Proofs and Prompts

A beginning for mathematics

Daniel Litt, professor at the University of Toronto

Three years ago, AI systems could not reliably add two numbers. A year ago, internal models at OpenAI and DeepMind received the equivalent of a gold-medal score on the IMO. Now, these systems are autonomously resolving major open questions. It’s hard to imagine this trend continuing for another year, but I expect it will. It is clear that this will require a radical rethinking of our profession.

A few weeks ago, I gave a talk titled The End of Mathematics. If you only read the title1, you might guess that this talk was about how, soon, AI will “solve” math. That’s not what it was about. The talk instead laid out a gloomy vision of the future, in which, despite the possibility of AI systems that are robustly superhuman at mathematics, the design of our institutions causes human understanding of mathematics, and possibly even mathematical progress in the abstract, to stall. I think we will avoid this future, but I also think it is plausibly the default if academic mathematics does not adapt. Despite my relative enthusiasm for the use of AI to do mathematics, I share this view with many of its detractors.

Here I want to lay out, instead, a positive vision of the future of mathematics, and the human practice of mathematics. I claim we can deepen human understanding even as the production of interesting mathematics becomes less dependent on it.

This essay will take as a premise that AI systems that are robustly superhuman at most or all aspects of mathematics will be here soon. But the concrete changes to our institutions I propose only require accepting the weaker premise that the production of mathematical text is becoming increasingly disconnected from mathematical understanding.

What are we even trying to do here?

I think it has now become clear that there is no consensus in the mathematical community as to what our goals are. Some of us want to solve problems; some of us think of mathematics as play or as poetry. For some: “Wir müssen wissen –\text{\textendash} wir werden wissen.”2 Some of us think we are penetrating the mysteries of the platonic realm. Some of us think the goal is to embody love of and understanding of mathematics,3 and to transmit that love and understanding to the next generation. 

My personal, if self-referential, answers are:

  1. We’re trying to produce and understand high quality mathematics.
  2. We’re trying to produce high quality mathematicians.

These goals should be construed broadly. What high quality mathematics consists of has changed quite dramatically over time; we come to its definition as a community. We are not just training PhD students to do research in mathematics. A substantial part of our job, though perhaps an underemphasized one, is to educate the general public about high quality mathematics and mathematical thinking.4

Whatever our goals are, we’ve operationalized them primarily through proving theorems. Almost all papers or PhD theses have a main theorem, and ostensibly a proof of it. But it should be clear that the goal of mathematics is not to prove theorems; if it was, it would be trivial to automate. A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules to them, with no attention whatsoever paid to their meaning. It has had particular significance when a theorem resolves an open problem, especially one that has resisted substantial effort. Again this is easily automated; our computer or monkey can simply conjecture all mathematical propositions in alphabetical order.

The general attitude of our community towards a technology that can prove theorems and solve open problems suggests that these operationalizations of our values are at best incomplete.

The prospect of automating mathematics by enumerating all conjectures, and all proofs of ZFC, is probably not so disturbing to you. But let us for a moment assume the computer or monkey is very smart; perhaps it understands the results it is proving, and writes beautiful expositions thereof. Perhaps it has a good sense of what we find interesting, and is primarily focusing on those questions. Perhaps it has, in the course of enumerating theorems of ZFC, answered many of our most pressing open questions, and is asking many more fundamental open questions. Is there still a need for human mathematicians?

I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers. In fact I think we are at the beginning of an incredible, wonderful explosion of mathematics, and if we value human understanding, there will be more need for human mathematicians than ever before. But the profession will have to change.

In the course of this change, we will have to decide what to hold on to and what to throw away. Some things I would like to preserve: learning seminars; serendipitous conversations that spark an idea; students knocking on a professor’s door to chat about math. A robust community learning exciting new mathematics. Thousands of people that, together, slowly start to resolve their confusion.

I worry that much of what has been written on this topic, including some of my own past writing, focuses too much on trying to preserve the precise shape of the institutions of academic mathematics, rather than our values. How can we preserve the journal and peer review system?5 How can we protect the arXiv? How can we keep our role as gatekeepers? If you have internalized the fact that existing AI systems can produce relatively high quality results for the marginal cost of a few dollars, the idea that any semblance of the current equilibrium can survive what’s coming is absurd.

As we try to find a new equilibrium, we could try to chase the edge of model capabilities. Right now AI systems arguably underperform us at theory-building, asking questions, exposition, …\ldots so we could prioritize and reward those skills. I think this is unwise: compare the speed at which the academy adapts to the speed at which model capabilities improve. We need to consider the endgame. If the models remain incapable in some domain, we can adjust later.

Before I propose some relatively concrete steps we can take, let me remark on what we’re trying to protect mathematics from. There is a lot of anger at AI labs, and certain individuals at those labs. But whatever our judgment of the labs, we need a plan that does not depend on AI capabilities disappearing. The basic issue is not the labs’ behavior, ethical or not.6 It’s the technology itself. I think there is some belief that the labs will “move on” from math next year, be nationalized or broken up, or that a financial bubble will pop, somehow returning things to normal, or…\ldots But there is no way our institutions can survive unchanged when anyone with a laptop and a few hundred dollars can generate what would have been an Annals paper last year. AI does not care if you are anti-AI.

Producing high-quality mathematicians

The most urgent question our profession needs to answer right now is: what should our students be doing? It’s now possible to produce a PhD thesis one hasn’t even read; in terms of demonstrating understanding, mathematical text is worth the paper it is printed on.7 The text no longer reliably conveys a signal about the person who produced it.

In my view we should welcome interesting mathematical results regardless of provenance. But our institutions have historically relied on the same signal to indicate both mathematical progress and mathematical expertise. These now must be distinguished.

I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied. While we might require the topic to be original, its provenance—\text{\textemdash}AI or not—\text{\textemdash}is irrelevant.8

How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently. The student would be trained to ask interesting questions and try to resolve them, by whatever means. To keep students on track, there might be regular meetings in which the student is asked to independently work through an unfamiliar example, apply a technique in a new case, etc.

The allocative aspects of our job (hiring, graduate admissions, etc.) are in dire need of reform if we want to retain human mathematical expertise. Broadly speaking I think we should focus on rewarding skill in the parts of our jobs that cannot be automated: the internal (e.g. understanding mathematics) and social-relational parts, and operationalizations that hew as closely to those aspects of the profession as possible. For example, talks and sustained mathematical discussion now demonstrate understanding much better than papers. Once AI systems improve at exposition and “digestion,” this will be even more the case. We already interview faculty hires; we must now do the same for graduate admissions.

