Every time I’ve organized my thoughts sufficiently to finally write a proper blog post about the impact of AI on mathematical research, something massive happens which causes me to delay. I was preparing to wrote a post last week when I heard about the formalization of Fermat’s Last Theorem in Lean. And I was preparing to write a post this week when the news broke about an AI-assisted solution of the Millenium Prize Navier-Stoker problem. Rather than delay again (what will the big breakthrough be next week?), let me just record some thoughts quickly while I have a few moments to spare before heading to bed. This will not be as organized, or as carefully thought out, as I had hoped, but the speed at which the mathematical landscape is changing does not seem to allow for a leisurely collection of thoughts. I imagine I will come back to many of these topics in the future.

1. It has been clear to me for at least a year now that LLMs (and other AI systems) are going to forever change the way mathematical research is conducted. And we’re not talking about a minor change here – I mean in a seismic and irreversible way. Some of my colleagues have been in denial about this, insisting that AI is overhyped and we can continue with the old ways for the foreseeable future. I don’t know if they still think this after the recent announcement about Navier-Stokes, but the controversy surround that announcement (more on this below…) probably gives that viewpoint more juice, for the moment. However, in my opinion this position will soon appear hopelessly naïve.

2. I am not going to engage here with the issue of whether AI is a good thing or a bad thing for math (or, for that matter, society). For one thing, it’s clearly both at the same time, and the Schrödinger-like balance between the two is impossible to fully conceptualize right now given how rapidly things are change. Perhaps more importantly, the world is not going back to the pre-LLM days, even though many of us, for various reasons, may want it to. So a debate centered on whether AI is good or bad seems to be missing the point to me. The real question is what are we going to do to protect what needs to be protected before it gets irreversibly destroyed, and how are we as mathematicians going to adapt to this new reality. (An analogy: it’s not particularly helpful to discuss whether nuclear weapons are good or bad – they exist, and the important thing is that we prevent them from being used to destroy civilization.)

3. For me personally, at this stage in my career, AI is a fantastic tool. I’ve used it to prove and formalize new theorems, to create new software (I’ll blog about that at some point in the near future), to assist with administrative tasks, to prepare teaching notes, to proofread midterm exams and drafts of math papers, and to summarize recent developments in mathematics. It’s also an amazing writing tool. (This post is being written without AI, for the record – I try to be judicious with when and how I use AI in my writing, and I find it both refreshing and good for the old noggin’ to write in the old fashioned way whenever possible). Most importantly for the purposes of this post, I’ve used AI to gain a better understanding of mathematical ideas that interest me – which, ostensibly, is the purpose of mathematical research in the first place. In a vacuum, then – and if we could just freeze this moment in time – I would say that I’m thrilled by the recent advances in AI capabilities and what they offer for mathematicians.

4. The problem, of course, is that we’re not living a vacuum. And we can’t freeze this moment. The new AI tools being developed are upending the way that mathematics is done in such a dramatic fashion that students and untenured research faculty are – understandably – freaking out. There is massive uncertainty in the future of our profession, and uncertainty creates anxiety. This uncertainty is what I’d like to focus on here.

5. In many ways, I feel that the worries are not completely aligned with reality. Partly, that’s because I don’t think there is actually a consensus within the mathematics community about what our value to society actually is, or why we value the specific things we do. Is our goal to foster human understanding of mathematics or to discover new mathematical truths? Do we do mathematics because it’s beautiful and (for a twisted mind like my own) enjoyable, or because it’s useful? Is our value to society based on the actual theorems and techniques we produce, or on the fact that we are training the next generation to think critically, logically, and analytically? It’s all of the above, of course, but the problem – as I see it – is that for a long time we’ve used some of these things as proxies for others without thinking clearly through the distinctions. In the past, developing powerful new mathematical theories has been inextricably linked to the ability to “think like a mathematician” and train the next generation of students. Proving important new theorems, or applying sophisticated mathematical techniques to important problems, has been viewed as something that only a well-trained human mathematician could do. But now autonomous AI systems are capable of doing many of the same things – and in many cases both faster and better. This means that some of the skills we’ve previously treated as synonymous will have to be decoupled.

