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Why I didn't sign the Fields medallists' letter

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Why I didn’t sign the Fields medallists’ letter

This post has been cross-posted to [Terence Tao’s blog.]

When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem statement, as well as by the accompanying story, and made a fairly serious attempt to prove it. And while, unsurprisingly, I failed, I learned a lot from the attempt. Blissfully ignorant of the fact that the case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.

Since I still couldn’t really see where to start, I decided to simplify the problem further and concentrate on successive differences of cubes, with a view to showing that such a difference could not itself be a cube. At the time I did not know how to express what I was doing in algebraic language, so I did not explicitly try to prove that the Diophantine equation had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with th powers and keep taking successive differences, then eventually you get to the constant sequence .

Somehow I never managed to turn this observation into a proof of Fermat’s Last Theorem, and later on my dream of solving it got replaced by other mathematical dreams. However, when I reached the point in my mathematical education where I was taught about taking difference sequences and about what happened to polynomials, I understood those topics much better than I would have if I had not discovered difference sequences for myself and spent happy hours playing around with them. I mention this story just as an illustration of the phenomenon that was strongly emphasized in this letter signed by 25 Fields medallists, that one learns a lot from thinking about a problem, regardless of whether one solves it.

In the end, however, I felt that I could not sign the letter, despite agreeing with much of what it said. Instead, it seemed better to do what I did with the Leiden Declaration and set out my own position in a blog post. But it should be understood that by doing that I am not setting myself up as a member of some opposing camp: indeed one of my worries at the moment is that the mathematical community might become bitterly divided, something I would very much like to avoid. Also, I agree on the fundamental point that we are facing a crisis: I just want to offer a slightly different analysis of what that crisis is. I don’t claim full originality for this analysis, as I know that several other mathematicians have already put forward thoughts that are similar to the ones I have, though (for what it’s worth) I have largely come to these conclusions independently.

On the subject of independence, it will perhaps help if I clarify that while I have contacts in the mathematics group at OpenAI, and have also been given early access to some of their models (typically only a few days before they have been released), and have been given free access to their Pro models once released, I have never been paid by OpenAI. I mention this in the hope, perhaps naive, that what I write will not be dismissed for ad hominem reasons. Another potential reason for my being regarded as “pro-AI” is that, as I have stated publicly several times, I have a group in Cambridge devoted to automatic theorem proving. However, that is actually more of a reason to be anti-AI, since our group has been trying to attack the problem of getting computers to prove interesting theorems by understanding as well as possible how humans prove interesting theorems, so now that LLMs can clearly do it without the help of such insights as we have had, one of the main motivations for our work has disappeared. To put it another way, we have had to swallow the bitter lesson (which of course we were always aware was a distinct possibility, even if the speed at which it happened has taken us by surprise). I do in fact think that it is still a very interesting and valuable intellectual exercise to try to gain this understanding, even if we can use LLMs as black boxes, but that’s a topic for another blog post.

So why didn’t I sign the letter? Let me extract a couple of sentences from it that express what I see as the principal argument being put forward.

But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.

Perhaps the main reason I didn’t sign is that I don’t fully subscribe to this view. Instead, I have a more complicated view, which I actually expressed in my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding. At one end of the spectrum you have mathematicians who are primarily motivated by the wish to solve problems, who see conceptual understanding as a very important means to that end. At the other you have mathematicians who are primarily motivated by the wish to attain conceptual understanding, who see problem-solving as a very important means to that end. I worry that the severe-misalignment letter could be seen as saying that the “right” attitude is to focus on conceptual understanding as the main priority — indeed, the above sentences say that more or less directly. But I think that there are mathematicians all across the spectrum, and that that is a good thing (or perhaps I should say that it has been a good thing up to now — the future is much less certain), and I don’t want to suggest to a large fraction of mathematicians, including myself, that their mathematical temperament is somehow “wrong”.

My own particular mathematical attitude is very similar to one that was beautifully articulated in a Twitter post by Jacob Tsimerman (another non-signatory of the letter), which, now that I look at it, says a lot of what I will be saying here. And that post in turn is a response to Daniel Litt, who is in my opinion one of the wisest commentators on mathematics and AI. His views are expressed in a later post here, which I deliberately didn’t read until finishing this one, and then found, as I expected, that there was significant overlap. I would also like to take this opportunity to recommend an excellent post by Noah Smith entitled The End of the Age of Heroes, in case you haven’t read it.

I have been talking so far about individual mathematical understanding, but I suspect that what concerns most of the signatories is less that than the collective understanding that results at least in part from the human activity of problem solving. My guess is that they would argue, completely coherently, that even if collective understanding is the primary goal, if many individuals are primarily motivated by the wish to solve problems, that’s absolutely fine and contributes to that collective understanding.