I think we should try to foster a robust seminar culture in which speakers are expected to explain their topic to the audience’s satisfaction. Much has been written recently (by myself among others) about the fact that we are primarily interested in understanding, not merely the truth value of mathematical statements. If that is the case, let us make sure we actually understand each other.

Right now the use of AI systems to do mathematics above some minimum bar relies on the fact that our community has produced many open conjectures, whose interest is evidenced by the existence of human mathematicians who care about them.9 The recent importance of this fact suggests to me our community plays a very important function that we have, arguably, underrated: namely, figuring out what is interesting. It is not entirely clear to me how to operationalize this, but one possibility might be to reward the construction of research programs (either with help from AI systems or otherwise) that persuade others of their worthiness.

To be clear, I am not saying that AI systems will not be able to ask interesting questions, make interesting conjectures, pursue interesting programs, and so on. I think they most likely will, resulting in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications. But others will primarily be of interest because they tell us something fundamental about basic mathematical objects, and accrue value only if we can and do engage with them. It seems to me that it will be up to us to build a community of researchers to do so, and we should reward mathematicians who do. And even if the AI is asking excellent questions, there is no reason to think it will ask the same questions we would. 

All of these changes are oriented towards increasing the amount we talk to each other about mathematics. It seems to me that this would be positive even in a world with no AI.

I think there is room in this world both for mathematicians who, like me, are enthusiastic about AI, and for those who do not use it. But as the models begin to produce huge quantities of mathematics, it will not be possible to avoid their outputs entirely.

Producing high-quality mathematics

As we think about how to reshape our profession, it’s important to understand that, whether one likes it or not,10 it’s impossible to stop people, amateur or professional, from pushing a button to produce mathematics. The idea that we will persuade people not to play around with math, or that we will be able to “reserve” problems for graduate students, is just not realistic.11 And we shouldn’t want to do this!

There is now more interest in math than at any other time in history. We should be ecstatic for mathematics’s sake, even as we are concerned about mathematicians and mathematical expertise. And by and large, the value of this button-pressing comes from the mathematical community. If a conjecture falls in the woods and no one is around to hear it, who cares?12 For the abundance of new mathematics to have value outside application, we will need an abundance of new mathematicians. And for results with applications, we will want people to be capable of understanding their assumptions and consequences.

I wrote above that solving problems and resolving open conjectures is an incomplete operationalization of our values. But nonetheless it is important to solve problems and resolve conjectures! The provenance of such solutions only matters insofar as it intersects with the existing structure of the profession (incentives, prestige, and so on). It is obvious that structure needs to change in any case.  

Mathematics used to be the cheapest of the sciences. I think the biggest change we are facing is that now, some portion of our questions will be answerable via a cash injection. I know some of my colleagues find this distressing. Previously those questions might have brought together a research community, led to interesting auxiliary developments, and so on. This contingent progress may now no longer occur.

But don’t you believe in mathematics!? There will always be more to learn. If a basic question can be resolved for the cost13 of a nice dinner, we should be delighted. But that’s only the beginning. We will ask what the answer explains, and what it helps us understand. It will lead to many more new questions, some of which can in turn be resolved for the cost of a nice dinner, and others which renew our confusion and lead to the development of a research community.

Our industrious new helpers will be churning out an unbelievable amount of math, pursuing our interests or perhaps their own. We will have our own questions, and confusions; sometimes they will be resolved by the models, and sometimes they won’t. Sometimes the answers will be complicated, and we’ll devote a learning seminar to them. Sometimes progress will be minimal, but the question itself will be so motivating it gives rise to a research community.

A student will be confused. They will knock on their professor’s door. Maybe the two of them will ask a model for help, or maybe not, but first they might spend some time at the blackboard thinking through the question. And the model might give them a beautiful explanation, but we all know that’s not enough; no one can understand mathematics for us. We have got to do the work. 

There is so much more to learn—\text{\textemdash}an infinite amount. We’ve always been at the beginning, and we always will be. 

Acknowledgments

I am grateful for comments from Mohammed Abouzaid, alz, Boaz Barak, Frank Calegari, Ben Church, Jennifer Cutler, doomslide, Elden Elmanto, Francesco Fournier-Facio, Tony Feng, Dan Freed, Peli Grietzer, Michael Groechenig, Stephanie Koh, Joshua Lam, Mark Sellke, Ravi Vakil, and Amal Vayalinkal.

  1. I regret choosing this title. ↩︎
  2. Hilbert’s full opinion is as relevant today as ever: ‘We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be Wir müssen wissen – wir werden wissen (“We must know –\text{\textendash} we will know”).’ ↩︎
  3. I owe this phrasing to Peli Grietzer. ↩︎
  4. Note that this list consists mostly of internal and social-relational functions (understanding, coming to a determination of what’s interesting, training, and so on). This is in contrast to our operationalizations (proving theorems, solving problems, etc.). ↩︎
  5. This system was already close to breaking before AI; it is overdue for radical reform. ↩︎
  6. Obviously some of it has not been ethical. But even if every lab had behaved perfectly, the capabilities of AI systems would still force us to radically adapt our institutions. ↩︎
  7. Which is not to say the text is necessarily uninteresting. ↩︎
  8. This is a practical necessity. There is no way to enforce restrictions on provenance, and attempting to do so will only create incentives to conceal use of AI. But I find it unlikely that someone whose only contribution was to push a button, and who did not engage deeply with the material, would be able to pass a rigorous defense. ↩︎
  9. To be clear, many open conjectures are less interesting than one might have hoped, post hoc, and are generally not an end in themselves. They are often meant to measure our failure to understand some object, but they are sometimes resolved without improving that understanding.  ↩︎
  10. On balance, I think I like it, though I am sometimes annoyed to find slop PDFs in my inbox. It took me some time to understand that these PDFs expressed a need for understanding; a person elicited them, often without being able to meaningfully engage with their contents, and needed to know that someone could engage, and that someone cared. ↩︎
  11. That we cannot reserve a problem for a graduate student does not mean we can’t give them the opportunity to work on it. This is compatible with the reconceptualization of a PhD outlined previously. ↩︎
  12. Some have suggested that interest in using AI to answer mathematical questions may soon fade. It is hard for me to see how this will happen as long as questions we care about remain unanswered. ↩︎
  13.  By this I mean marginal cost. Michael Groechenig points out to me that it is unclear that we should directly compare the cost of a machine proving a theorem to the cost of a human doing so, as the products of this work are arguably different. Only one of them produces understanding and expertise in a human being, which I think we might value independent of the result itself. ↩︎

Crossposted from my blog.

Received 9 September 2026.

68 responses to “A beginning for mathematics”

  1. JS Avatar
    JS

    I feel like when I read people excited about AI I get way more cynical about math compared to when I read people who are pessimistic about AI. I think this was well written and completely reasonable but weirdly it makes me not want to do math.

