SERIES · As application deadlines approach, Proofs and Prompts is publishing a series of posts from 19 to 23 October for people considering a PhD, doing a PhD or looking for a postdoc: guidelines, discussions and speculation about what to do and what careers in mathematics will look like. This post is one of them. Each is written independently and reflects only its authors’ views. All posts in the series
This is a response to a recent post on Proofs and Prompts titled “Who will become the new mathematicians of Math 2.0?” written by an Anonymous PhD student (who I will refer to as “Anonymous” below). I would urge particularly all tenured and tenure-track mathematicians to read it because there are many students that think the same way, and are planning to leave academia as a consequence.
I recently wrote a guest piece on Terry Tao’s blog (“What should we tell our students?”) where I warned that we are ‘about to lose an entire generation of mathematicians’ so I feel a duty to reply to Anonymous’ post, to try to offer some counterpoints to the arguments expressed therein. More importantly, I feel a duty to talk to my students about these issues, and I have shared Anonymous’ post with them, so that we can have a discussion. I hope my colleagues in the mathematical community reach out to their students as well.
In order to respond to Anonymous, I will quote some key sentences, and then elaborate with my own thoughts.
I recently heard a mathematician say that it is “paradoxical” to not want to study a field that is undergoing a scientific revolution in reference to October 6th. This mindset completely misses the point of why I do mathematics, and I suspect this is true for many other graduate students and even other established mathematicians.
I want to start here because I believe I may be one of the mathematicians that they are referring to. In a video discussing my post on Tao’s blog, I said
“And I think it would be in fact paradoxical that people would not want to join a field of study during a scientific revolution where we’re seeing this incredible acceleration.”
I will come back to the ‘paradoxical’ remark later on, but I want to first mention what I said right after that quote:
“It is somewhat chaotic. It seems somewhat confusing to be a mathematician right now. But we are trying to figure out together what should the math landscape look like in a year and in five years and how to navigate… how we can use LLMs in ethical ways and in a way that is going to construct solid mathematics along the way and in a way that the mathematical field is going to thrive.”
Let me elaborate on that. Historic upheavals of the status quo are usually chaotic while they happen, and in many cases many of those living through such times have been trampled and mistreated (with such dramatic and relevant examples as the industrial revolution). It is the duty of those in positions of power or security to fight for the rights and well-being of those who may be more likely to suffer through a revolution. In our case, we are living through what can be considered a technological revolution that has great potential to accelerate research and progress and, at the same time, it has a terrifying civilization-ending non-zero potential. Unfortunately, trying to pretend that the revolution is not happening, or trying to preserve the status quo, only delays the inevitable, and prevents communities from fully preparing and reinforcing their core values, so that their professions can survive said revolution.
If we limit our conversation to mathematics and the mathematical community, the LLMs have already revolutionized research mathematics, with the proposed resolution of several landmark conjectures, such as Navier-Stokes or the quasi Riemann hypothesis. This is undeniable and, simply put, we must learn how to coexist with this new technology. But they are also an existential threat to the way mathematics works, and the way the mathematical community operates. So it is absolutely normal and expected to be excited about the research advances, and also extremely concerned about the ways it may change our field, as Anonymous very eloquently points out.
In particular, while the OAI drop of October 6th is certainly of “historic” proportions, it clearly also has had a disruptive effect of historic proportions. Whether the global effect of the LLMs will be a net positive will be determined much later in the future. But it is clear that the “arms race” of frontier model companies to develop ever more capable mathematical models is a corporate publicity stunt and, very early on, even before the Fields medalists warned of a misalignment in an open letter, I asked mathematicians not to be complicit in such an arms race.
At the end of my video about the Oct 6 drop, I encouraged people to stay in the field to make changes from within the community. It is up to us in the profession, and more so up to the newer generations, to lead the way to ethical and sustainable research practices that will allow mathematics to prosper. How to do so is to be decided, so all of Anonymous points are well taken. Whatever we do, we need to make sure students do not abandon mathematics because they think it has become unrecognizable. But the key question is: who is in charge of making these changes and laying out a path forward? I claim that no one is in charge, and we all are. Precisely those that see the field driving towards a cliff, have a great role to play within the community to warn about such dangers! It is a chaotic time, the future is uncertain, and this is disconcerting, but I think there are many reasons to be optimistic and to continue in the mathematical profession.
