Mathematical Objectivity and Mathematical Objects
Field (Hartry)
Source: Laurence & Macdonald - Contemporary Readings in the Foundations of Metaphysics
Paper - Abstract

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Philosophers Index Abstract

    The most objectionable feature of traditional Platonism isn't its assumption of mathematical objects but its assumption that mathematics has a certain kind of objectivity. The objectionable kind of objectivity is the view that any sentence of mathematics has a determinate truth value even if it is undecidable by axioms we accept or are disposed to accept. "Nominalism" or "fictionalism1" is simply one form that a properly antiobjectivist philosophy of mathematics can take. The paper also argues that while structuralism contains important insights, the version due to Resnik and Shapiro faces a serious problem in dealing with structures that have symmetries.

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