Reasoning by Analogy in Mathematical Practice

TL;DR

This paper develops a descriptive theory of analogical reasoning in mathematics, distinguishing between superficial and deep analogies, and provides criteria for when an analogy offers genuine inductive support for conjectures.

Contribution

It generalizes existing criteria for analogical reasoning, offers a better account than Bartha (2009), and provides new insights into extending finite properties to the infinite case.

Findings

Deep analogies can provide genuine inductive support in mathematics.

The proposed criteria improve understanding of analogical reasoning over previous models.

The account explains how finite properties extend to infinite cases in mathematical practice.

Abstract

The testimony and practice of notable mathematicians indicate that there is an important phenomenological and epistemological difference between superficial and deep analogies in mathematics. In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of 'suggesting' a conjecture in the psychological sense). The proposed conditions generalize the criteria put forward by Hesse (1963) in her influential work on analogical reasoning in the empirical sciences. By reference to several case-studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha (2009). Moreover, our proposal offers novel insights into the practice of extending to the infinite case mathematical properties known to hold in finite domains.