We need to talk about AI and Math
Over the last couple of months, I had many conversations with people about the future of mathematics and the role of artificial intelligence (AI, in case that’s a brand new abbreviation to you). I would argue that we live in times of transformative change to our field that may well be without precedent in the history of mathematics. I have encountered many reactions to it, from complete indifference, which I have met most often among tenured colleagues closest to retirement, to intense engagement, both enthusiastic and hostile towards AI.
The “AI and Math” debate is a multifaceted problem that mathematicians are, unfortunately, very poorly prepared to take part in. It’s not that we are unable to participate; it’s just that our training doesn’t prepare us well to talk about problems of this kind. My proof collapses if someone points out a mistake, but my sociological or philosophical argument might be strengthened by people engaging with it, even just to critique it, and pointing out the other facets.
What I want to do here is gather some of the main points of this debate. There is no narrative running through this document; it’s just a collection of items to think about. They fall under several different categories, ranging from what I would consider undisputed facts to broad points I would like to think about and return to in a separate piece.
AI will change math. In fact, it already has. Some years ago I knew a mathematician who spent their tenure-track years on the Jacobian Conjecture, an 87-year-old problem in algebraic geometry. They published little in that time and did not get tenure. In July 2026, Levent Alpoge, working with Claude’s Fable 5 model, produced a counterexample to that conjecture in dimension three, in a matter of hours. Two things here are worth keeping apart. The mathematics was not done by a model on its own: Alpoge set up the search, and he credits Akhil Mathew with asking the question that started it. But the search itself, the part that had consumed years of a career, took an afternoon. It is also worth being precise about what was settled. The conjecture is now known to be false for n ≥ 3; the case n = 2 remains open, and journal review is still under way. Even with those qualifications, this is a big change in mathematics, and we are only at the very beginning of learning about the capabilities of large language models (LLMs), while Silicon Valley is already working on the next best technology that will make LLMs seem completely useless in comparison.
The models continue to get better. For a while, we’ve heard a similar story: “LLMs can only do X, but not Y.” Then the models did Y, so we said “they can do Y, but not Z,” and, of course, the models did Z as well, but the story we’ve told ourselves about what models can’t do keeps repeating. It is unclear whether there is anything that we can do that the models won’t. For a specific example, something I’ve heard many people say is: “even if LLMs get better than humans on solving problems, only humans can decide which problems are interesting.” To put it mathematically, this claim requires a proof. It is possible to imagine a future where models are far better than humans at coming up with interesting problems and entire research programs, solving them, and explaining the solutions to people and to other models. I’m not saying that this is the inevitable future, but that any exclusion thereof from the realm of possibilities requires justification. I believe that such justification might actually exist, since research programs are much harder for reinforcement learning. How would you create a reward function that grades interesting questions on a large scale? How would you generate the training data? These points suggest that human mathematicians are needed to answer such questions now, but it doesn’t show that human mathematicians will always be needed.
Environmental concerns are selectively applied. In 2023, U.S. data centers consumed about 17 billion gallons of water directly, for cooling. That is the figure usually quoted, and it is the wrong one to quote. The same Lawrence Berkeley National Laboratory study puts indirect consumption, the water used to generate the electricity, at roughly twelve times the direct amount, which brings the real total to something over 200 billion gallons. Set that against about 760 billion gallons a year for U.S. golf courses and between 1.3 and 1.6 trillion gallons for California’s almond orchards alone. Data centers use roughly a third of what golf uses and a sixth of what almonds use: a good deal more than the direct figure suggests, and still well short of either. Two caveats follow. The trajectory matters, since the same study projects direct consumption at 38 to 73 billion gallons by 2028, which on the current ratio would put the total in the range of golf within a few years. And national aggregates hide the problem that actually matters, which is local: data centers cluster, and a single facility can strain one watershed in a way that no national total will ever show. Those are the arguments worth having, and I don’t dismiss them. What I object to is the selectivity. Almonds and golf are also discretionary uses of scarce water, and neither draws a fraction of the scrutiny. I believe data centers provide more utility than both golf courses and the almond industry, although I’ve heard arguments in favor of almonds stemming from their being food and contributing to the economy as a trade export. Regardless of whether you agree with me or not, if we are going to weigh a technology’s environmental cost against its benefits, the same test should apply to everything else we irrigate.
