The Context of Proving

TLDR

Discussions of mathematical problem‑solving have focused on how known proofs are found, but this perspective suffers from limited insight into the discovery process, reliance on formulaic strategies, and a separation from the actual circumstances in which reasoning occurs. The paper aims to provide a detailed descriptive account of the process of finding a specific proof, highlighting shifting perspectives, wrong paths, mistakes, and outright errors. It does so by describing the work of discovering a proof, tracing the evolution of perspectives, missteps, and errors. The study finds that visual sketches or written drafts can reveal unexpected avenues for proof discovery, supporting the claim that the methods provers use shape the context for ongoing work and uncover the reasoning needed for a given proof.

Abstract

Discussions of mathematical problem-solving and heuristic reasoning have typically examined how proofs that are already known might be found. This approach has at least three problems: first, provers engaged in discovering proofs for themselves cannot have this perspective; second, if a proof is difficult, formulaic strategies quickly run out; third, beginning with a proof already in-hand separates reasoning about a proof from the actual circumstances in which such reasoning occurs. As an alternative approach to the study of mathematical reasoning, this paper presents a detailed descriptive account of the work of finding a specific proof, including the shifting of perspectives, the wrong paths, the mistakes and the outright errors. Even the appearance of a sketched diagram or of a course of mathematical writing can suggest unanticipated possibilities for finding a proof. This material is used to illustrate the paper’s central claim - that the ways that provers go about working on proofs provide the context for continuing that work and for discovering the reasoning that a particular proof is then seen to require.