By David E. Starr
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Additional resources for Entity and Existence: An Ontological Investigation of Aristotle and Heidegger
If these are granted, the epistemological challenge is to give the rough outlines of what this explanation could be. This is a hard task: for the nominalist, the claim that despite the lack of causal connections there are reliability links between the beliefs of mathematicians and the entities those beliefs are supposed to be about is as mysterious as if someone claimed to have reliable beliefs about 'the daily happenings in a remote village in Nepal' (Field 1989: 26). The analogy is gripping, but suspicious.
And this argument cannot be rejected unless the nominalist is willing to concede that in this context, mathematical sentences are not to be paraphrased. The obvious retreat is that sentences should only be paraphrased in contexts where such a paraphrase does not defeat the very purpose of their utterance. When the anti-nominalist says that there are abstract entities, the aim of his utterance is to make an assertion that is true just in case there are abstract entities. When the mathematician says that there are prime numbers that are larger than 100, her purpose must be something else.
But the failure of earlier proposals involving lexical decomposition should give us a pause. See Fodor (1998) for an argument that we are unlikely to find such equivalencies. 37 Hodes (1984) argues along a different line that no definite description we can come up with could adequately fix the reference of numerals. Benacerraf (1965) and Putnam (1967) already argued that reference-fixing via descriptions is problematic in mathematics. The idea behind Hodes's argument is that reference-fixing through a description cannot work, because we can systematically reinterpret whatever we can say about the natural numbers in such a way that (i) we preserve the truth-values of all our claims, but (ii) the numerals will now pick out different objects.