Will AI be cheaper? In 1965, Gordon Moore predicted that the number of transistors packed into a microchip would double every year, a rate he revised to every two years in 1975, by which time he was running Intel. Transistors are now beginning to approach the size of individual atoms, and therefore cannot be further subdivided, so it is worth being careful about how much weight that observation can still bear. The trend that matters for us is not transistor density but the cost of running a model, and that has been falling quickly for reasons largely unrelated to Moore’s law: better architectures, distillation, quantization, and competition among providers. It is reasonable to expect that AI will keep getting cheaper and that mathematicians will be able to integrate it into their activities at a low cost. Right now, whatever amount we are paying for a Claude or ChatGPT subscription, these companies are operating at a massive loss, and this will need to end. Whether it ends because the technology got cheap or because the price rose to meet the cost makes a great deal of difference to a mathematician’s budget, and I don’t think we know which it will be. One might speculate that prices will increase with the supply chains for advanced chips being geographically concentrated and politically exposed; a serious disruption there would mean we all pay more for access to technology across the board, including AI. If I were to guess, I’d still say that the times ahead are the times of abundant AI, mostly because the cost curve has so far moved faster than the supply risk.
Individual motivations vs societal motivations. Mathematicians often say that it doesn’t matter whether LLMs are better at solving problems, writing papers, or preparing slides, since we do mathematics for its inherent beauty. This conflates the motivations of individual mathematicians to do mathematics with the motivations of society to fund it. I enjoy playing The Settlers of Catan with my family; but neither I nor anyone else can make a living as a Catan player. Mathematicians may still choose to do math, but in an AI-dominated world, will they be able to do it as part of their job or as a relaxing weekend activity? If we argue that we do math for its inherent beauty, then governments should fund it accordingly, i.e., by viewing it as a form of art. The trouble is the level of funding that follows. Most math papers are read by no more than five people. I’m pretty confident that one of my most cited and most influential papers was only read carefully by exactly four people, the authors included. Governments do fund art with small audiences, opera houses and national archives, but they fund it accordingly; that means funding at a level well below what mathematics currently receives.
Problem of scale. Another argument I hear very often is that there will always be mathematics and there will always be mathematicians. For a parallel situation, we can look to the population of horses around the time of the invention of automobiles. Karl Benz built the first gasoline-powered car in 1886, Henry Ford began production of the Model T in 1908, and the first Ford tractor followed in 1917. The U.S. equine population peaked in 1915 at about 26.5 million horses and mules; by 1960 it was just over 3 million. While horses didn’t disappear, their numbers fell almost nine-fold in forty-five years. Two things about the parallel deserve to be said plainly. The displacement came as much from the tractor as from the car, so what did the work was not one technology but a family of them arriving together. And the analogy is imperfect in an obvious way: horses had no alternative occupations and no say in the matter, while mathematicians have both. The usual reply is comparative advantage, that displaced workers move to tasks where they still hold an edge. I don’t think this disposes of the question. It tells you that mathematicians will find something to do; it does not tell you how many of them will be paid to do mathematics. Depending on the estimates, there are between 100,000 and 250,000 active research mathematicians in the world, not including teachers or industrial researchers. Asking how this number will change is a legitimate question and it has to do with the availability of positions and grants that I describe below.