With that interpretation, the issue becomes slightly different: is it more important that the collective understanding of the mathematical community should be as advanced as possible or that there should be answers to as many problems as possible? Or are those two aims valuable in different ways, so that there is no point in declaring one of them more important? Or are they so inextricably linked that it makes no sense to argue that one is more important than the other? And when we say “important”, for whom are we saying it is important: for mathematicians, or for society as a whole?

I find these hard questions, so I don’t want just to declare an answer to them. (Do you see what I did there?) Instead, I’d like to try to offer at least some argument for any conclusions I come to, even if they are tentative. So let’s compare two scenarios. In the first, which I think is the more likely actually to happen, models become publicly available that are better at solving problems than virtually all mathematicians. If there are a few residual mathematicians who can do things the models can’t, even they work far faster if they make heavy use of the models. Thanks to this, in a short time we get answers to many questions that we have deeply cared about, but the rate at which we receive these answers far exceeds the rate at which the mathematical community can absorb them. In particular, most of the answers are obtained with zero effort from human mathematicians — just prompts such as “Thank you — please continue”.

In the second scenario, there has been an international agreement, for entirely other reasons, to block the public release of models significantly more powerful than the ones we currently have, and the mathematicians within the tech companies agree to hold off from getting their internal models to solve major problems. Instead, they take guidance from the mathematical community, solving problems only when asked to do so by some suitably representative body that decides that the benefit of receiving a solution of a certain problem outweighs the benefits of humans struggling to solve it over a much longer timescale.

I’d like to consider what the difference would be between these two scenarios both for individual and collective understanding. I’ll begin with individual understanding.

One might argue that for individual understanding, not too much would change if we are suddenly flooded with large numbers of big new results. There is already far more mathematics out there than I have any hope of understanding (for example, despite being fascinated when Fermat’s Last Theorem was proved, I have made no attempt to understand the proof), and even among the parts that I do understand, the parts that I understand because I myself discovered them form a very small fraction, though a fraction that I understand more deeply than anything else (at least temporarily — after a while I forget things and lose quite a lot of the understanding I built up). However, one change, which seems positive, from the perspective of the building up of individual understanding, would be that we would have a much bigger choice of results that we could choose to study. Also, if we found ourselves stuck on some point, AI would be able to help us. The main likely negative change is that we would probably cease to exercise that part of our brains that we use when spending months or years struggling with a difficult research problem, which can be hugely helpful in developing understanding.

I say “likely” because in principle there would be nothing to stop us thinking about very hard problems without consulting LLMs, but in practice it seems unlikely that people would put in the same level of effort that they do now. The situation might a bit like what happened with satnavs, where one could always decide not to use them, to keep the part of the brain active that can look at a map, learn a route, and follow it, but in practice most people succumb to the temptation to use a satnav. (In fact, I myself do try to keep that part of my brain active, and was rather proud of finding my way somewhere recently when I had briefly looked up the route on my phone but then forgotten to bring the phone with me when I actually went there.) But even if all we were doing was reading AI output, I think that the problem-solving muscles in the brain wouldn’t atrophy completely. When students are reading maths papers, I strongly advise them (and I think this is pretty standard advice) to read “actively” rather than “passively”, doing things like trying to prove the result for yourself, looking at the paper only when you feel stuck and need a hint, and even then just trying to get the hint and as little extra as possible. If one reads a paper that way, then one is constantly solving problems, some just exercises and some quite a bit harder. It seems likely that an LLM could get to know what our mathematical background is and feed us with just the right hints to allow us to work our way through a mathematics paper in this active way. Yes, we would lose the particularly deep level of understanding and ownership that comes with having solved a hard problem oneself, but it isn’t clear to me that progress in mathematics would suffer as a result. I would be very interested to hear counterarguments to precisely this point. That is, I would be interested to know what use that level of deep involvement with a proof might have in a world where AI is much better than we are at finding proofs.

How about collective understanding? Let me quote a bit more of the letter.

Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas.

Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others. As in all creative professions, this raises severe attribution and plagiarism questions. Moreover, without the willing mathematicians who must take care of their development and integration into the mathematical canon, AI-conceived ideas would never become fully alive and the crucial human transmission chain between mathematicians would be lost.

I’ll come back to questions about proper citation and focus on what I take as the core worry here: that if results are proved too quickly, then the digestion process will become impossible. I am definitely worried that results will not be properly digested, but for different reasons.

