That sentence, spoken by Dick Gross this past Tuesday, is a particularly effective epigrammatic expression of the value system that prevails in our corner of mathematics. It also fits neatly with the ontological commitment expressed in a sentence from my Princeton Companion to Mathematics article: “if you can steal ideas, then they are real” — itself a hack of Ian Hacking’s formulation of entity realism: “So far as I am concerned, if you can spray them, then they are real.”
Dick thought the reason number theorists are having such a hard time coming to terms (positively or negatively) with Mochizuki’s work on the ABC Conjecture is that they can’t find the answer to the first and most important question to address to any proof: “What’s the idea?” Many people, he predicts, will be asking that question at next month’s conference in Oxford.
No one should be surprised that Dick Gross and I feel the same way about ideas. As graduate students at Harvard we both heard Barry Mazur advocate proof by “pure thought;” we both attended the lecture (also mentioned in my PCM article) in which Jean-Pierre Serre praised the proof of a famous conjecture by saying that it contained two or three real ideas. I don’t know for sure but I suspect that Dick is familiar with the (possibly apocryphal) story that André Weil once defined a mathematician as someone who had two (real) ideas, then backtracked with the nasty comment that “But then Mordell would be a mathematician.” And more to the point, Dick was marveling at the persistence over the years of our generation’s group of number theorists trained at Harvard, Princeton, and Berkeley. Someone who finds it important to keep repeating that all knowledge is socially constructed could well claim that Dick Gross and I both talk that way because we were formatted to talk that way, and there’s really no way to contradict such a claim.
I was up at Harvard for the number theory seminar, but I also had the privilege of meeting two more of the Ambitious Young Historians I have been mentioning periodically — one who has been following this blog, presumably as a window on what mathematicians think about the pressing issues on the day, and a second whose name had been brought to my attention. The meeting took place at a festive dinner; one of the AYHs let slip that there are five of them, so that I can look forward to meeting two more, about whom for the moment I know absolutely nothing.
At the dinner I established to my own satisfaction that the flames of the Sokal affair have not completely died down, even though 19 years have passed since Sokal published his famous hoax. The historians treated it as an article of faith that support for Sokal’s side in the Science Wars was a reaction to the reduction of scientific research budgets at the end of the Cold War. I’ve never seen it proved convincingly that there was a connection; and mathematicians were never as uniformly hostile as our counterparts in the natural sciences to social constructivist approaches to history and sociology of science.
What is certain is that an AYH, even one as undoubtedly brilliant as the three I have met thus far, will have a much harder time finding a tenured university position than an Ambitious Young Mathematician. This circumstance was noted in the review of MWA by Ernest Davis:
To make the case that government funding for math should continue to be greater than that for history, comparative literature, philosophy, and so on, it’s necessary to argue that mathematics serves the general interest in some ways that these other fields do not.
The general interest, as I see it, has urgent need of the talents of every AYH I have met as well as the ones I have not yet met. Unfortunately, I have very little influence on the matter, which is naturally of much more pressing concern than the aftermath of the Sokal affair to the AYHs at the festive dinner and elsewhere. In this connection, I would like to quote Mark Kisin’s ruminations on “how the influence of billionaires has changed the way mathematics is done compared to the feudal way where you were sponsored by one of the Dukes,” but I can’t because I didn’t record his exact words and have only provided a paraphrase. The AYHs are well-equipped to address this topic in depth; I will share my own scattered thoughts on the question in the next few days.