Psychological Nature of Verification of Informal Mathematical Proofs

It is a pleasure to dedicate this article to Dov Gabbay on the occasion of his sixtieth birthday. I still have fond memories of him as a very young person, when I met him for the first time four decades ago at Stanford. That discovering or finding proofs is essentially a psychological process is widely recognized. Distinguished mathematicians such as Poincare and Hadamard have written very personal statements about their experience of discovery. There is, in fact, no serious body of opinion disputing this matter. The question of verification is very different. On the one side we have the important concept of a formal proof, well developed and used in logic and foundational studies of mathematics. The essential character of such proofs is their being verifiable by an algorithm, many of which have now been implemented on digital computers and used in teaching to check student proofs. There have also been important research applications. A well-known example is the computer analysis of a large number of cases in the proof of the four-color theorem. The complete proof was originally not, and I believe, there is still not, a published formal proof of the theorem itself. But there is already a smattering of research articles focused on theorems that have formal proofs. My focus here is on the informal proofs that still dominate the research literature and undoubtedly will do so for the foreseeable future. What is the basis for saying that an informal proof is valid? It cannot be that it has been checked by some familiar algorithm of formal verification or computation. Certainly some parts will often have this character, but all those informal proofs that have not been formalized, but are judged correct, must have a different basis. What is it? The familiar and almost standard answer is an appeal to understanding, a concept notable for its psychological vagueness. In saying this, I do not