Progress in mathematics

Among the many fond recollections that followed the recent death of William Thurston, I came across a reference to this article on proof and progress in mathematics.

Thurston, a geometer of great insight and also I think a great contributor to the popularization of mathematics, argues for a view of mathematical progress which does not restrict itself to just the production of completed proofs.

Thurston's view of mathematics and how mathematicians think is in contrast to that expressed by Feynman in his lecture on the relationship between Physics and Mathematics where he repeatedly identifies the mathematical approach as starting with fixed axioms as opposed to looking at the overall network of logical relationships. To some extent that is true but only as one of the exercises of mathematical thinking and the question of alternative axiomatizations is always in the air - with a view of the whole network being not just essential to this but also always being at the heart of truly "complete" understanding.

One point of interest to me is that Thurston's article was written in response to a suggestion by Arthur Jaffe and Frank Quinn that credit for the kind of work Thurston describes be more explicitly separated from that for actually completing rigorous proofs. I certainly share Thurston's discomfort with their use of the word "theoretical mathematics" for the more speculative and less fully locked-down types of discussion. (The fact that theoretical physicists do a lot of speculative math seems to be a very poor justification for that choice of wording.) But I find it ironic that there is a tone of conflict when both papers seem to be arguing for the same thing in one sense - namely more attention to the role of intuitive and speculative thinking in mathematics. I suspect that Jaffe and Quinn were being a bit tongue-in-cheek with their suggestion of a separate discipline, but of course Thurston had some reason to take it personally as he was cited as an example of someone whose incompletely documented speculative advances may have discouraged others from pursuing the same goals. Thurston, who was working hard to bring others up to speed with his ideas, seems to feel unfairly criticized, but it may well be that those outside his circle had good reason to suspect that whatever progress they made would gain little recognition because it would always appear that their ideas were already known in Thurston's group.

A further irony, especially considering Jaffe's frustration with the premature announcement of Dobrushin and Minlos, is that his own close colleague Thaddeus Balaban announced an impending proof of existence of the YM4 Quantum Field Theory which undoubtedly deterred a number of others from proceeding to investigate that topic in the late '80s.

Of course the issue of credit has always been fraught. And ever since well before Newton and Leibnitz, or even Cardano, Ferraro and Tartaglia, the question of fair attribution when one party appears to be ahead but holds cards close to the chest has been problematic. In one sense it would be much better if priority counted for less, but perhaps the overall rate of progress would be slower if it was dependent on dullards like me and those who take us ahead would fail to do so without the thrill of victory as a potential reward.

Perhaps (in fact almost certainly).
But I think I could live with that!

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