    Reply
    1. Anonymous Avatar
      Anonymous

      I agree the text is reasonable and yet I can’t say I feel delighted about the fact that the money for a nice dinner could produce a solution to an interesting math problem. I would rather just go have a nice dinner and ponder the problem the next morning.

      Reply
      1. JS Avatar
        JS

        Yeah, I think one of the frustrating things about reading a lot of these pro ai people is they say things like “people derive joy from math for a lot different reasons!” and then present a “hopeful” view of math that precludes the joy for the majority of people who don’t hold their view. They aren’t wrong but it’s just very depressing.

        Reply
        1. Sukessh Velusamy Avatar
          Sukessh Velusamy

          I don’t think that Litt’s point is that everybody will think his suggested way of doing math is ‘better’ than the current way, just that there is no possible way to maintain the current way of doing math, and his new system does have some improvements over the current system. One day, the mathematicians of the future might not even be able to imagine a world without superhuman math AI, the way many chess players today can’t imagine a world without chess engines.

          Reply
  2. Pádraig Daly Avatar
    Pádraig Daly

    Inspiring and credible vision!
    One worry is that it sounds a bit like mathematicians becoming akin to poets and critics of poetry. Which there’s nothing wrong with but how much funding would there be? Before we could claim there is some use for pure research as it leads to unexpected applications in computer science, cryptography, physics, engineering etc. With an AI that solves any problem they can just go straight to application without funding the “useless” mathematics. Especially with the AI solutions costing a fair bit of money this is an issue.

    I suppose this is an different issue than the one the post discusses as it’s more about how our society and economy is organised, and what we deem to be worthwhile activities.

    Reply
    1. MS Avatar
      MS

      This is a very important point (perhaps THE most important point), and I think most of the commentary regarding the impact of AI on mathematics has failed to address it. All of the discussion regarding how to train and assess mathematicians is pointless if the public decides not to fund mathematics. We have to do a much better job advocating for the continued existence of professional human mathematicians. A large proportion of the general public already thought academics should not be publicly funded, and it is surely becoming a much larger proportion every hour of every day.

      Reply
      1. a Avatar
        a

        Presenting mathematics as “beauty, joy, etc.” will only make people more cynical. Instead, its prospects (if any?) for applications (e.g. AI safety?) could/should be emphasized as it always has been.

        Reply
        1. Sukessh Velusamy Avatar
          Sukessh Velusamy

          I think mathematicians should emphasize there is no way advancements in math from a chatbot could ever be applied unless there is a community of people who can understand the math (unless the chatbot itself can do the application, but that would probably be AGI anyway).

          Reply
      2. BCollas Avatar
        BCollas

        I couldn’t agree more (see my reply to Pádraig Daly above) — in UK 2023: math ~£495 billion ($670.0 billion) gross value added.

        An essay that (hopefully) doesn’t fail to address the issue: https://collas.perso.math.cnrs.fr/math-and-ai.html

        Education, structure, economy and producing sciences 🙏

        Reply
    2. a Avatar
      a

      Indeed, if there is no prospects for applications, one should expect less funding. Then this beginning for math will be to end up as some kind of poetry.

      Reply
    3. BCollas Avatar
      BCollas

      Agreed, it is excellent point and it must be part of the discussion.

      As a fact, in the UK, for 2023, mathematical sciences are credited with £495 billion ($670.0 billion) gross value added and 4.2 million jobs, so 13% of UK employment (similar figure for France).

      With the new AI-Math, mathematics now audits a trillion dollars AI industry — classical music audits no one

      Reply
  3. Sash Avatar
    Sash

    > it’s impossible to stop people, amateur or professional from pushing a button to produce mathematics.

    This is not true, you can just ask the frontier model companies to refuse answering research mathematical questions. They already do that for cyber security, bioweapons, copy righted music and art, I don’t see why we can’t add some areas of maths to it if we want to.

    This is not like invention of calculator or even a chess engine. It costs millions to even acquire hardware to run a frontier model if you had their weights and it costs billions to train said frontier model. These costs are still going up, I reckon we won’t see GPT 3 (a model from 5 years ago) running on our smart phones in my lifetime. It still can’t run on my beefy 5k Desktop PC.

    In the present we live in, it is totally an option to prevent AI from destroying some areas of mathematics if there is political will, a move I strongly support.

    If anything this will create a great randomized control trial. Let’s have some areas of mathematics with AI allowed and some with no AI and then 5 years later we can compare both the fields and judge what’s better for the health of mathematics and its mathematicians.

    Reply
    1. Benjamin Andersson Avatar
      Benjamin Andersson

      Seems to me that nothing prevents people from downloading a local verison of the model on their computer that does not contain these guardrails. Compare with banning pirated material online, or drugs.

      Reply
    2. Irreverant Avatar
      Irreverant

      Qwen 3.8 is 5 times smaller than GPT-3, orders of magnitude smarter, and can definitely run on your PC, unless you mean 5K from 10 years ago ….

      Reply
    3. Sukessh Velusamy Avatar
      Sukessh Velusamy

      Two problems with your argument:

      1. No government is going to ban chatbots from doing math:
      Even for cybersecurity and bioweapons AFAIK there is no law forcing these companies to avoid answering those questions, and answers in these areas could lead to a lot of real world harm! How, then, is anyone going to convince Congress of the grave harms of chatbots solving math problems? And even if you managed, mathematics and AI are global. I really don’t think it’d be possible to convince even America to prevent their models from answering math problems, let alone every nation on Earth.

      2. This is very much like the invention of the chess engine:
      According to Deep Blue project manager C. J. Tan, the system’s hardware cost at least $2.5M dollars in 1997, or $5.2M today. 30 years later, it would be crushed by Stockfish running on an iPhone. You may not be able to run GPT-3 on your PC, but you can run (highly quantized) GLM 5.3 Flash on the most expensive Macbook Pro, which is WAY better. In fact, Qwen 3.8-27B is better than GPT-3, and much smaller. Considering all of this happened in ~6 years, do you really think local models will remain behind for long?

      Reply
    4. sash Avatar
      sash

      Response to my replies.

      The AI companies would like you to believe that this is going to be ubiquitous and omnipresent but last I checked, the Chess Engine did not involve a trillion dollar investment building massive data centers. The fact that they are building these data centers shows that they themselves don’t believe you will be able to run these models on your home PC’s anytime soon. Let me give a few comments.

      1. To do research level maths, you probably need at least 1 Trillion Parameter Models, Qwen 3.8 and any sub 100B parameter models will never do anything useful in maths and can be ignored.