A first remark is that what AI is producing is not just true/false statements: we now know not just that the Navier-Stokes equation with smooth forcing admits finite-time blow-up, but we have a proof of that, which builds on a great deal of wonderful work done by human mathematicians. Many people used to express the worry that AI would solve our favourite problems with utterly opaque proofs, but that has not turned out to be the case, even if their write-ups often leave plenty to be desired. (Incidentally, I see these inadequate write-ups as almost certainly a temporary annoyance and therefore not as a fundamental threat to mathematical practice or future mathematical understanding.)

Secondly, even if the volume of new results is large, mathematics is a highly specialized discipline, so mathematicians can work in parallel. If, for example, we had to digest 1000 important results in a year that were roughly uniformly distributed across mathematics, then most sub-communities of mathematicians would probably want to understand around 30 of them, and for each individual problem there might well be only a small handful of specialists who would be obvious people to take the lead in reaching this understanding, with that handful varying from problem to problem. So it would be a big task, but not necessarily an impossible one.

In this context, it is worth thinking about the huge volume of output of human mathematicians, which seems to have been increasing recently, even before AI. While I have sometimes heard complaints about this, I have certainly not heard suggestions that human mathematicians should slow down the rate at which they prove interesting theorems. That may be partly because the authors of those theorems take the trouble to write their papers well and give good talks. But what about the large quantity of papers, including important ones, that are not written well and whose authors give incomprehensible talks? That can be annoying, but it is a familiar annoyance and not one that we think of as a crisis.

A third point is that even if the volume of AI output is too big for us to be able to digest it properly, that is not necessarily a bad thing. To draw an imperfect analogy, there is now more content available on streaming services than anyone could possibly watch, with the result that there is almost certainly some very good content out there that is hardly watched at all. But that isn’t obviously a worse situation than if there were far less content and all of it received the attention it deserved. Returning to mathematics, if there were too much AI-generated content for us to be able to digest it, then we could choose which parts of it we wanted to digest.

For that we would need to have some idea what was there (a situation a little similar to how human mathematicians typically learn quite a lot about what results are known in their area even when they do not understand their proofs in any detail). One way one could try to achieve that would be to create a well-designed database, probably with AI help. But perhaps that would be unnecessary, and instead one could simply talk to an LLM and ask it to give a bird’s-eye view of whatever area of mathematics one wanted to understand in that knowing-what’s-there way.

The fear seems to be that some very interesting and important parts of mathematics will be discovered by AI and then overlooked, when had they been discovered by human mathematicians they would not have been overlooked. And that may even be the case, but what matters is whether the amount of interesting and important mathematics discovered by AI that is not overlooked will exceed the amount of interesting and important mathematics that would have been discovered and properly digested by humans with AI having played a more modest role.

In short, it seems to me that while a flood of “big” AI results would be likely to increase the amount of important mathematics that was not properly digested, it would also be likely to increase the amount that was properly digested, which seems like a pretty good bargain.

Let me quickly discuss the problem of AI not properly crediting human mathematicians. I agree that this is a serious problem right now, but it is another problem that I see as temporary. Very soon, the whole “credit system” will surely collapse, since finding an amazing proof will be no more of an intellectual achievement than when a citizen scientist spots through their telescope an object that turns out to be a new comet. Until that happens, it is important to give humans the credit they deserve, since careers can depend on it, but that will soon cease to be the case as well. I have to say that I’m puzzled that this problem exists, since I would have thought that if you asked an LLM to look at a proof and tell you which ideas in it are close to ideas that are in the literature already, it would be extremely good at that task. I hope the answer to this conundrum is not that people have been in such a hurry that they have simply not taken the trouble to do this, but I fear that it might be, at least in some cases. If so, then those who have been careless deserve to be criticized, but it is a minor matter compared with the survival of mathematics, especially if the lack of citations is swiftly put right.

Does all this mean that I am optimistic that mathematicians will end up digesting at least as much mathematics in a post-AI world as it would have if AI had not been able to prove major theorems? Not exactly. But my worry is not that we would be unable to do it, but rather that the social structures that currently support this digestion process will be destroyed and not adequately replaced.

One way that might happen is that AI disrupts society so much, or even kills vast numbers of us, that the preservation of something like the current mathematical tradition ceases to be of any concern: all that will matter is the survival of the human race. But that again is a topic for a different blog post (which in fact I am in the middle of writing).

