16 07 2005

In chapter I.2 of Naturalism in Mathematics, Penelope Maddy suggests that the role of set theory in mathematics is not ontological, metaphysical, or epistemic. Instead, she suggests that it unifies disparate areas of mathematics (allowing us to see that Zorn’s Lemma in algebra and the Axiom of Choice in analysis and the Well-Ordering Principle of set theory are all in fact the same thing), helps coordinate division of labor, and eases explanation of intuitive results like the Jordan Curve Theorem. To play these roles, the axioms just need to be fairly certainly consistent and need to be sufficient to provide surrogates for the rest of mathematics. There is no need for them to be seen as an ontological analysis of what things exist (ie, just sets), a metaphysical analysis of mathematical entities (ie, natural numbers are von Neumann ordinals, or Fregean extensions), or epistemologically more certain than the intuitive premises of any given area of mathematics.

This all seems right to me, but I think this doesn’t obviate the need for some ontological, metaphysical, and epistemological foundations for mathematics. While Benacerraf has shown (in “What Numbers Could Not Be”) that set theory at least hasn’t yet answered the metaphysical questions about natural numbers, I don’t think this means that no solution is necessary. In fact, Benacerraf is often taken as the start of an argument either for structuralism about mathematical entities (which is still somewhat mysterious, as far as I can tell) or an argument that numbers are their own sort of entity apart from sets. Similarly, though the axioms themselves don’t need to be certain, it seems that they can acquire inductive justification from their explanation of various observed regularities throughout science and mathematics, and can confirm these higher level results by showing that they are part of a unified theory.

Of course, set theory as currently practiced may not end up being the right way to solve these problems. (I’m sympathetic to a Fieldian sort of fictionalism myself.) But it can still be socially useful for mathematics.