6. Take journal publications, for example. I’m a member of several editorial boards and we’ve had some intense discussions on these topics in recent months. The consensus I’m seeing emerging seems to be that journals should primarily judge the quality of the mathematics itself, not the relative contribution of AI to authorship, when deciding whether to accept a paper for publication. I agree with this, but it punts a major question, which is how we will evaluate scholarship moving forward. If journals do not distinguish between human and AI contributions, who will? Hiring committees? Tenure committees? Prize committees? Traditionally, such committees have relied heavily on the prestige of certain journals as a proxy for quality, and the quality of one’s publications has been a proxy for the quality of one’s mathematical abilities. But some of these things are rapidly diverging from one another as we speak…

7. More generally: what is the point of doing math research in an age where machines are in nearly all respects better than humans? Some of my colleagues would argue here that there remains something special and inherently non-mechanical about human creativity and ingenuity, but in my opinion the evidence points decisively away from this conclusion. I think that pretty soon AI will be as creative – or likely much more so – as the most creative human mathematicians alive. On this note, my colleague Mohammad Ghomi recently wrote: “These days I am reminded of a story from Rumi that I read when I was a child. It is about a destitute bedouin who one day comes across a puddle of rain water in the desert. Thinking that it is precious, he puts it in a jug and takes it to the caliph in Baghdad. The caliph orders the jug to be filled with gold coins and given back to him. But he also orders that he be shown the Tigris river, flowing behind the palace, on his way back. The story ends with the bedouin staring at the river in astonishment. In some versions of the story, he tosses the jug into the river.”

8. Enough questions – let me offer my own personal answers to a few of these things. For one: to me, the main point of doing mathematics is to improve human understanding, and I think that AI is already a tremendously useful tool in this regard. To the extent that the goal of mathematics is to discover what’s true and what’s not – to discover, prove, and certify theorems – AI will also be extremely useful. (There is also the delicious irony that AI itself was developed by mathematicians and computer scientists with a background in theoretical mathematics, and LLMs would never have seen the light of day without sophisticated mathematical tools and lots of people who understand them.) So why is it, then, that so many mathematicians are so pessimistic about the future?

9. I think it’s primarily because what many in the math community ACTUALLY value are not theorems but the sense of illumination that comes with proving a new theorem or discovering a beautiful new mathematical theory. And we value the math community itself, which is made up of smart, hard-working, people who dedicate their lives to uncovering truth and beauty in its purest and most abstract form and communicating the joy of mathematics to others. We value teaching and mentoring, and we value our position as the curators of a body of mathematical knowledge that is thousands of years old. And these things really are threatened by AI.

10. It seems, then, that we will have little choice but to reaffirm and re-embrace the more humanistic elements of mathematics. Math has always been both a science and a liberal art. To the extent that some mathematicians end up as managers of AI systems, it will become more like a lab science, and to the extent that others end up trying to digest and communicate AI-assisted discoveries, it will become more like a liberal art. What doesn’t seem sustainable, however, is the view of mathematics as something whose development rests crucially on human ingenuity. And that is difficult for many of us to grapple with.

11. For some, the response to all of this has been anger. People are angry – rightfully so – at what they view as unscrupulous behavior by the big AI companies. They are angry at the casualness with which these companies toy with the livelihood of mathematicians and hype up their systems’ accomplishments while simultaneously downplaying the role of the mathematicians who paved the way to these breakthroughs. They are angry that our students are using LLMs to circumvent the struggle and friction which build the mental toughness needed to persist through hard problems. They are angry about the massive energy usage of data centers and the lust for money and power that appears to be driving many of the decisions being made by companies such as OpenAI, Anthropic, Google, and Meta.

12. For others, the response has been grief. My colleague Josephine Yu wrote in a recent group chat (I asked her permission to quote this): “When I read math papers, I marvel at the human brains that came up with these beautiful ideas, and I’m thankful that my brain and their brains can communicate these abstract thoughts through the language of mathematics.  When I’m doing research math, I feel I’m in community and in conversation with many people who came before us, being a part of a slowly unfolding story, of humans developing mathematics over hundreds and thousands of years.  Sadly, now the conversation is going to be dominated by machines talking to machines… You may say it doesn’t matter, a theorem is a theorem no matter who proved it. I understand the point of view, but AI-generated math doesn’t bring the same level of awe or joy for me. If my favorite band is releasing a new album with AI-generated songs, would I buy it?  I wouldn’t, but I understand that many people would… Grad students need time and space to struggle. I am sad that AI is burning the ladders for the next generation.  I’m sad my daughter will likely not have the space to struggle with and learn mathematics the way we did.”

13. And for some, the response has been to try to look at the bright side. My response to Josephine’s message was: “Josephine is focusing primarily on mourning the loss of what we all love about mathematics, and I think that’s really important. Things are happening so quickly that it’s difficult to find the space to grieve what this is going to cost the discipline of mathematics. And there really is much to grieve… It may be that my focus on trying to find the good in all of this is really just a coping mechanism — I’ll need to ask my therapist about that and get back to you. But part of what’s happening in my mind is that the issue on the table is unfortunately NOT whether we think this is a good tradeoff for mathematics as a whole. Perhaps all of us would say no, but that’s not really the point. AI is here. It’s happening. It’s not going away. So we are going to have to learn to adapt to it. I’m personally trying to focus on the adaptation part rather than the feeling sad part, but I wholeheartedly agree that we need to make space for appreciating what we’re losing.”

