Enhanced If-thenism

Abstract

This essay attempts to revive if-thenism (or deductivism), the view that contemporary pure mathematics is the study of what follows from what. Historical antecedents are traced in Russell and Putnam, some traditional objections dispatched, and an account of applications different from Putnam’s deployed to undercut the indispensability arguments. The central challenge is that mathematics is not indifferent to what goes in the ‘if’ part of the if-then; this is where Simple If-thenism is enhanced to include an account of the rationality of one choice over another. The connection between the ‘if’ and the ‘then’ is accounted for in terms of the second-philosophical view of logic in Essays ##8 and 9, and arithmetic is treated as in Essays ##9 and 10. The final section deals with matters of meta-mathematics, especially Gödel’s second incompleteness theorem.