The Crisis in Mathematics and the Prospect of AIcademia

The Crisis in Mathematics and the Prospect of AIcademia

Popular Philosophy

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A continuation of the old blog, found here

Mathematics is in crisis. Or, at least, mathematicians are. Mathematics itself, on the other hand, might be entering a golden age. This strange tension is likely to confront a number of academic disciplines in the coming few years, and so, even academics who aren’t mathematicians themselves should probably be thinking about the situation currently facing mathematics, since something like it will may face them soon too. I am myself a philosopher rather than a mathematician. However, as a philosophical logician (among other things), I am more mathematician-adjacent than most of my colleagues whose work fits more squarely in the humanities. So I have been impacted by the crisis in mathematics more than most in my field, and it has led me to think about the future of academic research, especially in the sciences. The conclusion I’ve come to is quite a humbling one, at least for us humans.

The Crisis in Mathematics

When LLMs like ChatGPT first burst onto the scene just a few years ago, many were impressed with their wide range of linguistic capabilities, for instance, their poetry writing abilities. This was, in some sense, unsurprising; they were, after all, language models. While the linguistic abilities of these systems were impressive, they were widely mocked for their utter mathematical incompetence. Mathematicians, it seemed, were in the clear. However, since the release of “reasoning models,” first with OpenAI’s “o1” in September 2024, then with “o3” in April 2025, the writing has been on the wall that these systems were coming for mathematics.

These new “reasoning models” are trained through large-scale reinforcement learning to engage in an extended internal “chain of thought” before producing a final answer. That is, they are trained to break problems into steps, recognize and correct mistakes, and abandon unsuccessful approaches for new ones, doing all of this “internally” before they submit a final answer to the user. Their performance can then be improved by scaling both the training compute used to reinforce successful reasoning behavior of this sort and, crucially, the amount of inference-time compute they are permitted to spend working through a problem. These models quickly became very very good at tasks in which success could be verified, most notably, coding and math.

Just over a year ago, both OpenAI and Google announced their models achieving gold medal performance in the International Mathematical Olympiad, a set of competition problems designed to challenge the most mathematically-gifted high school students. Since then, model capabilities have progressed beyond self-contained problems to genuine research mathematics. Over the last few months, a number of notable conjectures—most notably, the Unit Distance Conjecture and the Jacobian Conjecture (for dimensions greater than 2)—which had stumped mathematicians for decades have been solved by large language models, the former by an internal model of ChatGPT and the latter by Claude Fable 5. I will not go into the details of what these mathematical conjectures say, but I will note that they were very significant open problems in their respective fields.

In both of the two cases just mentioned, the LLM did not prove the conjecture. Rather, it produced a counterexample, disproving the conjecture. This has been the general pattern of the most prominent results in the last few weeks since the newest class of models have been made public. Each day now, it seems, more and more conjectures are falling at the hands of LLMs. The prompts for some of these results are quite comical, and, to mathematicians, I’m sure depressing.

Last week, Dmitry Rybin posted a ChatGPT-generated counterexample the Dinitz-Garg-Goemans conjecture along with the prompts he used to get ChatGPT to generate it, the first of which included the simple command “You should do a breakthrough.” When it came back from its attempts an hour later with no such breakthrough, Rybin simply urged it “Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure.” Ninety minutes later, still no conclusive counterexample, only a partial result that did not suffice to refute the conjecture. Rybin urged it again: “enough of partial results. Let’s finish with a complete unconditional counterexample.” Ninety minutes later, it came back with one that has now been verified by the mathematical community.

The results that AI models have achieved in solving open math problems are currently being tracked on the website vibemathed.com, a cheeky reference to “vibe coding,” which is now the norm for many software developers. Just a few weeks ago, there were around 60 problems with AI solutions tracked on the site. Now, at the time of writing this, there are 220. In a week or two more, perhaps that number will triple. In the last few weeks, twitter has been busy with different users, at various levels of mathematical ability, sharing prompts for cracking conjectures, with one particularly prolific user, Christopher D. Long, jokingly declaring himself “mayor of Conjecture City.”

Given that the most prominent results that LLMs have been producing are counterexamples to conjectures, it is natural to dismiss these results as a product of mindless brute force search rather than genuine understanding. However, this would be to greatly undersell what they’ve been doing. The idea behind the counterexample to the Unit Distance Conjecture was described by multiple prominent mathematicians as “beautiful", bringing deep ideas from algebraic number theory to bear on a problem in combinatorial geometry. With respect to the Jacobian Conjecture, the counterexample was simple—short enough to fit in a single twitter post—and easily checkable. However, that did not mean that the generation of it did not arise from a deep understanding of the problem.

