How Mathematicians Determine if an Argument Is a Valid Proof

Abstract

The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.