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Mathematical realism is the claim that mathematical objects exist and are capable of being referred to Mathematical platonism is the claim that mathematical objects exist, are capable of being referred to, and are abstract in that they o Are not spatiotemporally located o Are neither physical nor mental o Have no changing intrinsic properties
Various versions of platonism:?Standard or object-based platonism (Quine, Goedel). Structuralism (Resnik). Full-blooded platonism.
Benacerraf's dilemma In his 'Mathematical Truth' puts forward a challenge for any philosophy of mathematics, namely that they should provide an adequate account of both mathematical semantics, and how we can come to know mathematical truths. He presents two conditions:?
Semantic condition: the semantic account given to mathematical discourse ought to be part and parcel of the semantic account given to the wider natural language. The account of meaning, truth and reference given to mathematics must be "adequate to any propositions to which these concepts apply". In particular, mathematical statements and natural-language propositions must be susceptible to uniform logicogrammatical analysis. Hence claims like 'there are at least three large cities older than New York' and 'there are at least three perfect numbers greater than 17' ought both to be analysed as 'there are at least three FGs that bear relation R to A'. Epistemological condition: an explanation of how we know a mathematical proposition p must include a suitable relation between the truth conditions of p and the purported knower's state of belief: it must be possible to "link up what it is for p to be true with my belief that p". For Benacerraf, this requires an Alvin Goldmanstyle causal account.
For the platonist, then, the challenge is to explain how the semantics of numbers, sets, and so on ties up with an epistemology that explains how we can be in the right kind of contact with such objects. On the other hand, the formalist and other 'combinatorial' views must explain how their association of truth with provability (or some other syntactic property) connects the truth of a proposition to the merely syntactic property. Benacerraf takes it we know some mathematical truths. As such, his stance does not affect a nominalist of the Hartry Field-school, who rejects the idea we know any mathematical truths.
Possible platonist responses Platonists have tended to offer two responses to Benacerraf's challenge.
1. Attempt to fill the objectionable gap between human knowers and abstract platonic objects through some kind of 'intuition'. Commonly associated with Kurt Goedel and Penelope Maddy.
2. Reject Benacerraf's challenge outright.
3. Adopt an empiricist attitude with respect to mathematical knowledge, and argue that mathematics is compatible with empiricism. Commonly associated with WVO Quine.
The Goedelian response Goedel's intuitionism Kurt Goedel postulated a kind of 'intuition' of mathematical objects. He wanted to hold that the objects of transfinite set theory exist, but that we do have a perception of the objects of set theory. He took the fact that we feel as though we are manipulating real objects when doing mathematics as evidence of a relationship between us and sets. Goedel drew an analogy with sense perception. For Goedel, both sense perception and 'set perception' allow us to build theories, to expect that future perceptions will agree with them, and to believe that questions not decidable now will be in the future. "Despite their remoteness from sense experience, we do have a perception also of the objects of set theory, as is seen from the fact the axioms force themselves upon as true." Goedel argued that both types of perception represent an aspect of objective reality, and that we therefore have no more reason to be confident in sense perception than in mathematical intuition.
Rejecting Goedel's intuitionism Goedel's account is, if not flawed, at least insufficient. As Benacerraf points out, the analogy is without content. Without an explanation of how this intuition is supposed to work - analogous to the scientific description of sense perception - the account does not fill the gap between knowers and known objects at all. It does not tell us how the purported knower is supposed to be in the relevant kind of contact with mathematical objects. Rather than explaining away the problem, it simply 'makes unpalatably clear what other theories hide', to paraphrase JL Mackie's view on moral intuitionism.
Maddy's more sophisticated approach Penelope Maddy takes Goedel to be on the right lines. Maddy is a set-theoretic realist: accepts the conclusions of the Putnam-Quine indispensability argument and believes in the existence of sets. However, she accepts Benacerraf's critique of Goedel. Her response is to accept Goedel's account of mathematical intuition, and appeal to psychological literature to try to provide an account of how we might be in the right kind of contact with sets. Maddy takes her account to have four attractive features:
1. It allows for a straightforward Tarskian semantics for set-theoretic discourse.
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