Confirming Mathematical Conjectures by Analogy

TL;DR

This paper explores how analogy can be used to confirm mathematical conjectures, distinguishing between incremental and non-incremental forms, and proposing a hybrid Bayesian framework for understanding this process in pure mathematics.

Contribution

It introduces a novel hybrid framework combining Bayesian confirmation theory with non-incremental analogy-based reasoning in pure mathematics.

Findings

Distinction between incremental and non-incremental confirmation by analogy

Bayesian account of incremental confirmation

Role of analogy in informing prior credences without new evidence

Abstract

Analogy has received attention as a form of inductive reasoning in the empirical sciences. However, its role in pure mathematics has received less consideration. This paper provides an account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting 'hybrid' framework captures many important aspects of the use of analogical inference in the realm of pure mathematics.