On proof and progress in mathematics

Abstract

This essay on the nature of proof and progress in mathematics was stimulated by the article of Jaffe and Quinn, "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics".Their article raises interesting issues that mathematicians should pay more attention to, but it also perpetuates some widely held beliefs and attitudes that need to be questioned and examined.The article had one paragraph portraying some of my work in a way that diverges from my experience, and it also diverges from the observations of people in the field whom I've discussed it with as a reality check.After some reflection, it seemed to me that what Jaffe and Quinn wrote was an example of the phenomenon that people see what they are tuned to see.Their portrayal of my work resulted from projecting the sociology of mathematics onto a one-dimensional scale (speculation versus rigor) that ignores many basic phenomena.Responses to the Jaffe-Quinn article have been invited from a number of mathematicians, and I expect it to receive plenty of specific analysis and criticism from others.Therefore, I will concentrate in this essay on the positive rather than on the contranegative.I will describe my view of the process of mathematics, referring only occasionally to Jaffe and Quinn by way of comparison.In attempting to peel back layers of assumptions, it is important to try to begin with the right questions:1. What is it that mathematicians accomplish?There are many issues buried in this question, which I have tried to phrase in a way that does not presuppose the nature of the answer.It would not be good to start, for example, with the question How do mathematicians prove theorems?This question introduces an interesting topic, but to start with it would be to project two hidden assumptions:(1) that there is uniform, objective and firmly established theory and practice of mathematical proof, and (2) that progress made by mathematicians consists of proving theorems.It is worthwhile to examine these hypotheses, rather than to accept them as obvious and proceed from there.

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