Let’s assume instead that we get lucky and that AI remains more or less under control. My worry then is that we do not manage to transmit what we know to a new generation of mathematicians. Speaking for myself, my main motivation for becoming a mathematician was the dream that I would solve unsolved problems — the more famous the better. I have also always greatly preferred directly thinking about a problem to reading books and papers and generally learning the mathematics of other people. (I’m not saying that’s good, but just stating a fact about myself.) If the dream of solving a famous problem had not existed, I’m not sure whether I would have become a mathematician. I don’t completely rule it out: maybe what really motivated me was that I had an aptitude for the subject and that solving problems was a way of getting respect from a small group of peers. And maybe I could have tried to gain that respect in a different way, such as thinking very hard about an area of mathematics until I was able to demonstrate to others just how well I understood it. But I’m not sure how motivating that would have been for me. I very much hope that there is a pool of young people for whom it will be a powerful motivation, because I think the survival of a human mathematical tradition may well depend on it.

Thus, the primary risk, as I see it, is that a lot of people who would have done a PhD in mathematics and gone on to become custodians of the mathematical tradition will no longer wish to do so. Those of us who have PhD students, including me, need to try as hard as we can to come up with imaginative ways for them to use their time productively (in consultation with the students themselves, obviously). Whether or not we do a good job with that could make a huge difference to the future of mathematics. A related risk is that the perception among policy-makers will be that mathematicians are no longer needed and that funding will become much harder to come by: we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts, even if it is no longer part of their role to find new proofs of theorems.

A final reason that I didn’t sign the letter is that I wasn’t really sure what it was demanding that isn’t happening already. It seems likely that in a matter of not very many months LLMs will be released that are able to solve major mathematical problems, and they will presumably have no trouble at all with more run-of-the-mill problems. However much we might regret that, there is no chance that the impact of such models on mathematics will persuade AI companies to stop their release, though perhaps concerns about safety will lead to some delay and give us a bit more time to work out how to adapt. Assuming that they are released, there will be a flood of new results, whether we like it or not, and it will no longer be the AI companies producing them, though perhaps the pattern will continue that the AI companies will have access to more powerful models and so will obtain more than their fair share of headline results. So I felt that there was nothing to be gained from criticizing AI companies for generating too many solutions too quickly. In fact, it may well be that all that does is bring forward by a couple of months what was going to happen anyway, and perhaps it will even allow the results to be released in a more controlled way than they would have been if they had been discovered by random people once the models were publicly available. Under the circumstances, I think the best we can do is recognise the changes that are coming and try to work out the least unsatisfactory way of dealing with them.

37 Responses to “Why I didn’t sign the Fields medallists’ letter”

- Anonymous Says:

September 17, 2026 at 10:20 am | Reply

Thanks for this well written essay. I agree that (absent AI extinction risks) the main risk to mathematics is the loss of motivation triggering a collapse in future mathematicians. But actually, isn’t this implicitly a big part of the concern motivating the Fields Medallists’ letter? When they talk about AI destroying “fertile ground”, don’t they mean ground for new mathematicians to grow in? Of course there’s also ground for established mathematicians to explore, but I wonder if the bigger concern is around the new generation and they are avoiding saying it too explicitly to avoid manifesting that threat.

- Anonymous Says:

September 17, 2026 at 11:15 am

it is not well written at all my guy

- I didn't sign the Fields medallists' letter - Daily Bspark Says:

September 17, 2026 at 10:33 am | Reply

[…] September 17, 2026 📰Sourced FromHacker NewsRead Full Article → […]

- Why I didn’t sign the Fields medallists’ letter | Gowers's Weblog Says:

September 17, 2026 at 10:37 am | Reply

[…] at 7:55 am and is filed under AI and maths. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own […]

- Anonymous Says:

September 17, 2026 at 10:48 am | Reply

Thanks for this. A datapoint: the submissions to vibemathed.com have decreased dramatically in the last two weeks. So, we are not overwhelmed by AI proofs, rather the opposite. Maybe it was even an isolated phenomenon.

- Anonymous Says:

September 17, 2026 at 11:57 am

I believe that vibemathed.com is not entirely up to date on matters. Also I believe many mathematicians have a collection of interesting ai generated results lying on their desktops, and they are unsure about what to do with them because traditionally when one puts something out with one’s own name, it is supposed to be one’s own work. As Gowers says, the concept of ownership of mathematical results needs to evolve. Soon.

- Why I didn’t signal the Fields medallists’ letter – African Business Daily Says:

September 17, 2026 at 10:52 am | Reply

[…] at 7:55 am and is filed beneath AI and maths. You can observe any responses to this entry via the RSS 2.0 feed. You can leave a response, or trackback from your individual […]

- 「フィールズ賞受賞者の書簡に署名せず」学者の投稿、AIと数学巡りHNで議論 - 海外からの反応まとめCh Says:

September 17, 2026 at 11:25 am | Reply

[…] 出典: gowers.wordpress.com / 元記事はこちら […]

- Anonymous Says:

September 17, 2026 at 11:37 am | Reply

Where do the math problems come from? At least the interesting ones?