      2. This technology will continue to remain highly centralized in the near future (at least a decade). Again you can just look at how the companies are behaving, there would not be a 100 Billion + investment in model companies if they did not believe that these technologies will remain highly centralized for the model companies to reap a profit.

      3. I don’t want to go into this, because it is not my technical specialty, but Moores law has stopped a while ago. GPUs are not on that curve, there is not much miniaturization left, and there is no reason to believe we will ever be able to run a 100B model on a smartphone. CPUs have stopped getting faster a while back. What is left is building more and more parallel compute, maybe price of compute will go down but it will probably always require a container of GPUs to run a trillion parameter model for a single person.

      To mathematicians, this centralization is a huge advantage. Forcing the model companies to post train their models to not solve mathematical research problems is only a political problem and is a clean solution. I think mathematicians should bring this up, because as Hugo said, if Mathematicians delegate their creativity to the machine, what resistance do you think other intellectual disciplines will put up?

      Reply
      1. Irreverant Avatar
        Irreverant

        Also you’re right that the frontier almost by definition will be dominated by big players, but all I see is a growing pie

        > The fact that they are building these data centers shows that they themselves don’t believe you will be able to run these models on your home PC’s anytime soon. Let me give a few comments.

        Keep in mind the data centres are meant to serve millions of people and have the added benefit of batching efficiencies, these are not mutually exclusive facts at all.

        Also the models being run on local hardware is largely a matter of priorities and politics. A company like Qwen could have chose to never release small models, yet they did …
        It’s also very likely that OAI’s faster models like Luna are in fact small, they just don’t want to share them.

        Also NVIDIA sells entire devices for local inference https://marketplace.nvidia.com/en-us/enterprise/personal-ai-supercomputers/dgx-spark/

        (yes I’ve said “also” a lot lol, didn’t want to number for the sake of making it seem like I was referencing your numbers)

        Some nitpicks :
        CPU’s single threaded performance is still improving https://www.cpubenchmark.net/single-thread/, I think you meant clock speed.
        Also ASICs in this space can dramatically improve the speed https://chatjimmy.ai/ (try this)

        Reply
  4. Marcin Kotowski Avatar
    Marcin Kotowski

    “anyone with a laptop and a few hundred dollars can generate what would have been an Annals paper last year. ” – isn’t this a hyperbole? I mean, no idea what capabilities will be available a year from now, but today a laptop and a Pro subscription are not enough to churn out Annals-level papers.

    Reply
    1. Sukessh Velusamy Avatar
      Sukessh Velusamy

      Maybe he means GPT 6 Astra is at the level of capability where it could have generated the Unit Distance Conjecture paper, which Timothy Gowers said he “would have recommended acceptance [to the Annals] without any hesitation.”

      Reply
  5. Vincent again Avatar
    Vincent again

    What if not only proving the conjectures, but deciding what is interesting, what is a deep notion, a fruitful point of view or theory is better done by the push of a button?

    What if it becomes a common experience for a mathematician that she think about a question, and then, when prompting an AI model succintly about said question, she find a wiser, deeper and better formulated prose than hers?

    I understand this may sound too naive to actually happen. Yet this is what the current trend suggests for the near future if you open your eyes a bit. At least it should be treated as a plausible development, something to consider.

    And while this hypothetical situation still leaves room for math to be done for the beauty and pleasure of it (maybe for more people and in more quantity than before,who knows?), and AI math to be read by humans in order to understand it, it would drastically change what math is by removing a lot of initiative from the activity.

    If you prefer, imagine that all serious mathematics is like what physics (or whatever) is to you currently. You can follow some discoveries made by real physicists (powerful AI in this analogy) and enjoy them, but there is little sense in you contributing to them, and your activity as a “physicist” can be at most that of a science journalist.

    Reply
    1. Marcin Kotowski Avatar
      Marcin Kotowski

      How the hell did suddenly an AI model become “she”?

      Reply
      1. Vincent again Avatar
        Vincent again

        I meant “she find[s]” as in “the mathematician finds on her screen” ^^

        Reply
    2. Phil T Avatar
      Phil T

      If you read Daniel’s essay, that is the situation he believes will occur shortly.

      Reply
      1. Vincent again Avatar
        Vincent again

        From what I understand, Daniel sees human initiative enduring in the future of mathematics, a (mysterious) mystery of AI-produced math seamlessly slipping into the place of the mystery of the mathematical universe.

        That’s different from taking seriously the drastic shift in the system of incentives that makes confused students knocking on their professor’s door possible in the first place.

        I understand one wants to give the idea of mathematicians becoming science journalists a more comforting appearance. I am mainly reacting to this because I wish mathematicians would take AI risk more seriously, and comfortable narratives don’t seem to be conducive to that, though I may be wrong.

        Reply
        1. Phil T Avatar
          Phil T

          I agree that aspects of this proposal are unsatisfying (at least to me) and don’t fully address the implications of this world. Just pointing out that this is the explicit premise.

          Reply
          1. Vincent again Avatar
            Vincent again

            I see, thanks for the comments, indeed I should have removed half of my post.

            Reply
  6. taken_aback Avatar
    taken_aback

    Hi Daniel,

    It is no secret that many people are not happy with your so called ‘AI-realist’ views on twitter/X, including me.
    I think you are a bit too confident in ignoring various aspects of human agency, in particular, mathematicians agency in controlling their future. However, that is not what concerns me most (to each their own opinion).

    What concerns me is why you attended a secret summit at OAI in early August to discuss the ‘future’ of mathematics.

    I have seen you compare that summit with Oberwolfach for example, but note that Oberwolfach is a summit *for* mathematicians and it is **strictly** not a summit on the so called ‘future of mathematics’, a topic in which no one person can comment on.

    What exactly made you think that you were qualified to represent all of us (your twitter/X micro-celebrity status notwithstanding)? If you knew this was a secret summit, why not work to invite people with opposing views and of equal or more academic reputation as you?

    Reply
    1. Jake Levinson Avatar
      Jake Levinson

      I think this accusatory comment is totally inappropriate, darkly insinuating that “many people” aren’t happy with his “so-called” views. At the moment, the mathematics community does not have a unified view, so it is no secret that *anyone* with clearly articulated views is in disagreement with some fraction of our community.

      As for accusing him of some kind of secret conspiracy: Daniel posted his “secret” talk on the internet, the same day he gave it!

      Yes, Litt is indeed not “qualified” to “represent” all mathematicians — what would it would even mean, to be qualified in that way? Neither is Terry Tao, who despite his obvious stature is in no way the spokesperson, President or Pope of mathematics. Nor are any number of other mathematicians qualified to speak for all of us. But we are all entitled to comment on the future of AI in mathematics, with all our viewpoints.