In an attempt to understand Fable 5’s disproof of the Jacobian Conjecture, Terence Tao, widely regarded as the world's the greatest living mathematician, turned to ChatGPT, just as anyone else would. In his blog post on the topic, which acknowledged his indebtedness to ChatGPT, he posted his chat log. In it, ChatGPT speaks to him as an advisor would speak to a student, letting him know that he’s on the right track and patiently explaining things to him. For instance, in response to some question asked by Tao (which I will not pretend to understand), ChatGPT says “Exactly" and offers a thorough explanation. Tao responds “Ah Ok,” asks a follow up question, and the conversation continues. It was reading this exchange when the gravity of what was happening really hit me.

Now What?

Mathematicians are now facing the question of what the discipline will become in the age of AI. Last week, at the 2026 International Congress of Mathematicians, Tao gave a talk addressing this question. The basic issue motivating the talk was what Tao called the “AI Capability Conjecture,” which is not itself a specific conjecture, but, rather, a general schema for more or less optimistic specific conjectures about the mathematical capabilities of future AI systems:

At some point in the near future, some AI tools will, at some expense, and with some level of human supervision, be able to correctly accomplish some research-level mathematical tasks in some fields of mathematics, with some non-trivial success rate, and at some level of correctness and quality.

Here, each “some” is a variable, to be determinately specified in order to yield a determinate AI capability conjecture. The question, then, is how we should fill in these variables in order to yield a specific AI Conjecture, and what we should do to prepare for the truth of such a conjecture. The question is particularly pressing if we fill in the variables in a particularly strong fashion, for instance, as in this version of the conjecture proposed by Aldo Corsi:

“Within 3 years, commonly available AI tools will, effectively for free and with zero human supervision, be able to accomplish any research-level task in any field of mathematics at a super human rates of success, correctness and quality.” Now what?

The real question is at the end: now what? What will mathematicians do if this is indeed the reality in three years?

Tao’s lecture focuses on problem solving, where LLMs really seem to accel. Even here, however, he takes it that there is a still a lot of work for human mathematicians to do. Tao describes the problem-solving pipeline as having the following steps:

Open problems ----- proof generation ----> unverified solution ---- proof verification ----> verified solutions ---- proof exposition ----> well-written solutions ---- proof publication ----> accepted solutions ---- proof canonicalization ----> definitive solutions

Whereas mathematicians have typically spent much of their time and cognitive resources on the proof generation part of the pipeline, Tao’s suggestion that, in a period of “proof abundance” due to AI, human researchers will now spend much more time latter parts of the pipeline, digesting proofs, clearly explaining them, and turning them into mathematical canon.

One metaphor, suggested by Grant Sanderson on the Dwarkesh Patel Podcast, is that mathematicians of the relatively near future will be more like art museum curators than artists themselves. That, is, of the vast space of AI-generated mathematical results, they will select the ones that are most worth learning, organize them into a coherent progression, and clearly present them in terms that are digestible to other people. This is already, in large part, what textbook writing amounts to, as well as the sort of popularization work that Sanderson himself does with his YouTube channel, 3Blue1Brown. Part of the proposal, as Sanderson elaborates it, is the essentially human element of curation. The thought is that, even if, in three to five years’ time, AI systems are better at explaining results than human beings, we still trust the taste of humans to curate what is worth learning.

This is an interesting proposal, and it might sound nice to some, but I still find it a bit depressing. Imagine telling an artist that they could no longer do art themselves, but not to worry—they can serve as a curator of the work of other artists. Few artists would be happy with this, and it’s hard to see why mathematicians would be happy with the analogous thing either. Though explaining things to non-experts is an important part of mathematical practice, mathematicians and other academic researchers typically aspire to push the frontier of research forward, not to simply curate the research of others who have done so. Textbook writers, for instance, are often also leading figures in the field, and it is often the case that many of the results that they canonize in their textbooks are ones that they have themselves established or contributed to establishing. The prospect of the erasure of mathematicians from that whole aspect of mathematical practice, relegating mathematicians to mere curators of research done by AI is, once again, a bit depressing, to put it mildly.

Now, one might think that there is in no reason to despair just yet. In Tao’s talk, he distinguishes between two aspects of mathematical practice: theory building and problem solving. These two aspects of mathematical practice correspond to two kinds of propositions which figure in mathematical papers: definitions, which are stipulated, and theorems, which ar

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