Deep Disagreement in Mathematics

TL;DR

This paper explores the concept of deep disagreements in mathematics, linking them to mathematical revolutions, and examines the potential for resolution through virtue-theoretic approaches, using Mochizuki's abc conjecture controversy as a case study.

Contribution

It introduces the idea that deep disagreements can occur in mathematics and connects this to revolutionary changes, proposing a virtue-theoretic framework for understanding their resolution.

Findings

Deep disagreements can exist in mathematics.

Mathematical revolutions are related to deep disagreements.

Virtue-theoretic approaches may help resolve such disagreements.

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.