Reasoning by Analogy in Mathematical Practice.

This article develops a descriptive theory of analogical reasoning in mathematics, identifying conditions under which analogies provide genuine inductive support to mathematical conjectures beyond mere heuristic suggestion. Building on Mary Hesse's framework for analogical reasoning in empirical sciences, the authors defend three necessary and jointly sufficient conditions for strong analogical inference in mathematics: Materiality (similarities must be pre-theoretic and significant independently of the argument), Relevance (the source's mathematical connections must be robust and plausibly extend to the target), and No-Essential-Difference (no known essential differences should undermine the analogy). Through detailed case studies—including Euler's characteristic, the Basel problem, and analogies from finite to infinite domains—the paper argues that these conditions better capture mathematicians' intuitive distinctions between weak heuristic analogies ("hookings") and stronger evidential analogies ("relay-results") than alternative accounts, such as Paul Bartha's articulation model. The authors emphasize the context-sensitivity of these conditions and present their account as a unifying descriptive framework that clarifies expert judgments on analogical reasoning in diverse mathematical areas without addressing the ultimate justification of such inferences.