数学已经改变了，让我们也随之做出改变

[This is a guest post by Frank Calegari. This blog post was initially written in a different file format and converted using AI. — T.]

Why we should not be sad

The release by OpenAI on October 6 of over 700 research papers was a watershed in the history of mathematics. AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics. So why are some mathematicians so unhappy?

Some of it is the apparent disregard by AI companies for the practice of mathematics. The OpenAI papers — despite some tremendous ideas — are atrociously written and violate our community expectations of attribution. Some of them are even wrong. We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.

Much of the reaction seems to reflect a feeling that the subject that they love is under attack, and is being “hollowed out”. The recent OpenAI dump made barely a dent on the general problems of functoriality and reciprocity in the Langlands program. But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry. Mathematicians should do a better job explaining that when we talk about the central problems in any area, we are merely discussing the most natural boundaries of our understanding — and that unlocking them is not the end but the first step to finding new playgrounds to explore. Of course, this takes time. I looked for areas of mathematics which were “harder hit”. It is hard not to be struck by Hugo Duminil-Copin’s lament:

Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.

I expected to see a few of them in the list. But not all of them. Not all at once. Not with such nonchalance.

How would I respond to that?

The classification of finite simple groups.

I can’t say much about percolation theory, but there is another historical precedent I think mathematicians should take note of. In 1972, Dan Gorenstein announced a program towards the classification of finite simple groups. After a flurry of activity over around a decade, the “classification” was announced by Gorenstein in 1983 (somewhat prematurely, with the case of quasithin groups not being completed for another two decades by Aschbacher and Smith).

The classification of finite simple groups was in many ways a huge success, but it was also a failure in that it — apologies to my group theorist friends and collaborators — cooled a lot of the interest around the subject. The “famous problem” had gone, and mathematicians on the whole turned their attention elsewhere. And yet, there remain very rich questions in finite group theory and even about the classification itself (to name just one: John Conway’s favorite question is why does the Monster group exist?). Perhaps for some fields, the OpenAI dump feels like how a group theorist may have felt in 1970 if the classification was dumped in their office, not in the form of the actual papers, but printed out on a spirit duplicator, smudged a little, then spread randomly like confetti all over the floor.

Right now, we need efforts like that of Gorenstein, Lyons, and Solomon and their work on the revised classification. If this was all we had to do, then it would be a fairly thankless task. But we also have the responsibility not to pronounce the subject “done” but rather — with these new tools available — ripe for further investigation. When mathematicians lament the winding paths missed by the construction of a galactic bypass, they need to realize that those old paths still exist and seek them out. And we need to reward that behavior when it comes to offering people jobs.

The promise of AI, and the (dis)-unity of mathematics:

Since I was 15, I’ve wanted to know what was. First, there was the childhood fantasy of finding a formula that Euler missed. But soon one learns more: how Grothendieck’s period conjecture, together with a sufficiently fine understanding of the geometry of mixed Tate motives (following work of Don Zagier, Sasha Goncharov, Francis Brown, and others) gives a completely satisfying structural, conjectural explanation of what’s going on. The irrationality of , or even its transcendence, is a mere shadow of the actual interesting underlying problem. Apéry’s proof of the irrationality of has inspired me for over years not because of its conclusion, but for how it reveals the tiniest hints of some still secret story relating algebraic -functions to arithmetic.

I hope AI can help mathematicians break out of the straitjacket of being constrained by the first natural conjecture they cannot prove, and work on unraveling the deeper mysteries woven throughout our subject. In the last thirty years, progress in mathematics by humans has been remarkable. But at the same time, I have felt a growing unease; the number of times I have seen mathematicians of all levels being unable to explain to someone even slightly outside their field why what they are doing is interesting has been distressing. The natural grooves of our profession have very much been bent toward more narrow technical expertise than a bigger vision. AI allows us an amazing opportunity to broaden ourselves as mathematicians.

I have tried (not always successfully) to be broad as a mathematician. I had some help by the fact that, while I knew since I was 17 that I wanted to work in algebraic number theory and the Langlands program, I grew up in the crucible of Melbourne University, which was a home to many Thurston students and low-dimensional topologists. This meant I was bathed in the sweet liqueur of hyperbolic 3-manifolds. That exposure has helped my career immeasurably. As with any two deep areas of mathematics, hyperbolic three-manifolds and arithmetic are much more closely connected than would first appear. But it’s still hard to be an expert on both, let alone pursue a crazy idea combining them. Let me get technical for a moment. For every imaginary quadratic field , there are many commensurability classes of compact arithmetic hyperbolic three-manifolds indexed by non-split quaternion algebras . Manifolds in each of these commensurability classes give rise to elements in via the decomposition of into hyperbolic tetrahedra, and their volumes are always rational multiples of . I have always had this idea that one could conceive of elements in these different commensurability classes as forming some type of geometric Kolyvagin system (“Heegner points”) associated to , which one could yoke together by some unknown integral arithmetic version of the Jacquet–Langlands correspondence. This is not a question I have pursued, for two reasons. First, there were so many underlying conjectures involved that any result I could prove was going to be conditional on many of them, something that’s not always appreciated by the mathematical community. Second, assembling all the technical background to formulate such an idea precisely would have taken a huge amount of background work. Now, with AI, the barriers to trying such a problem are much lower. It’s not trying to prove a theorem. It’s trying to explore some completely new direction that may or may not be nonsense. There are so many problems in mathematics that are still so fresh and vital but were previously untouchable that the opportunity to approach them is massively exciting. We need to be more ambitious, to create bold research plans that weren’t even conceivable a year ago.

This is not to say that everyone must (or must not) use AI. But we have to rethink and readjust what’s important. Proofs of major conjectures, using AI or otherwis