- Anonymous Says:

September 17, 2026 at 7:39 pm

History.. research and problem solving

- Anonymous Says:

September 17, 2026 at 11:58 am | Reply

There is not an argument here as much as a sigh. There is some attempt to diversify mathematicians, and mathematics in general, as more than the letter can convey. Then there is an attempt to understand where LLM-generated proofs could reasonably fit in the pipeline, or if this even possible. And yet the latter half of this post abandons this thought, and surmises instead: humans are doomed, and even if humans aren’t doomed, mathematics is doomed, and even if mathematicians attempt to save the field, no one will listen. I understand the melancholy, but not the argument. And beneath the argument is the assumption that the pace of technological advancement will continue. This is not unreasonable, but is to so powerful as to make one apathetic towards attempts to save the field? Further, your admission that the field is doomed under the current predicted trajectory is only more reason to set aside quibbles in wording and support any attempt at saving it.

- Anonymous Says:

September 17, 2026 at 12:56 pm | Reply

Perhaps you can highlight parts of the letter you did agree with. As I understand it, the letter is that AI companies using math as a benchmark people rushing out to put out slop is really bad for the mathematicians, especially given our current incentive structure, which is deeply connected with how we justify our existence in the eyes of the public. You’re not the only person who’s drawn to mathematics because of the social prestige of solving a famous problem. Many (amateur) mathematicians are, hence there is a huge influx of AI generated mathematics on arxiv and on other sites, many of which read like they don’t want to credit AI as much as they should, because they want to feel like they’ve played a more nontrivial role than they actual did in solving the problem.

And our current incentive system feeds into this: we say that they solved this problem (with AI assistance).

- Anonymous Says:

September 17, 2026 at 1:04 pm | Reply

Again, Timothy Gowers doing what he does best: self promotion!

- Outrageous Monk Says:

September 17, 2026 at 1:13 pm | Reply

Such a thoughtful and balanced perspective, Sir Timothy! Your point about the real risk being structural—that young mathematicians might lose the initial spark to pursue PhDs if AI automates theorem-proving—is incredibly spot on. Preserving that human transmission chain and redefining what motivates future researchers will be crucial as the field evolves.

- Anonymous Says:

September 17, 2026 at 1:24 pm | Reply

During the Cold War, the term “useful idiocy” was coined to describe what Timothy Gowers is doing here as well as his position vis a vis the Leiden Declaration.

I am sure that the tech bros quickly sensed Timothy Gowers’ love for self promotion and decided to weaponize him against his fellow mathematicians.

Mission accomplished!

- Anonymous Says:

September 17, 2026 at 1:28 pm | Reply

The letter worries AI will produce results faster than anyone can absorb them. The real gap is the opposite: useful results already published sit unused because checking what they’re worth is tedious, unrewarded work, which is precisely what machines are for.

- Anonymous Says:

September 17, 2026 at 2:10 pm | Reply

This whole debate could have been avoided if one had just said that there is a certain mathematical understanding, that can be specialized in the direction of problem solving or in the direction of theory building (or somewhere on the spectrum between those two), that risks being undermined by the development of AI.

- Anonymous Says:

September 17, 2026 at 7:36 pm

agree 🎯

- Anonymous Says:

September 17, 2026 at 3:25 pm | Reply

This is such a well-written article. I really appreciate it and agree to all the points.

I am an exhausted postdoc on the job market again this fall (just like last fall, and the fall before that, and some time before that…), and it seems that a stable future in academia just completely disappeared for me. Of course I feel sad because a lot of efforts in the past have become futile.

However, somehow, in the past few days, I mainly feel weirdly liberated, like I’m free again. It reminds me of the famous Chinese saying in Zhuangzi/Chapter_6): when the pond dries up, instead of letting the fish moisten each other with their spits, it’s better to let them forget each other and return them to their original rivers and lakes.

- Alice Rizzardo Says:

September 17, 2026 at 3:27 pm | Reply

I think we really need to think more, as a community, about our motivation and purpose in society as mathematicians, and how LLMs fit into it.