Supervision in the times of AI. Given the rapid progress of new models on solving open problems, a graduate student working on a research project right now might see their problem solved by AI in a year or two. I grant that “scooping” has always been a threat, and every researcher should do thorough literature review before embarking on a new project. This, however, is qualitatively different – a student might “scoop themselves”: work on a project for two years, only to prompt a 2028 model LLM that will produce a solution within minutes. What does the student have to show for their work over those two years? A single prompt? While our profession will eventually adapt, it is the current transitionary period that we need to worry about. We can’t simply sacrifice our current students, postdocs, and tenure-track faculty by sleepwalking into the new age. Jacob Tsimerman, one of the 2026 Fields medalists, has stopped taking graduate students, because he doesn’t “know what to do with a grad student in math who’s not very AI motivated”; he expects AI to surpass mathematicians within two years and is leaving the University of Toronto for AI safety work at OpenAI. Tsimerman and I might come to different conclusions about whether to take on graduate students, but I ought to have an answer to these concerns. As a community, we need to be clear minded on this: this is not business-as-usual anymore. The calculation has changed, the threat to the careers of graduate students is already upon us, and those of us with the responsibility of mentoring need a serious answer to the new reality; our mentees deserve at least this much. What strikes me is that the AI enthusiasts and skeptics I talk to mostly don’t.
Will AI be good for math? If AI will change mathematics, it makes sense to evaluate this change. Here, we can point to proving new conjectures, but at the same time to potentially massive unemployment of mathematicians. We have seen that already happen in other areas; I wouldn’t want to be an engineer, a data scientist, or a lawyer entering the job market in the next couple of years. The important distinction this points to is between mathematics and mathematicians. I find it easier to talk about mathematics, by which I mean the totality of mathematical knowledge, a proxy for which can be the literature on the subject, along the lines of David Hogg’s essay [arXiv:2602.10181]. If there is a new tool that can do mathematics better than humans, then arguably the change is for the better. This is not entirely true, and I hope to return to the point in a separate piece. But what about the mathematicians? Are horses better off now than they were 150 years ago? There are way fewer of them, but at the same time, they no longer pull massive streetcars or participate in military combat. If we end up with a fraction of the current number of mathematicians, none of whom have to write referee reports, are we better off?
Math and society. Should society continue to fund math research? First, I want to make clear that my remarks are limited to research and not teaching or outreach. Funding these is a separate question, but let me point to a massive salary difference between research and teaching/outreach positions: if the argument for funding math research is that it informs teaching and outreach, then the research salaries should be adjusted accordingly. The interesting question to me is related purely to funding math research. This includes positions at universities, government grants, etc. I note that this is the key driver behind possible changes in the number of research mathematicians. Suppose that, using AI, we can get the same amount of math innovation with only 10% of the funding. Should the budget of the Division of Mathematical Sciences at the NSF shrink by 90%? I think this gets the matter backwards. A productivity gain is a reason to buy more of something, not to buy the same amount for less. The current pace of mathematical discoveries far outpaces that of the 1700s, and nobody proposed that we should have stopped once we matched the earlier rate; there is no reason the future pace should be pinned to the current one either. If it makes sense to fund mathematics research at all, as I believe it does, then making it cheaper is an argument for more mathematics, not for a smaller budget. That said, most mathematics Ph.D. students don’t take academic jobs, and instead go to work in industry, where their skills are not fully utilized. Many of these jobs don’t seek maximum innovation; they just need someone or something good enough, and AI might be able to replace them in these job completely. With shrinking employability, will students continue to pursue their graduate degrees in mathematics, and how might that impact the field?
Mathematicians of the future. What will successful mathematicians look like in the future? Here is a vision that a friend shared with me – mathematicians will have two choices: use AI or leave the field. You can’t be a delivery driver without a driver’s license. You can’t be a mathematician right now without using LaTeX. And LaTeX is worth dwelling on, because it was not only a tool we picked up; it was also a job that vanished. Mathematics papers used to be typed up by departmental secretaries. Where are the secretaries now? Fewer people, doing better work, with technology. That is the pattern, and I have not yet seen a convincing argument for why AI should break it. I would like to see one.