I didn't sign the Fields medallists' letter

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.