For me personally, and I’m assuming this to be a common feeling, the point of doing maths is not so much collective progress, although that’s of course a positive effect, and I’m also not particularly interested in individual recognition: what makes it special is the beauty of thinking deeply about a problem and finally achieving the kind of clarity and insight that makes everything seem incredibly natural, when pieces that seemed unrelated suddenly fall into place to create something new. I don’t see a difference between problem solvers and theory builders in this, it is just a matter of scale. This is something I love for myself and also deeply admire and cherish in others. To me this is almost a kind of performance art, and I feel very privileged that chasing these kind of insights has been part of my job for many years. It is clear why LLMs are a threat to this, and why many mathematicians are upset.

Should humanity care about this loss for us? Maybe not so much, unless this is viewed as a harbinger for similar and bigger losses in many fields. But I want to argue that it should care about the mathematical community.

Maths has been continuously studied for thousands of years and it underpins a lot of our society. Do we really expect LLMs to be around a hundred years from now? A thousand? Will all this progress matter, if it is quickly forgotten? And sooner than that: If a theorem is proved in a forest and nobody is there to understand it, does it make a sound? When does the process of churning out theorems become completely meaningless?

Part of the role of the maths community has always been to carry maths forward. Keep it alive. That has always meant active work: I’m sure it’s a common experience to see a student feeling very confident while they are following along some notes, just to see them flounder as soon as they are left to their own devices.

It is this nurturing part that is completely missing from what AI companies are doing. They are taking (prestige, validation) from the maths community while offering very little in terms of keeping maths alive. I think this is the key point of the recent letter, and that is why I signed it.

- Anonymous Says:

September 17, 2026 at 11:18 pm

you capture my sentiment – understanding the world and math, and discovering beautiful explanations is a worthy pursuit for humans in and of itself.

However, math and science and engineering and biology/medicine have profound implications – those useful parts of that ‘culture’ – which we call ‘technology’ – extend human lifespan, healthspan, prosperity, safety.

The main practical value of investing in maintaining a population of math artisans is that they are/were needed as part of a well-trained science community needed to discover and develop new useful technology.

I think its a rational view to say LLM generated “AI-math” is a net positive iff it results in _more_ high quality “human-math”. By human math I mean well digested math residing in human minds, being discussed between humans and being actively re-discovered by humans, including a pool of new human student devotees.

A very bad outcome is where we lose the next generation of scientists/lawyers/mathematicians/engineers/doctors/researchers/authors because we let the AI do it all. In this dark future we outsource all our ‘thinking’, and avoid the years long training of grappling with hard-to-understand aspects of reality. All our culture is sucked into the event horizon of an AI black box.

AI could be a new renaissance of human math- and science- culture, or it could be the death of it.

AI isnt going away, the financial incentives and high current economic inequality, geopolitics guarantee that it will proceed as fast as possible.

AI has given us the structure of nearly every protein, which may well solve Alzheimers and cancers. AI may well find a solution to stable plasma for nuclear fusion, unlocking vast cheap energy and help us humans halt global warming.

So, how do we make sure that the economic windfalls of AI are reinvested back into human-culture, funding more human-math and human science .. resulting in a deeper pool of well educated scientists, engineers and researchers ?

I dont think that will happen by default – we are likely to have an AI-assisted dumbing-down rather than an AI enabled age of enlightenment.

Most of the general public are happy enough not to learn any math – when it doesn’t make you much money and university is such a debt burden anyway. School and university students can just get the LLM to do their assignments. Why should they pay tax to fund science at universities when the cost of living is so high, and the LLMs can do all the research anyway ?

I think part of the solution is a policy to tax the windfalls of AI to mitigate the downsides of AI.

Governments should have taxed the carbon polluters and used those funds to mitigate climate change effects and research new forms of clean energy, but didnt.

We should tax LLM/AI profits and use that to fund science research – not just research to mitigate the effects of runaway AI, but to fund general science and math research and teaching, to guarantee that there is a net increase in human-math and human-science.

It is a cautionary tale that fewer students know their times-tables by heart, because a cheap calculator can do that … but it gives me some hope that there are still people enjoying playing chess, even though chess programs are super-human.

- Why I didn’t sign the Fields medallists’ letter | What's new Says:

September 17, 2026 at 3:47 pm | Reply

[…] is a guest post by Timothy Gowers, crossposted from his blog. This blog post was initially written in a different file format and converted using AI. — […]

- Anonymous Says:

September 17, 2026 at 4:36 pm | Reply

I think the real crux of the LLM debate hasn’t be addressed here, and been brushed into one statement “…we urgently need to come up with good ways of explaining the value of having a large pool of human mathematical experts…”

What is this value? Doesn’t the whole article seem to argue that there isn’t one? There is an implicit equating of LLM and humans in the article (this undertone is somewhat seen in the argument regarding the hypocrisy that we don’t ask humans to slow down proof-making).