14. And, to be clear, I am also both angry and in mourning – in addition to feeling excited about all these new tools revolutionizing mathematics.

15. Take the recent announcement by OpenAI that they have solved the Navier-Stokes problem. I’m angry that they may have acted unethically – perhaps even illegally – in scooping a rival team. (I don’t want to get into the particulars of this debate, however, or to take sides, as I don’t know all the relevant facts and it seems too early to reach a conclusion.) Even if OpenAI is not guilty of academic misconduct in this instance, I’m angry that they are cavalierly racing ahead with developing technologies which they themselves characterize as extremely risky. The same day the Navier-Stokes solution was announced, an Anthropic employee (who had previously worked at OpenAI) specializing in AI safety quit the company, warning that “the two AI companies are more focused on beating each other and global competitors in developing the most advanced model possible than they are on safety… [they] are racing straight to self-improving superintelligence and gambling with our lives.” He warned that AI is advancing so quickly he believes there is a greater than 10% chance it “could kill all humans” within the next decade, and apparently this is not an isolated view among AI safety experts who work at these companies.

16. I’m also in mourning for the mathematical culture that we’re now in danger of losing, for all the reasons Josephine so eloquently described. Conquering one of the seven grand millennium challenges should feel like a triumph, but instead it feels to me like an indescribable loss. This is in part because a number of humans who contributed key ideas to the resolution of Navier-Stokes (especially Diego Córdoba and Luis Martínez-Zoroa) are not even mentioned in much of the recent flurry of news coverage (for example this NY Times article). But also because the proof which OpenAI published was the result of millions of dollars worth of “compute” on internal models not accessible outside of the company – an effort and expense which they seemingly undertook only to establish marketplace dominance over their commercial rivals. This is not the idyllic vision of mathematics which attracted my colleagues and I to devote our lives to the subject. And it raises the specter of a future in which the ability to do cutting-edge research is dictated by access to expensive computing resources whose availability and distribution will almost surely be highly inequitable.

There’s also the following point which Terry Tao recently made: “In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field through the efforts to solve such problems, and then to digest any partial or complete solutions that emerge for further insights… Prematurely solving the problem by purely AI-powered methods—particularly without full transparency into the solution process—can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.”

17. Yet, despite all of this, I remain cautiously optimistic. I mean, it would definitely suck if AI kills all humans within the next decade. There’s also all the other reasons mentioned above – as well as lots of reasons I haven’t even gotten to here – to be fearful of and sad about the change this technology will bring. But mathematics will continue to be a crucial component of progress in science. Fantastic new mathematics is going to emerge, and we will still want – and hopefully even need – humans to understand and explain it. Mathematicians will continue to do mathematics, even if computers can do it better, for the same reason that humans still play chess and watch Jeopardy! And for a while, at least, human mathematicians will remain an important component of the mathematics research process, even if our role transitions over time to being facilitators and curators of knowledge, rather than the primary creators of it.

18. In the short term, there’s a lot we mathematicians can and must do. We must, for example, resist the pop culture characterization of mathematics as now “solved”. I have dozens of problems in my notebooks that I’d like to solve which AI (at least the publicly available models) cannot currently crack. And if those problems are solved, it will be relatively easy for me to come up with new ones that are even harder. There will always be mathematical problems which even the best AI system in the world cannot solve (this is, in some sense, an actual theorem of Gödel). But more importantly, solving hard problems is only a small part of what mathematicians actually do. We also find good definitions. We create new mathematical theories. We build bridges between different subjects. We look for proofs that we, as humans, find elegant and aesthetically pleasing.

19. And, of course, we teach real-life students. Particularly in peril at the moment are young mathematicians-in-training – graduate students, in particular – who are being blitzed with a new reality that is confusing, frightening, and perhaps not what they thought they were signing up for. We must do everything in our power to help them flourish and give them wise counsel. And it’s no secret that most universities employ lots of mathematicians primarily because lots of undergraduate students take math classes. This will hopefully continue to be the case, perhaps even more so, in the age of AI. And I would argue that the people best suited to teach mathematics will continue to be the ones who devote their lives to studying it and chronicling the vast array of knowledge that’s been built up over the millennia. That being said, it seems quite likely that good teaching and good exposition will be more highly valued than before, and that research prowess alone is not going to cut it in the age of AI. This, I humbly suggest, could end up being one of the positive outcomes of the AI revolution.

20. Let me conclude on an optimistic note, of sorts, with a short poem by the American writer Wendell Berry (who died just nine days ago):

“It may be that when we no longer know what to do
we have come to our real work,
and that when we no longer know which way to go
we have come to our real journey.
The mind that is not baffled is not employed.
The impeded stream is the one that sings.”