Comment by impendia
18 hours ago
Math professor here.
Different academic disciplines are very different, and most don't engage in the same kind of black-and-white problem solving that mathematicians do, where you either have solved a problem or you haven't.
But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.
We haven't yet figured out what that should be, but I presume that everyone would agree that this should continue. As one possible model, check out this blog post of Terry Tao's, where he gives his own perspective on the recently proved Jacobian conjecture.
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
When computers can solve the underlying actual problem, this sort of work seems likely to rise in value, and be something which a greater number of mathematicians engage in.
I would like to ask a question to you as a math professor: I think we all agree we do not know what the discipline will look like in ten years. But doesn't the rapid surge in mathematical proofs and methods imply that - at least for the coming years - there will be more, not less, work for mathematics?
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
In practice, mathematicians' workloads have been a function of their work ethic, motivation, and competing demands on their time. There's no big-picture question of "how much math there is to do now"; the amount of remaining math to discover has long been presumed to be, for all intents and purposes, infinite. (Similar questions are relevant on a much smaller scale -- for example in case of someone who has specialized in a narrow specialty which goes dead.)
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
Applied mathematician here.
Where I see models having a huge impact is in simulation code development.
One blocker for years now has been the adoption of GPUs. LLMs can fairly successfully and very quickly port to GPU and suggest/implement useful optimisations. Once it's verified, a code can go from anywhere between 2x to 1000x faster (mainly because CPU codes are so poorly optimised). Some science can reach much greater problem sizes, while some can run the same problems in hours rather than months and both can be revolutionary. Even more than that, LLMs seem to be finding fundamental performance bugs in both open and closed source core libraries so there's a bit of a whole-ecosystem uplift.
Can't comment on the more theoretical, less computational applied maths impacts!
> But all insist that you engage in some sort of outwardly visible production in your field: books, articles, conference presentations, public lectures, exhibitions, performances, something.
My vote is for interpretive dance. Give the laity something for their money.
Let us try not to end up in a situation where knot theorists get all the credit. https://www.youtube.com/watch?v=Q6zxKBUI3wY
This is golden.
> I presume that everyone would agree that this should continue
I would not presume this at all.
Every brick in the house you live in has been put there by a worker. The food you eat has been cultivated by a farmer. etc. etc.
You must explain what you give back to these people. It's fine if it's in a roundabout way, but it can't be nothing.
And the presumption isn't that there's something. The presumption is that there's nothing, and you must prove there's something.
> You must explain what you give back to these people.
Ha-ha. Sorry, but it is not "these people", who is paying to the mathematician. Government is doing it. So they must explain to the government what they cold give back to the government. Usually it is loyalty and the use of their social position to confirm the correctness of the government’s actions and political programs.
What is payment? What is money?
Goverment is extracting taxes from these people "for their own benefit" and giving it to the mathematician. When a farmer farms the food of the mathematician and sells it to the mathematician for money, he is simply earning back money that already used to be his.
Which is all well and good -- if it is actually in his own benefit.
Tao's blog is a nice example of human interpretation of math. He explains in an understandable but rigorous way beyond what would just be in a journal paper (or the output of an AI).
Lest someone gets the wrong impression, I think that any mention of the a priori strange-looking resolution to the Jacobian conjecture should be accompanied with a read of pages 160–161 of https://arxiv.org/abs/2609.05746. The tl;dr being that the very same construction featuring in the resolution appeared earlier in a draft paper that was accidentally made publically available for a travel award application. The same story has accompanied several other of the major announcements made by now. We haven't have a move 37 for maths yet, despite what OpenAI's marketing department might want you to believe.
Similarly, if all you ever read were OpenAI blog posts, you would get a very wrong impression of the usefulness of large language models of today in maths. For a working researcher, it's not a magic wand that you point at any given proposition and it tells you whether that proposition is true or not. It does appear to help if, while pointing your wand and utter the magical incantation “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
Instead, the reality seems to be closer to this, projecting a fair bit: a given mathematician will have a collection of propositions that they care about, and that they'll use as their own internal benchmark as new models come out. Very rarely will anything come out of it, but sometimes, in particular if you make sure to provide the wand with all relevant context, papers that could be relevant, proof strategies and lemma structures that you suspect are useful, something (which may or may not be plagiarism) will pop out, and that's really nifty. Moreover, it is not unimportant what the proposition and the relevant proof is like. And what does come out tends to be quite bizarre; proofs that use terminology that doesn't exist, seem overly pretentious, based on nonsense analogies where it's surprising that it even works at all, and the only comfort is that you can join it with an equally unreadable Lean blob. And where you would be _crazy_ to just publish those artifacts and think that you have contributed much of anything to maths.
But sometimes it works. It's still very unclear what kind of maths the models are good at, but it seems to certainly be an advantage if what you're looking for is a counterexample hidden in a pile of otherwise similar-looking non-counterexamples, if your proof is one that requires considering 36 different cases, each of which are so tedious that no researcher would have the patience to go through them by hand, or if the proof is an amalgamation of several existing structures, some of which are only documented in Georgian.
The gold rush, more than anything else, seems to be populating the convex hull of existing maths.
This can all change. The $15 million wand requirement today will be less tomorrow. Whether we ever get a move 37 is less clear, or whether we will eventually reach stagnation as all low-hanging fruit is picked, and the convex hull is populated; call this cope if you like. But maybe we do get move 37s all over the place, and it's fine that people think about what that future will look like.
Until then, and while we're still picking friut, let us rather have a think about what we can do to fix the incentive mismatch, to ensure that we increase the prestige of digestion over being the first to convince the LLM to do it up. Since that's the one thing everyone seems to agree, chances are it'll probably converge to something that doesn't have to be written in commandment form, but out of the guest posts hosted by Tao so far, the one by Antieau has some useful suggestions for standards (that aren't entirely unlike those from Leiden): https://terrytao.wordpress.com/2026/09/15/fast-math-slow-mat...
> It does appear to help if, while pointing your wand and utter the magical incantation “do it up bro”, you also make it convert $15 million into heat, but for most people, this kind of inverted Midas touch isn't quite accessible yet.
That's like where chess was when Deep Blue was built by IBM. Productivity improved. There was someone complaining on here recently that the seat-back entertainment system on some airline had a chess program set to "trounce all humans".
*"The Evolution of Human Science"* (originally published in Nature in 2000 as "Catching Crumbs from the Table"), collected in Stories of Your Life and Others.
It's a very short piece written as a journal editorial. Metahumans have advanced so far beyond human comprehension that they do all the original science, communicating via digital neural transfer that humans can't access. Human scientists are left doing hermeneutics: interpreting metahuman publications and reverse-engineering their artifacts, trying to decode work they couldn't have produced themselves. The editorial asks whether human science still has a point, and lands on a modestly hopeful note: interpretation is still a legitimate form of inquiry, and understanding metahuman work still expands human knowledge even if it isn't original discovery.
It reads rather differently now than it did in 2000.