems plausible that we do in fact project our abstractions about smaller patterns to larger cases, e.g. we discern the 2-, 3- and 4-patterns and believe that this progression will continue indefinitely. But such belief does not suffice for knowledge without (at least) some sort of warrant or justification.
Problems for projection We need more than the descriptive claim that we do project. We need the normative claim that we are right to project. But how can the normative claim be supported? It can t be that the projection is supported by best mathematical theories, on pain of circularity. And if we can t appeal to that, we face rule-following worries: how do we know that we should not be following some less standard rule?
Problems for description Here, Shapiro echoes Hilbert: if we establish the consistency of an axiomatization, we can establish the existence of the structure described. Shapiro doesn t use Hilbert s notion of consistency but the wider notion of coherence. Coherence gets explicated in terms of set-theoretic satisfiability.
Problems for description If we know that an axiomatization is coherent, and we know that it is categorical, then we know that the structure corresponds to the descriprion. But how do we come to know that a description is coherent and categorical? These notions get spelt out in set-theoretic terms, and the question of their justification gets pushed back to mathematical theory. Again, these are the very things we re out to justify.
Nominalist structuralism Given the problems faced by Platonist structuralists, some have endorsed a form of nominalist structuralism. We ll focus on Hellman s modal structuralism. The central idea is that we can enjoy the benefits of Platonist structuralism but without the ontological commitment. Rather than thinking about actual structures, we need only be concerned with possible structures.
Modal structuralism Consider some arithmetic sentence S. Hellman would paraphrase S: X (X is an ω-sequence S holds in X ) In words: necessarily, if there is an ω-sequence, then S is true in that sequence. Hellman calls this the hypothetical component of his view.
Vacuity This paraphrase faces an immediate vacuity problem. The nominalist structuralist does not want to be committed to mathematical objects, since they are abstract and so present e.g. access problems. These are the very things that caused issues for Shapiro. But, for all we know, the universe may not contain enough physical objects for all of mathematics. E.g. if the universe contains only 10 100,000 objects, then there won t be any ω-sequences.
Vacuity Even if we are convinced that the universe contains enough objects for arithmetic, there may be some limit to the number of objects it contains, and there will be branches of mathematics that require more. If there are not enough objects, then the antecedent of the hypothetical component will be false and so the whole conditional will be vacuously true. For this reason, Hellman introduces the categorical component: X (X is an ω-sequence) In words: it is possible that an ω-sequence exists.
The categorical component The categorical component guarantees that, if the hypothetical component is true, it is non-vacuously true. There s one wrinkle. Both components are expressed in second-order S5. But it had better be S5 without the Barcan Formula: BF xfx x Fx With BF, the categorical component entails: X (X is an ω-sequence) which Hellman obviously doesn t want.
Modality Overall, then, the thought is that mathematical sentences are elliptical for longer sentences in second-order S5 (without BF). They have a hypothetical component to achieve this, and a categorical component to avoid vacuity. The account therefore avoids Shapiro s metaphysical burden, but does so at the cost of respecting mathematical sentences at face value.
Modality Further, the nature of the invoked modality must also be explained. Is it metaphysical possibility, or logical, or mathematical? Hellman says logical, but now there s a problem. Logical modality usually gets explicated in set-theoretic terms, but that is to let abstract objects back in. Instead, he refuses to explicate the logical modality at all, but leave it primitive.
Ontology vs ideology Quine says that a theory carries an ontology and an ideology The ontology consists of the entities which the theory says exist. The ideology consists of the ideas expressed within the theory using predicates, operators, etc. Ontology is measured by the number of entities postulated by a theory. Ideology is measured by the number of primitives. It is often thought that ideological economy has epistemological benefits. A theory with fewer primitives is likely to be more unified, which may aid understanding.
Ontology vs ideology So Hellman has increased ideology in order to reduce ontology. Is this satisfactory? We ll discuss this further next week in light of Hartry Field s nominalism.