      Litt is simply a mathematician who has thought and written about AI in math, and has done so in the public sphere. That seems like a fine justification to invite him (along with dozens of other people) to an event about the future of AI in math, and for him to accept the invitation.

      Reply
  7. anon Avatar
    anon

    If in fact AI systems become more capable than human mathematicians at the majority of their cognitive work, isn’t this whole discussion just of minuscule importance compared to the broader implications for society / knowledge workers?

    Reply
  8. just different Avatar
    just different

    > A substantial part of our job, though perhaps an underemphasized one, is to educate the general public about high quality mathematics and mathematical thinking.

    Here’s where the money is, literally. All of the high-minded pronouncements about mathematical understanding sound like navel-gazing copium to an awful lot of people. It won’t matter if the field sorts itself out and adapts to AI if the general public assumes it’s all pointless bullshit anyway and that we’re just among the many who are getting displaced.

    BTW, I wish we had a less off-putting term than “digestion” (although I understand the metaphor Tao was getting at). Is the coinage “synegesis” (by analogy with “exegesis”) any better?

    Reply
  9. Jules Verne Avatar
    Jules Verne

    I would rather quit math than subscribe to this utterly dystopic vision of mathematics that you launder as a utopia just to secure funding. Screw this.

    Reply
    1. sukessh velusamy Avatar
      sukessh velusamy

      I am not a mathematician, but I do like math. I’ve learned most of the more advanced math myself, by reading textbooks, and I find that enjoyable, even if I didn’t come up with the proof myself. To me, it wouldn’t really matter if the proof I was learning was written by another human, or by an AI, as long as it was, for lack of a better word, ‘enlightening’. Would it really be so bad if top mathematicians did the same thing as me (at a much higher level, of course)?

      Reply
      1. Sash Avatar
        Sash

        Well said, I too would rather quit. And frankly if this is what happens to maths, then they should stop calling it “mathematics” anymore. It’s not like a sweatshop employee who built clothes calls himself a tailor? It’s laundering the name of a noble tradition for something completely different and frankly much worse !

        Reply
    2. Pierre Menard, Author of the Quixote Avatar
      Pierre Menard, Author of the Quixote

      Same. I am seriously considering quitting my postdoc; the sloppers certainly will not miss me. The OP, too, reads like slop: the “new” PhD “concept” is nothing more than a fancy repackaging of what is common practice in oral examinations at the bachelor’s and master’s level, except that instead of reading Hörmander’s “The Analysis of Linear Partial Differential Operators” one would be forced to eat the slop and enjoy it. How innovative. I am very much looking forward to a future in which PhD students are tasked with pushing a couple of buttons and then parroting back what they have learnt from the machine.

      Reply
      1. Jules Verne Avatar
        Jules Verne

        Clever username. Quite appropriate given the stated vision of professional mathematics in the post.

        Reply
  10. BlaineTheMono Avatar
    BlaineTheMono

    >> “it’s important to understand that, whether one likes it or not, it’s impossible to stop people, amateur or professional, from pushing a button to produce mathematics. The idea that we will persuade people not to play around with math, or that we will be able to “reserve” problems for graduate students, is just not realistic. And we shouldn’t want to do this!”

    Hi Daniel! I need to push back on this. Under the current status quo in AI use in mathematics, not being able to stop people from pushing the button means that the days of speaking on work in progress at a conference are over. How could one feel safe announcing unpublished results in the abstract of a talk, knowing that there *will* be people, whether colleagues or strangers on the internet, prompting the hell out of it possibly before the talk even takes place?

    I don’t have an answer to this problem, but I do think it’s a serious issue. Wanting to prevent the morons to push the button is not just reasonable. It’s a necessary consition for trust and free communication within our specialized mathematical subcommunities.

    Reply
    1. Sukessh Velusamy Avatar
      Sukessh Velusamy

      I think Daniel explains the solution well enough, stop assigning prestige to being the first person to prove something, and to slop proofs which leave you more confused after reading than before. Instead, assign prestige to people who have demonstrated a deep understanding of the mathematics and the capability to explain it well. This is much more feasible than banning the entire population of the world from using chatbots.

      Reply
  11. Jonathan Noel Avatar
    Jonathan Noel

    I agree. Our existing systems of training, credit, and assessment in math will require serious changes. They were flawed and we should not mourn their loss. We should redesign them both to address the new issues posed by AI and to reduce existing bias, unfairness, and other shortcomings. It will be a heck of a lot of work and we will make mistakes, but the biggest mistake would be try to keep everything the same.

    Reply
  12. Novum Organum Avatar
    Novum Organum

    This is the first essay I’ve read from inside the profession that takes robustly superhuman AI as a premise and then asks what mathematics should become, rather than what should be defended. That’s why the title is right: a beginning, not an end.

    And the golden era is going to be much bigger than mathematics. Math is the canary: it’s the cheapest, most closed-loop discipline, and its bottleneck was pure human-years. Once that bottleneck is priced in dollars and tokens instead of decades, the same dynamic extends to every pure and theoretical science and to engineering. There will be more math done than at any point in history — especially applied math: whole classes of problems that used to be cost-prohibitive, including the systems problems sovereign programs actually care about (rigorous verification of the software and hardware we build, the math of AI models themselves, control, logistics, materials). The “cash injection” line is the key sentence in the post. It used to be that a question either had a research community attached to it or it never got answered; now any question worth a datacenter afternoon can be attacked. And when AI starts running experiments — with humanoid labor close behind — the same cash-injection logic reaches all of empirical science. Math is the preview.

    Which is exactly why there’s no reason to fear a stall in mathematical progress. The supply of open problems is effectively infinite — by your own closing words, there is “so much more to learn — an infinite amount” — and solving problems generates the next questions. The worry that the labs are “mining non-renewable problems” treats the map as if it were the territory. The engine is just finally fast enough to explore it.

    What changes is the profession — the same way software engineering has already changed. In SWE, the shallow work — boilerplate, routine features, straightforward implementation — has simply disappeared, and the bar for being a competent hire went up, not down: you now have to understand more, faster, in order to steer and verify what the machine produces. The same is coming to math. Shallow math (routine lemmas, bookkeeping, standard formalization, literature navigation) goes to the machines; deep math (conjecturing, framing, understanding, synthesis, teaching) survives and becomes worth more. Mathematicians will always be needed — you’re right that an abundance of new mathematics requires an abundance of new mathematicians to digest it — but the skill level required to be a competent research mathematician is going sky-high. People who can’t make that jump won’t be “replaced”; they just won’t be in the profession. That’s fine.