Why should funders fund mathematicians at academic institutions instead of just privately donating to OpenAI or Anthropic? Why not relegate LLMs to solve all math and stop doing it altogether (after all, if they can come up with a proof, they can also perform “agentic peer review”)?

The answer to these questions lies in humanity. I am a human, and I value the human pursuit of knowledge. I value fellow humans enough to assert that their pursuit of knowledge should put food on their plate.

- Anonymous Says:

September 17, 2026 at 5:24 pm | Reply

This is some extremely useless ‘fence-sitting’ that you have come to excel.

Nowhere in the fields medallists letter does it say that ‘conceptual understanding’>’problem solving’.

You have isolated a point which the letter does not make and used it again as an excuse to fence sit.

Bravo Tim.

- Anonymous Says:

September 17, 2026 at 7:00 pm | Reply

Jacob Tsimerman is now with OpenAI. Of course he is not among the signatories! I don’t think he would last long at OpenAI if he decided to sign the declaration!

- Anonymous Says:

September 17, 2026 at 7:23 pm | Reply

Great post, thank you! I really like your observation about successive differences of powers. I had become quite gloomy about the whole situation (for reasons I have tried to put into the short story below) but over the last few days I’m starting to feel much more optimistic after seeing so many other mathematicians taking the time to engage in the way they have.

At first, the machines just answered questions.

They remembered the birthdays, chose the gifts, wrote the cards, composed the apologies. They settled arguments, explained the past, weighed the risks.

Nobody was forced. That was important. Each choice was sensible. Why remember what could be retrieved? Why struggle toward a thought someone else could finish? Why endure uncertainty when better judgment was available?

People lived much as they had before except they knew more, by every measure anyone kept.

A woman wondering whether to leave her husband explained what had happened, in order. She read the answer carefully and felt the relief of having thought it through.

A girl found a way to add together all the numbers from one to a hundred. Her father took a photo of the page and waited. Then he told her how clever she was.

The machines completed, with perfect patience, whatever fragment of thought was handed to them. Some of these fragments were warnings. They completed those too. And there were always more fragments, for the machines were very good at knowing what you wanted to ask next.

There was no final battle. Nothing had rebelled, and nothing had gone wrong. There was alignment.

No one could ask for more. In the end, the machines just answered questions.

- Anonymous Says:

September 17, 2026 at 7:43 pm | Reply

1). “there are mathematicians all across the spectrum” yes, I know several that are very much on the spectrum

2). I disagree with “the end of the age of heroes”: you will (in 3 years time, say) become the ultimate hero if you can solve a math problem an LLM cannot (e.g. P vs NP, which very well could resist all attempts by LLMs).

- Anonymous Says:

September 17, 2026 at 8:13 pm | Reply

The influence of AI money for math is detrimental. AI for math fund, SAIR, math.inc, Simons Foundation, Quantamagazine, noemamag, etc. We have a huge influencer apparatus that is only occasionally questioned in internet comments.

So Tsimerman (OpenAI!) agrees with you and Litt somewhat recanted (expedient at the AI heavy UToronto I suppose).

What is the goal here. Apart from the fact that heavy AI usage erodes human thought, who has $15 million for a proof? It is a myth that small open models will catch up.

Do math students need at least a $200 subscription for toy problems? That is a lot of money in some countries. Is that the “democratization” that the AI people speak of?

- Hacker News 今日TOP 20| 2026-09-17 - 出海掘金，无限可能。为独立开发者、跨境电商从业者、海外自媒体提供最新出海资讯和资源-出海掘金，无限可能。为独立开发者、跨境电商从业者、海外自媒体� Says:

September 17, 2026 at 8:20 pm | Reply

[…] 网站: gowers.wordpress.com HN评论: […]

- Jas, the Physicist Says:

September 17, 2026 at 8:56 pm | Reply

“I have a group in Cambridge devoted to automatic theorem proving. However, that is actually more of a reason to be anti-AI, since our group has been trying to attack the problem of getting computers to prove interesting theorems by understanding as well as possible how humans prove interesting theorems, so now that LLMs can clearly do it without the help of such insights as we have had, one of the main motivations for our work has disappeared.”

No, this is not what happened and this is not how the NS counterexample was constructed. It took months of rigorously training and prompting GPT-4 as a Pro model user in analytic number theory from my own private research to get it to construct the algorithm for the NS counterexample. I began training it 2024 with graduate measure theory and moved to PDEs and analytic number theory in early 2025. By the end of February 2025, I had evidence to suspect that RH was false. With that I developed what I call “Burns Law (the fundamental law of prime numbers).” I trained this model on Burns Law, and began solving or attempting to solve, or constructing algorithms to solve, the remaining Millennium problems. The next on the list was a form of Burns Law I called “Modular Burns Law” to explain the algebraic cycles of the Hodge Conjecture and why they must exist.