Now let me backtrack in Anonymous’ post to the very first line:
To the mathematicians shaping the future of our profession,
The mathematical community is not monolithic, and this is a feature, not a bug, of the community. There is no single person, group, or entity that dictates what mathematicians do or how they should operate. In particular, Terry Tao is not a spokesperson for all of mathematics and, of course, there are many out there that do not agree with his views and his predictions for the future of mathematics. Personally, I highly value and respect many of the things he has said, and I disagree with others. I single out Tao here because Anonymous does, but we could similarly agree or disagree on Tsimerman’s or Duminil-Copin’s points… or any other non-Fields medalist point of view, from one of the many recent blog posts on the matter of AI.
Right now, there are many different groups trying to pull mathematics in very different directions. For example, while some mathematicians are completely embracing LLMs, other mathematicians completely refuse to use LLMs in their work. Some of us are choosing the middle ground, and we use LLMs with caution for certain tasks, we avoid them for other more intellectual tasks, while respecting the decisions of some of our colleagues (I include students in this category) who cherish them, or do not wish to use them at all.
When students choose graduate programs, and PhD advisors, many of us encourage them to ask lots of questions (to the staff, to faculty members, to students, to friends) about the atmosphere in the department, and their practices, and about the personalities of the potential advisors. Similarly, I would encourage students who are thinking of a career in mathematics to reach out to their current or prospective programs and advisors, and enquire about their stance on AI/LLM use. Does it match the student’s own attitude towards AI, or is it antithetical to their ideals? Since the community is far from monolithic, there are plenty of departments and potential advisors that will be aligned with whatever views the student holds, and it would be wise to choose programs based on that type of alignment, if the issue of AI is key in the decision making.
I love technical work. I love spending days, weeks, months working on a lemma or an idea without success. […] As AI has now become highly capable of performing this kind of work, I see shockingly little from the mathematics community about how this kind of crisis in education will be addressed.
There was a recent summit on “PhD Math Education in the Age of AI” and the recommendations for graduate programs can be found here. TPSE is putting together a panel discussion on the Future of Graduate Education in Mathematics. Many departments are having internal discussions on how to approach these issues. As I said above, the future is uncertain but we are trying to think of the best possibilities in education as well as in research.
What is the goal of a graduate program in mathematics? How do we define success in a PhD program? In my opinion, the PhD is an apprenticeship, and success is measured by whether the candidate has been trained and is proficient in all aspects of the mathematical profession. At the end of their degree, several mathematicians write letters of recommendation. A (good) rec letter or set of letters evaluates the PhD candidate’s aptitude in all of these aspects: research potential, teaching abilities, service to the community, potential as a colleague, etc. And then colleagues in other institutions read these letters and determine whether the candidate will be a good match for the research and/or teaching needs of their department. Many of the conversations surrounding AI concentrate on the research aspects, but there is a lot more to being a mathematician than just research.
On the subject of research aptitude, will letters in the future refer to the fantastic LLM skills of the candidate? Will we compare students and say such and such student is in the top 2% of LLM-prompters that we have ever met? Will we say wonderful things about their slop-paper output in our letters? Will we praise the many slop-papers that they have been able to de-slopify? I do not think so – we will still need to have a way to measure their research potential based on their own capability of thinking deeply about problems, and coming up with innovative ways to attack interesting problems. It is absolutely true, as Anonymous points out, that students need to spend long hours thinking deeply about a problem, and I do not think LLMs should change that. So PhD advisors should encourage students not to use LLMs for certain tasks such as proof generation of key lemmas, because these are the tasks where intellectual development actually happens (see the Specific Recommendations in the section on Research and AI in the recommendations from the Summit). In a recent conference I was at, someone suggested to grad students to always be working on at least one project that is completely LLM-free.
As a result, I do not think anyone should or will rob the students from their long hours working on a lemma by “pure thought” because most of us recognize the irreplaceable value of the time spent on that activity. It is precisely the outcome of those thinking sessions that sets students apart at the time of evaluating their intellectual and research ability in research letters.