    So the institutions must change, and the shape of the change you sketch is the right one: the rigorous defense that actually signals understanding, provenance irrelevant; seminars where the speaker has to teach the audience; research programs rewarded, because deciding what is interesting is now the scarce skill; interviews for admissions. Versions of the university that can’t do this will be disbanded — or bypassed by registries, labs, national programs, and events like the Mathathon that do. The 2027 “slot machine” is a real risk, but it’s an incentive problem with exactly this kind of fix, not a capability problem. One more thing worth naming: the duplication we’re seeing right now (three groups proving Feige’s conjecture independently) looks like waste, but it’s just transition noise — like bitcoin mining. The marginal value of the first solution collapses, while the information it produces (which problem classes are solvable, at what cost) is precisely what re-prices the field and points human effort at what matters.

    To the trad-math apologists, including the very good ones: you will be bulldozed by the advance, and most of you will know it was right the moment it happens. That is not the end of mathematics. It’s the beginning.

    Reply
    1. just different Avatar
      just different

      > research programs rewarded, because deciding what is interesting is now the scarce skill

      The missing link here is that at present, someone’s credibility about what is interesting is proportional to how many “interesting” theorems they’ve already proven. It’s not at all clear how that sort of status will be reorganized.

      Reply
    2. Yemon Choi Avatar
      Yemon Choi

      Was it really necessary to get an LLM to write this for you? At the very least, you could ask your machine to trim it down and remove the attempted rhetorical flourishes that just make those of us with English as a first language think of people like Boris Johnson.

      Reply
      1. MM Avatar
        MM

        A highly practiced LLM user isn’t capable of much else.

        Reply
      2. Novum Organum Avatar
        Novum Organum

        You could ask me the same question, and nobody could answer it — which is precisely the point. “This smells like a machine” is the exact test I argued is breaking down, and you just used it to skip the content rather than engage with it. I predicted the reply in this thread would be aimed at the messenger rather than the message; it’s now playing out in real time. The essay’s own argument is that provenance is irrelevant to whether an idea is true or useful. You did the opposite move. The content is still standing — feel free to address it.

        Reply
        1. Morgan Thompson Avatar
          Morgan Thompson

          Why would someone engage with the content when you can’t even be bothered to engage with it yourself?

          Even if what you posted wasn’t generated by an LLM, the point would still stand. If you don’t care enough about what you’re writing to write with brevity and purpose… Why should anyone else care about what you have to say?

          Reply
  13. Marcin Kotowski Avatar
    Marcin Kotowski

    The problem of all such optimistic future visions is they assume that young people will want to enter the profession in its new incarnation. I’m not so sure they will. Imagine you’re a bright and curious 20-year old – would you rather join a profession that in a couple of years may decline or be reduced to exegesis of machine content, or do something else? The prestige of mathematics has already suffered a serious blow. When young people choose alternative, a field dwindles and dies.

    Reply
    1. MS Avatar
      MS

      I fear the same. It all seems so hopeless. The only question that remains is what will kill mathematics first: (1) a complete collapse of bright young people who wish to become mathematicians, or (2) the government and general public deciding that there is no longer any need to fund professional mathematicians. Both seem inevitable. When (2) happens, mathematics will all but die out — it cannot survive as a hobby for people in their spare time, and there will be nobody left interested in prompting the machine and understanding its output.

      Reply
    2. sukessh velusamy Avatar
      sukessh velusamy

      I don’t really see any other option. Would the average person really hold mathematicians in high regard if they tried to ignore/avoid all proofs written with the help of AI, and if all the conjectures they are working on could easily be solved by the chatbot on his phone?

      Reply
      1. BlaineTheMono Avatar
        BlaineTheMono

        Well many average persons right now are supporting real human artists against the rise of AI generated “art”. Mathematics does not hold the same place because the average person isn’t quite as informed about it. If we want to keep our departments funded and don’t want to spend our careers checking AI slop, we definitely will have to go talk to the person on the street and explain to them what exactly we do. But if a whole community of people doesn’t want to frame AI “art” on their walls or read AI “novels”, I don’t think they’d be so hot about AI math either.

        I think the average person hates AI much more than the average mathematician does.

        Reply
  14. Back to the fields Avatar
    Back to the fields

    Seconding MS’s comments.

    “Hiring of people should be based on requirements that cannot be automated, ie, which require people to do them.” Yes, that is how all people have always been hired to every honest job in history.

    No one who you have ever thought of as remotely “strong” would be interested in doing a “PhD” in regurgitating A.I. output to an audience that would surely eventually come to always include at least one A.I., and would eventually consist entirely of “mathematicians” who had never done anything themselves, but all certify (to each other, no else cares) they all have great “taste” and are “very strong understanders.” And not even a really thoroughly set of referee reports, just a talk! (There will be no one at these departments who really knows what a referee report is, or who could write one independently. Well they course they totally could, but they won’t. Or come back in three minutes, they’ll have ten written for you).

    You’d never have joined something like this as a student. There’s no exploration, there’s no risk of failure, there’s no room to excel, and there’s no mystery. And there’s no safety net, because if there are still employers, they will not value this the way finance and tech used to value us. And no one else needs it, either. The physicists will never again be curious about us. This is just such an obviously worthless and artificial activity to devote one’s life to. It will attract the dregs of the dregs if attracts anyone at all.

    And no, it is nothing like being a literature department (all in big financial trouble, as we all know), because those people are the best at critiquing literature. And whatever it is they do, it is not yet coping over scraps, like our alleged leaders think is the best we have to look forward to.

    Possibly humanity’s oldest or second oldest intellectual endeavour and we fold in the space of two months. If this is the best we can do, we deserve what’s coming.

    The feeling of having utterly wasted my entire adult life up to now, on a scale unequaled by all but few other knowledge workers so far, is quite shameful, though. There is a real feeling of betraying the enormous sacrifices of one’s ancestors. Skip my parents’ generation, and it’s full circle back to the fields.

    Reply
    1. Daniel Litt Avatar
      Daniel Litt

      Why would you join something like this? Well, it might be because you’re interested in math, and would like to understand some. Personally I find myself using AI quite a bit, but then also putting it away to think through things on my own. I see no reason this practice can’t continue.

      Reply
      1. Back to the fields Avatar
        Back to the fields

        For people from before AI, interacting with ai can offer the residual experience, the simulation of parts of what our professional lives were before, back when they existed.

        But I think this proposal presents something very different to the generation that would actually become these “students.” I don’t believe “understanding” is really on offer to them here. We all know what it means when a certain type of student says “I understand the lectures but can’t solve the exercises.” For a the first generation, there is some check to this in terms of professionally chatting at tea time with pre-ai people. But this dies after one generation at most, and many of the pre-ai people will have left to the ai sinecures they are now preparing for themselves.

