Deep Disagreement in Mathematics

Abstract

Disagreements that resist rational resolution, often termed "deep disagreements", have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki's work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. Mathematical RevolutionsPerhaps the earliest "fully modern and explicit statement concerning a revolution in mathematics" was made in 1720 by Bernard de Fontenelle, permanent secretary of the Académie Royale des Sciences (Cohen 1985, 90). In his eulogy for Michel Rolle, Fontenelle characterized the innovations of the infinitesimal calculus as "une révolution bien marquée" (quoted in Cohen 1985, 214). Bernard Cohen identifies many subsequent mathematicians and philosophers who have spoken in terms that imply the occurrence of mathematical revolutions, including