Hilbert, Gödel, Instrumentalism and Inductive Reasons

Abstract:

Hilbert's mathematical instrumentalism sees the more theoretical (ideal) parts of mathematics as an instrument for proving truths in the more elementary (real) parts. Gödel's second incompleteness theorem shows that one cannot prove the consistency of ideal mathematics from within real mathematics and is therefore generally thought to destroy Hilbert's instrumentalism. This paper argues that though Hilbert's own instrumentalism may have been destroyed by Gödel's theorem, a broadly Hilbertian instrumentalism has not. By looking at inductive reasons in mathematics, we explain why Hilbertian instrumentalism survives Gödel's theorem. The talk is relatively untechnical and aimed at a general philosophical audience.