        I think the students will be aware of this too. Will they really believe that just passively and selfishly “understanding” something that no one else even ever needs explained (as you point out. MO activity has cratered! We don’t even seem to need each other!) can be a profession, or be perceived as a profession. You are describing what everyone will see as a hobby: chatting to LLMs and then never doing anything with the output.

        As a test: everything we claim to enjoy, computer programmers claim to enjoy, and clearly did. But CS applications to Ontario universities are down 50% from 2022. Why aren’t the students lining up for the overwhelming pleasure of having code explained to them and then parroting back they were told?

        Society will not go along with this vision and will be right not to.

        I promise I am not a bad, angry person. I am just not really, or not at all, able to deal with this destruction of our reality in the space of two months. Or with our coming expulsion from the middle class, just as climate change gets really really bad. Everyone who has contributed an essay here is doing a better job of handling it than I am. Thank you, all of you.

        Reply
        1. Grigori Avramidi Avatar
          Grigori Avramidi

          A friend of mine likes to say that the main reason society should fund math is to give clever people something to do and keep them from getting into trouble (it is not a universally held opinion, but he went to moscow state and is from a generation in which a lot of people went the other way, so he has examples). I suspect if math grows as stale as you fear, society will look for other ways to keep clever people out of trouble. They will figure out stuff to do with their mind, and there is no reason to artificially try to funnel them into math. (I read an interview with serre once, and he insisted that one should discourage more, so that the people who end in math are the ones who really want to be there for some reason…) To be honest, math has been getting stale for a long while, maybe that is how we got the ai revolution to begin with … We do live in interesting times, but calling it a curse seems a bit cliche.

          Cheers.

          Reply
          1. Back to the fields Avatar
            Back to the fields

            Moscow State in the 70s is precisely the culture that will be achieved by the proposal of the essay, yes. Or, maybe even more relevant to this proposal, the culture of Moscow State in the 20-teens: totally irrelevant, because not much was going on, but persisting in some zombie-state.

            Reply
            1. Grigori Avramidi Avatar
              Grigori Avramidi

              This would be Moscow state ca. 1990, with some of my friend’s friends becoming criminals. ”Do you mean ‘businessmen’?” “No, honest criminals.”

        2. Nilima Nigam Avatar
          Nilima Nigam

          Perhaps I am reading more into what you write than you intended. But I share your sense of loss.

          There are to parallels of this vision of a future mathematics, in art. There are relatively few well-renumerated positions for students to develop as apprentices of artists, eventually becoming artists. There are perhaps more positions in related areas [art criticism, art history, art pedagogy, etc.] These are valuable in their own right, and we could do worse as mathematicians than becoming familiar with the structure and nature of MFA/doctoral programs in the arts at our own universities. Then there are the programs teaching art-in-application, say for advertising, graphic design, etc. But humans like creation and admire the artist instinctively more than the critic. We know Frida Kahlo’s name, not Andre Breton.

          To me, the natural parallel of valuing and assessing ‘mathematical understanding’ is that of art criticism. You have to be familiar with techniques and the corpus enough to be able to say something meaningful about a piece of art created or generated. You can have competing interpretations of what the art is saying. I imagine art critics have views on interpreting or critiquing AI-generated art, but that’s a different comment.

          Of course people pursue art still. It is very accessible. In fact, because it is so accessible, a taxpayer may wonder: what is the point of supporting the endeavour of understanding art? We actually see the outcome of this.

          If you asked me to predict: based on what mathematicians are saying of their field today, what will math programs look like 10 years from now? I’d probably end up predicting something like an MFA.
          Almost certainly we’ll have to give up any pretence of ‘objectivity’: mathematical statements are true or false. The quality of an explanation does not follow the law of the excluded middle.

          You make a very good point about the speed of this transition [dictated by the pace of the tech market]. Maybe our profession deserves the demise of its current incarnation, if it falls apart in the span of 3 months. For my part, I refuse to cede my imprefect cognitive agency, or to make rapid changes on someone else’s timelines. I’m slow, and prefer to incorporate tools (or not) at a deliberative pace.

          Reply
    2. just different Avatar
      just different

      Finance and tech never valued us. They valued the one aspect of our skills that was also the easiest to automate. If you think that’s the only worthwhile skill that mathematical training imparts, maybe you really did waste your entire adult life.

      Reply
      1. Back to the fields Avatar
        Back to the fields

        The aspects of our skills that are easiest to automate turned out to be: proving theorems, writing good papers, and soon by the admission of the author himself, building theories. This plus giving decent talks were the core of the profession up until three months ago and were universally agreed upon as what really mattered. You certainly had to be good at least one of these things to convert mathematical training into any kind of paid work, academic or otherwise (in the “otherwise,” case, proving theorems was indeed a proxy, for work which has now also dried up).

        People who have not lived in corrupt settings: these substantial objective and honest criteria are a big part of why our community is/was so nice to belong to! There are alternatives! They are worse, and you’ll come to learn all about them now!).

        Now, we are supposed to all forget what what we knew 90 days ago and pretend that chatting at tea time was always the core of the profession. This is just not respectable, and people are not going to respect it.

        Reply
        1. just different Avatar
          just different

          So….any mathematical activity that isn’t specifically theorem-proving is “corrupt” and beneath contempt? There was absolutely no deeper motivation for caring about those theorems in the first place, other than providing an “honest” signal to someone else? Really?

          And if you think criteria in mathematics were ever objective, do I have news for you. “Objective” is not a synonym for “valid and reasonably well-defined.” Daniel Litt’s proposals do point the way towards valid and well-defined evaluative criteria. Maybe you personally won’t be suited to whatever form professional mathematics eventually takes, but I’m pretty confident there isn’t going to be a central committee checking your party membership.

          Reply
  15. John Goodrick Avatar
    John Goodrick

    “If a basic question can be resolved for the cost of a nice dinner, we should be delighted.”

    Well, it depends on who is on the other side of this transaction! I would be happy to treat a friend and co-author to dinner for helping me with a nice result. But should we really be “delighted” to give money to rent-seeking corporations whom we have reason to believe care about neither mathematics nor mathematicians, and who have already accumulated more wealth than the annual budget of the National Science Foundation?

    I have no doubt our math community could survive LLMs, if it were just that. I’m more pessimistic about it being suddenly flooded by Silicon Valley venture capital money and how that will warp its priorities and values.

    Reply
    1. BCollas Avatar
      BCollas

      Agreed, we should not be delighted to give money for producing science-ersatz and depleting the pipe that produces the proper one.

      A proposal? New interface-structures where compute and academia meet.

      We already have some successful models (see Lean FRO and the Mathlib Initiative) and the economy holds.

