Descriptivism about Mathematical Language

Descriptivism is a metasemantic theory according to which our expressions get their meanings and referents by being associated with certain descriptions or theories. In this talk, I will develop a descriptivist theory of the meaning and reference of mathematical language and defend it against two of the most serious objections against it, viz., the problem of indeterminacy and the problem of existence. The problem of indeterminacy is that descriptivism about mathematical language seems to entail that the reference of mathematical expressions is radically indeterminate. The problem of existence is that descriptivism about mathematical language seems epistemologically idle, because it seems only capable of explaining our knowledge of conditional mathematical truths of the form ‘if there are sets, then sets are such and such’, without doing anything to explain our knowledge of the relevant mathematical entities. My solutions to both problems stem from an account of what it is to associate a description with some expressions that allows that the descriptions we associate with our mathematical and logical expressions are much richer than they are standardly construed, and that they are “anchored” in ordinary language.

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