On the other hand, if an advisor demands results with a quick turnaround, and expects that the student will access an LLM for help to be able to move on – against the wishes of the student – then it is time to choose a different advisor. The ideal scenario is that the student will ask the advisor for help or a hint on how to proceed, even if an LLM could provide such ideas, because those conversations are actually also part of the apprenticeship: students learn how to collaborate with other mathematicians through those interactions. Plus the advisor may have a much better intuition than the LLM does anyway.
Many celebrate the possibility that AI will relieve them of the burden of difficult calculations, technical lemmas, and other supposedly tedious aspects of mathematical research.
I do not think that is a common sentiment. For the most part, we all value the time spent in understanding the technical difficulties of key lemmas. On the other hand, LLMs can be fantastic tools in helping us write code and run numerical experiments. I am not great at coding, and I find that the LLMs are allowing me to run computations that I only dreamed possible. It would have been very tedious for me to learn how to write code in Python, html, javascript, etc., but now I am able to create code in all those languages, run experiments, and later verify the results mathematically, all in hours instead of weeks (or instead of never!).
Terence Tao has proposed a vision of a new way of doing mathematics, Math 2.0, where the emphasis moves away from solving open problems and toward a more holistic approach. […] But I feel like I am asking the obvious question that has not been addressed: What if I do not want to spend my professional life doing these activities?
Two points here. One (1), some of us have always had a holistic approach to a PhD: I have always emphasized to my students that I expect them to become well-rounded mathematicians, not just great researchers, and that means that I try to push them not just to write a great thesis, but to be better instructors, better members of the community, better at networking, better colleagues, better at giving talks at all levels, etc. There is nothing wrong with a holistic approach, and in fact it makes candidates more appealing to a broader range of colleges and universities at the time of applying for jobs.
And two (2), many do not agree with Tao’s vision either, and that is ok. In my research, I want to solve problems. The coolest the problem the better. I do want to think about a problem and come up with an ingenious solution, and then tell the world about it. I do not want to inherit a slop-solution to a problem that a company dropped in a repository, and try to make sense of it. I would not want to do that, and I would not expect a student to want to do that either, so I would not ask a student to do that as part of their duties to complete a PhD. I simply do not think we will move in the direction of “moving away from solving problems.”
However, I have thought of problems that an LLM could solve, prompted them, and now I have LLM-prepared drafts on problems I care about. Instead of posting these papers directly to Hexagonmath.org, say, my current strategy is to introduce students to research by explaining the problem I am interested in, explaining why others are also interested in the same problem, discussing the idea that led me to prompt an LLM, and sharing the paper with the student. Then use that paper and that topic as a thread for an independent study on the particular area, so that the student learns enough background to understand the motivation for the paper, and the techniques that are used to solve the problem. Later, we will read together the LLM draft, understand the proof, and rewrite it in a way we understand and can make sense of it. While we think about the proofs, I hope we come up with ways to extend the arguments to strengthen the result, or apply the techniques to other similar problems. And finally polish the paper so that we can submit it to a journal, so that the student has an early experience in writing a research paper, and so that they have published a paper early on to talk about during research conferences. A paper they understand and they can feel is their own, because they have contributed to make it their own.
Will this strategy work? I do not know! I am just trying things out, honestly, and I am listening to any other mathematicians who are trying out different strategies to coexist with the new technology.
I do not expect to follow this early-LLM-draft protocol for a thesis problem, however. It would only be appropriate just as some first steps into research.
There is a problem with the proposed future of mathematics, and it is more concerning than my personal dissatisfaction. How exactly do you expect to train the next generation of mathematicians? You know how to formulate new research directions. Where did these skills come from? I doubt they were developed by reading successful proofs or listening to an explanation of an idea.
As a matter of fact, I do not think intuition and mathematical taste is learned by thinking deeply about a problem for hours on end. I have learned much more attending lectures and conferences, and talking to other mathematicians during lunches and dinners. I have had my best ideas reading other people’s papers and proofs. As an anecdote, early on in my career I signed up to write brief reviews for MathSciNet. I am not sure exactly why, but I was assigned to read and review several articles by Bo-Hae Im (some solo papers, and some co-authored by Im and others), and after reading the third or fourth paper by Im dealing with a conjecture of Larsen, I had an idea of how to prove a related result. I wrote to Im and we ended up collaborating on a research article that eventually was published by the Journal of the London Mathematical Society (and it had a significant impact in getting tenure).
