ASPECTS OF PROOF IN MATHEMATICS RESEARCH

Without having a clear definition of what proof is, mathematicians distinguish proofs from other types of argument. This has become increasingly difficult in the last thirty years, as mathematicians have been able to use ever more powerful computers to assist them in their research. An analysis of two types of proof (mathematical proof and formal proof) and two types of argument (mechanically-checked formal proof and computational experiment) reveals some aspects of proof in mathematics research. The emerging framework builds on the distinction between public and private aspects of proof, and revises the characterization of mathematical proof as being formal, convincing, and a source of understanding. What is proof in mathematics research? Hersh (1997) differentiates between two meanings of “mathematical proof”: what it is in practice and what it is in principle: Meaning number 1, the practical meaning, is informal, imprecise. Practical mathematical proof is what we do to make each other believe our theorems. It’s argument that convinces the qualified, skeptical expert. It’s done in Euclid and in The International Archive Journal of Absolutely Pure Homology. But what is it, exactly? No one can say