      Reply
  16. Pol van Hoften Avatar
    Pol van Hoften

    Fantastic and thought provoking essay!

    Reply
  17. Dominique Manchon Avatar
    Dominique Manchon

    Two (very) critical recent viewpoints on the recent Navier-Stokes episode can be found here

    https://rogueesr.fr/un-moment-dabjection/

    and there

    https://rogueesr.fr/un-resultat-sans-chemin/

    (in French).

    Reply
  18. Anonymous Avatar
    Anonymous

    Hi, I am a young student and aspiring mathematician.

    1) On the one hand it sounds like the author says prioritizing exposition is unwise, but most of the article boils down to prioritizing exposition more (seminar culture, knocking on professors door, talks for graduate admissions etc). Any insight on what is meant by this portion in relation to the main points:

    “Right now AI systems arguably underperform us at theory-building, asking questions, exposition, … so we could prioritize and reward those skills. I think this is unwise“

    2) I have noticed the young people in my math circle mainly want to go into ai or robotics. Few want to be mathematicians. I read the comment that math should become more applied to attract funding. That could be, but I wonder if that happens would that just end up sustaining more AI and EECS which are consumers rather than producers of math, rather than sustaining more math. Math is already a service/teaching field doesn’t an applied pivot overweight that position more?

    3) it seems whatever weight problem solving gets now it will be reduced to make an increased weight on seminars and exposition (and possibly covering more ground with AI). Isn’t that moving more towards generalists? My understanding is that human performance is usually spikey (spike in problem solving, or exposition but doing both at the top level is rare). Moving towards generalists might be an anti pattern in academia

    4) at the end of the day as a young mathematician, I am trying to reverse engineer the tastes of future tenure committees and their downstream impact shapes the incentives at every other level. I love exposition, but to a young person it just doesn’t sound credible of a message to say you will be able to get tenure if you get good at exposition. There is no evidence to suggest that. This is high minded but strikes me as speculative if taken as career advice for young people.

    5) Culture is set top down. We get fields letters post NS but we actually get little math exposition. Why aren’t fields leaders publishing exposition pieces like videos weekly? When pressure tested the actions and culture set from the top seems to not value exposition very much.

    6) why is poetry considered bad in this context as an analogy to math. I get that funding is scarce for poetry but funding aside isn’t math research already a kind of poetry: a search for the truth or signal hidden in complex noisy world? I just never understood the poetry framing used in math

    I would love to get insights on these! As I said I am student here to learn so any feedback is a gift!

    Reply
    1. Daniel Litt Avatar
      Daniel Litt

      These are good questions! I definitely do not think we should prioritize exposition in itself; as I wrote, I think the models will arguably be better than us at this soon enough. What I think we should do is prioritize activities that actually force us to engage with and understand stuff. These typically involve talking to other people, so have some expositional component.

      Reply
  19. Novum Organum Avatar
    Novum Organum

    If you want to see the “beginning” already happening in practice, watch this: Javier Gomez-Serrano (Brown), Harvard CMSA, Sept 11 — “New Math + AI Developments in Fluid Mechanics.” A fluid dynamicist dissects, in one sitting, the newly announced AI-assisted resolution of the Navier–Stokes problem, the week it dropped. URL: https://cmsa.fas.harvard.edu/event/navierstokes/

    Three things make it worth your attention. First, the workflow it demonstrates is exactly the division of labor this essay describes. The machine proposed a two-scale, three-layer construction that a specialist who has worked on this problem for decades says he had never personally thought of — and his job was to digest it, place it in the 90-year lineage (Leray, BKM, Escauriaza–Seregin–Šverák, Tao’s averaged model, Buckmaster–Vicol, Córdoba–Martínez–Zoroa), interpret the mechanism, and point it at the next direction. And the loop is closed on both ends: he prepared this very talk with a frontier model helping him digest the proof and build the slides — “without the assistance of AI, I wouldn’t have been able to prepare these slides in such a short notice.” So this isn’t a two-track “machine finds, human explains” setup; it’s one integrated workflow where both the human and the model are inside the loop. The result also sits squarely on top of decades of human groundwork — it’s a synthesis, not a substitute.

    Second, it defuses the scoop drama with actual precision: the independent Monday papers prove blow-up for forced Euler, Boussinesq, and IPM — the same program, different equations, no Navier–Stokes claim. Different problems, not stolen ones.

    Third, his “message to OpenAI” (5:20) is the most constructive document in this entire month of shouting. Not “stop doing math,” but a spec for how the collaboration should work: rewrite the paper so it’s comprehensible, with advice from field experts; release the full process — prompts, transcripts, the Lean project with its statement audit; cite generously and early; give the community time, compute, and access, not only announcements; and if you actually want to accelerate mathematical research, more researchers need access to state-of-the-art tools. That’s the norm-codification this essay calls for, written by someone inside the process, in one slide.

    And the Q&A answers the question everyone keeps asking — “if NS is solved, what’s left?” Plenty: turbulence, where even the definition is unclear; the unforced problem, which is technically a different universe; the stability of the construction itself. One blow-up construction generates a list of follow-ups that will occupy the field for years.

    You’re right that an abundance of new mathematics requires an abundance of new mathematicians to absorb it. This talk is what that abundance looks like from the inside — and what it means is we can now do in months what used to take decades. The direction of travel is set. It’s up to us to define the culture, explore it, and codify the norms, while the understanding is still being built. That’s not the end of mathematics. That’s the beginning.

    Reply
    1. Dominique Manchon Avatar
      Dominique Manchon

      A less idyllic (and not anonymous) point of view on this story: today’s official statement of the Société Mathématique de France can be found here :

      https://smf.emath.fr/actualites-smf/position-bureau-smf-navier-stokes-et-openai

      That’s not the end of the story. That’s the beginning.

      Reply
  20. Douglas Silva Avatar
    Douglas Silva

    What I find contradictory is this: we have computer scientists like Leslie Lamport, who have made progress in the field using the same modus operandi as mathematicians. Even though computers and advanced technology already existed, he revolutionized computing by writing algorithms on paper and constructing formal proofs. There are even artificial intelligence researchers who work this way and AI isn’t exclusively about LLMs. I don’t understand why pure mathematics should embrace this approach when even computer science research an applied field doesn’t fully embrace it.

    Reply
  21. Saikat Maity Avatar
    Saikat Maity

    I think the important question is not only whether AI can solve mathematical problems, but what it means for humans to understand mathematics when AI can produce solutions so easily.

    Mathematics is also about intuition, asking meaningful questions, finding connections, and understanding why a result matters. Perhaps AI will not end mathematics, but change what it means to do mathematics.

    Reply

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