As I mentioned above, we should make an explicit effort to allow students to think deeply about problems, but we should also make it clear that students should be reading papers and proofs. And whether we like it or not, some of the proofs dropped by OAI contain some interesting arguments that we should absolutely study and think about! A particularly valuable example is the proof of the non-existence of Siegel zeros, which several mathematicians (Banks, Kerr, Green, Barreto) have improved on, and now it seems the papers have converged to a remarkably simple argument that many of us would highly benefit from understanding.
Back to the question on how to train the next generation of mathematicians… I am not sure we have the answers yet, but I do not think it will look dramatically different from what we do now. There is a new tool in town, yes, but the guiding principles should be the same: train students to be well-rounded mathematicians.
If the future consists of interpreting AI generated results, organizing knowledge, and directing automated research, why should I remain in academia?
The way this question is phrased, it assumes that the worst, most dystopian possible scenario will come to pass, where AI can solve any single problem humans could solve, and that we would reach a point where only machines do math research. As I mentioned in my piece in Tao’s blog, the future of LLM capabilities is completely unclear. There are many ways in which LLMs could hit a limit in mathematics, for example from the theoretical or feasibility points of view. The frontier model companies are driving the narrative and trying to convince us all that their products are already super human when this is far from the truth. I wish mathematicians had a multi-million PR team behind them to build praise and support for our fantastic research record!
The reality is that many of us have started using LLMs to aid our research, as a tool or as a collaborator, and it turns out that it can be very exciting and fulfilling to do so. As one of the best recent examples of LLM use I know of, one of my colleagues (Kyu-Hwan Lee) joined a team of mathematicians to prove that the sporadic simple group M_23 is Galois over Q. Their team succeeded with the help of LLMs in a task that many had tried and failed, and made some really interesting progress in the inverse Galois problem. Here is an excerpt from their declaration of AI use:
“No text in this article was written by AI. Our work relies on longstanding domain knowledge and techniques for computing curves of positive genus [KMSV14]. The key strategic decisions and the conceptual analysis for this project were made by the human team.
AI served as a tool that accelerated a large part of our work, particularly the computations. We used Claude Fable 5 and 5.1, Claude Opus 4.8, and ChatGPT 5.6 Sol for searching the literature, generating code, checking results, and proofreading. The workflow that eventually succeeded was a tightly controlled collaboration in which the computational work was much faster with AI, while the human collaborators continued to direct the research and verify the results. AI systems increased our confidence in tackling difficult problems and helped us communicate more efficiently across our different areas of expertise.”
I find this result fantastic (the paper is a gem by the way), and I think it exactly illustrates the way senior mathematicians (Lee, Poonen, Pries) can use LLMs to accelerate research and understanding, and include graduate students (Shaowu Zhang) and postdocs (Xiaoyu Huang) in the process!
Mathematics is never going to be the same and I recognize that the way I did research and found fulfillment from it are no longer here.
Mathematics will be different, yes, but I firmly believe it will be similar to what we do now, in all the best ways. As a final example, very recently three mathematicians (Elkies, Klagsbrun, and Li) have found an elliptic curve of rank 32. I find this discovery astonishing and wildly exciting. In a private communication, Zev Klagsbrun mentioned to me
“It would be a mistake to view this as a triumph of humans over AI. In fact, it’s a great example of humans using AI to push the boundary of what is known.”
I could not agree more. Immediately after their curve was found, my student Dylan Costa and I started working on finding a quadratic twist of their curve with rank bigger or equal to 3 (in order to find a quadratic field over which their curve achieves rank at least 35), and with the help of LLMs (in coding and literature search) we succeeded to find one, conditional to BSD/Parity. Both Dylan and I found the work on this little contribution (certainly little compared to the giant discovery of the rank 32 curve) to be exhilarating. It would not have been possible just a few months ago without the help of LLMs. This is why I would find it paradoxical to leave mathematics now.
I am not asking you to agree with my personal preferences.
I just want to conclude by thanking Anonymous for writing such a great piece that should make everyone in math think long and hard about the future of the profession. Thank you. And a final message just to Anonymous: I would love to record an interview with you and post it in my channel. Or send me an email message if you want to chat. Bye and best of luck!
Received 22 October 2026.
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