Mathematical Naturalism and Revisionism About Mathematics

Anti-revisionism, according to which any revision to a practice should come autonomously from within the practice itself, is a key component of all naturalistic approaches to science and mathematics, yet it is rarely discussed in detail. In this paper, I focus on naturalistic anti-revisionism about mathematics. After clarifying the reasons why naturalists should be anti-revisionists, I examine two different ways of spelling out a criterion for mathematical anti-revisionism. The first criterion maintains that we should not revise mathematical practice on external (i.e. extra-mathematical) grounds. On the basis of case studies, I argue that this criterion is not supported by a sufficiently sharp distinction between revisions prescribed on internal vs. external grounds. The second criterion maintains that we should not countenance philosophical accounts of mathematics which involve abandoning accepted theorems. I argue that while in line with naturalistic tenets, this criterion does not fully capture what is important to the mathematical naturalist. This leads to the formulation of a third criterion, which adds to the second criterion the clause that we should not abandon established mathematical methodology.