Groundwork for a Fallibilist Account of Mathematics

Abstract

According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.

Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture

“…Mathematical justification should therefore not be connected with proofs, but with what in previous work I called simil-proofs [DT21]. These are arguments that look like proofs to the relevant subjects but may contain significant errors.…”

Transferable and Fixable Proofs

“…As should be clear from the above discussion, I use the term "proof" throughout the paper to denote a proof attempt or "simil-proof" (De Toffoli 2021Toffoli , 2022. 5 A proof in this sense may contain errors, whether fixable or not.…”