The Local Conception of Mathematical Evidence: Proof, Computation, and Logic

Abstract

Abstract The fact that mathematics is ordinarily practised as an autonomous science with its own, peculiar type of evidence constituted mainly by deductive reasoning (proofs and computations) is often taken as evidence that mathematics and science have specifically different evidential supports and specifically different subject matters. I argue against this conclusion by first analysing deductive proofs, and the type of evidence that is usually required for axioms, and claiming that most of the evidence for the most elementary and fundamental parts of mathematics is empirical. I then appeal to the role of computation to argue that non‐deductive inference from empirical premises is part of the contemporary methodology of mathematics, and so some of our proofs turn out not to be purely logical deductions. Finally, I discuss the relation between mathematics and logic and argue against logical realism by denying that statements attributing logical properties or relations are true independently of our holding them to be true, our psychology, our linguistic and inferential conventions, or other facts about human beings. In the end, both mathematics and logic turn out to be a priori only in the sense that some mathematical and logical truths are obtained through deductive proofs, and for pragmatic reasons, are insulated from experience; but neither mathematics nor logic are a priori in the sense of being immune to empirical revision.