How the Standard View of Rigor and the Standard Practice of Mathematics Clash
Mathematical proofs are rigorous – it’s part of what distinguishes proofs from other argument types. But the quality of rigor, relatively simple for the trained mathematician to spot, is difficult to explicate. The most common view, often referred to as the standard view of rigor, is that “a mathematical proof is rigorous iff it can be converted into a formal derivation” (Burgess & De Toffoli 2022). Each proponent of the standard view interprets “conversion” differently. For some, like Hamami (2022), conversion means algorithmic translation while others, like Burgess (2015), interpret it as just revealing enough steps of the formal derivation.
In this talk, I aim to present an overarching concern for the standard view. I’ll argue that no extant version of the standard view makes sense of how mathematicians make rigor judgments. Both Hamami (2022) and Tatton-Brown (2021) have both attempted to account for mathematicians’ rigor judgments using the standard view. I’ll argue that both still fail to adequately account for mathematical practice by positing that mathematicians engage in either algorithmic proof search and/or extensive training in formal rigor.
We seem to be left with two options: continue trying to amend the standard view or introduce a new account of rigor which is practice-focused. I’ll argue that issues with the two accounts are general issues that will likely occur for future formulations of the standard view. Thus, we should aim to introduce a new account of informal, mathematical rigor. I’ll close by sketching a new account of rigor related to the audience-based view of proof introduced in Ashton (2021).