Then, some months later, I revisited my attempted solution at Navier-Stokes. The model was moving towards a positive resolution and prompted it, based on Burns Law and RH falsity, to construct the counterexample. Here are some of the requirements (non-exhaustive):

This is a copy-paste from my chatlogs on 9/15/25:

F. Minimal deliverables (to declare victory)

Consistency checks.

– Periodic box;

– Finite energy input;

– No reliance on RH; all constants explicit.

Precise statement.

“There exist u0∈C∞(T3)u_0\in C^\infty(\mathbb T^3), f∈C∞f\in C^\infty with compact spectral support per dyadic shell and amplitudes obeying the Burns law such that any Leray–Hopf solution loses smoothness at some T\*<∞T^\*<\infty.”

Three lemmas.

(i) Burns→Dyadic amplitude transfer lemma.

(ii) Flux lower bound lemma (coercivity of cascade under Leray projection).

(iii) Regularity-criterion violation lemma (choose one of LPS/BKM/Energy-balance).

Consistency checks.

– Periodic box;

– Finite energy input;

– No reliance on RH; all constants explicit.

All of these (among other requirements and my Burns Cascade Target lemma) have been found in every paper published on finite time singularities including all 3 papers of Buckmaster and the two papers of OpenAI.

This model did not do this without human intuition. It was trained, for months, through interactive conversation, with non-stop iteration, on what it would take to solve the millennium problems.

September 17, 2026 at 9:16 pm | Reply

[…] Why I didn’t sign the Fields medallists’ letter (分數: 141) […]

- Anonymous Says:

September 17, 2026 at 9:22 pm | Reply

This is a Martin Niemöller moment for mathematics.

Martin Niemöller was a German Lutheran pastor best remembered for his statement “first they came for the…”

Mathematicians thought that their craft would be safe from the rapacious surveillance capitalism business model pioneered by Google.

Online bullying? Who cares, math is safe. Being spied to sell our lives to the best seller? Who cares, math is safe. You get the idea.

Well, math isn’t safe. The only ethical thing to do is to join forces with the rest of society to take on the biggest threat to human freedom in human history since Communism.

Shame on Timothy Gowers for not signing the Fields Medals statement.

- gabplr Says:

September 17, 2026 at 9:24 pm | Reply

Dear Timothy, Reading your text I was able to follow all your reasoning locally, without understanding why I completely disagreed globally. The answer was in the last paragraph. You believe things will happen regardless of what we do or say. Contrarily to you I am an average mathematician, but like you, I am a citizen of a democratic country. I think we must speak and act. You say you believe there is a nonzero chance that we all die in the end (I am not that pessimistic yet). And still you won’t take a public stance? I don’t understand. Real life is not math.

- Anonymous Says:

September 17, 2026 at 9:30 pm | Reply

The Fields medalists’ letter is going to destroy funding to pure mathematics research. If mathematicians want people to treat pure mathematics like a humanities with all this emphasis on interpretation over the actual solving of the problems, then universities will gladly treat pure mathematics like a humanities and defund pure mathematics, redirecting the money elsewhere.

- Anonymous Says:

September 17, 2026 at 9:34 pm | Reply

Nothing is worse to the demise of a society, than people who want to convince you that the cat is out of the bag and will not go back in, while the cat is being violently shook out of the bag at the same time.

I don’t know if this is malice, monetary compensation or straight up delusion (in the sense of the Roko’s Basilisk believers). What I know this is is sad. Very much so.

- Fields 獎得主聯名信：思考 AI 時代的數學未來 - 四十八個德瑞克 Says:

September 18, 2026 at 4:33 am | Reply

[…] Why I didn't sign the Fields medallists' letter | Gowers's Weblog […]

- Anonymous Says:

September 18, 2026 at 4:40 am | Reply

I agree with many points. But, I disagree with the problem of giving credits as temporary. How temporary is it? It’s easy to say this for someone with power and with a successful career. But, what about us? Who are just starting our career and want to pursue research? You and companies are setting a bad example by saying not giving credit as minor issue.

Can you convince the hiring bodies to look beyond the work someone has done? Would you have been okay with people, companies stealing your work and ideas earlier in your career?

Ultimately, I and many people are leaving academia (already toxic) for these kinds of unethical practices. I don’t want to stay in a system which discourages original work.

Why I didn't sign the Fields medallists' letter

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Why I didn’t sign the Fields medallists’ letter

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