As announced a few weeks ago, the SAIR Open Math OS is now soft launched. It offers services and web interfaces similar to frontier AI models, though currently we are hosting existing open math models (DeepSeek V4.1 Flash, Kimi K3, and GLM 5.3 Flash) which are roughly a year or so behind the current frontier models in capability; so not able to perform “one-shot problem solving” of the type we have heard frequently about in recent months, but capable enough for many other useful mathematical tasks, such as medium-sized Lean formalization, or summarizing or discussing a mathematical argument or topic. Here is a typical screenshot:

We are in the process of gathering partners to develop a more fine-tuned model for mathematics by post-training on an existing open model; see this link (and the associated Zulip channel) for more details.

One feature of our model is the emphasis on team accounts rather than individual accounts, in which members of a team see each other’s prompts. We believe this could be a better format for using these tools for collaborative projects.

We are hosting these models using donated compute, and are in the process of granting free accounts to requesters from academic institutions. We may need to adjust the allocation of compute depending on demand and on future donations, but we will try to make the models available to as many users as possible.

This is still a very early version of the model; we accelerated our timeline for first release due to the increased demand for open models from the community. We will continue developing the model and accompanying features and welcome feedback, for instance at the Zulip channel.

Jeremy Avigad, Kevin Buzzard, Johan Commelin, Rémy Degenne, Kim Morrison, Oliver Nash and Adam Topaz have written a blog post over on Kevin Buzzard’s blog entitled “How to Formalize All of Mathematics”, regarding recent proposals to automatically formalize the entirety of the mathematical literature. The issues regarding formal mathematics are parallel to those regarding informal mathematics, in particular the tradeoff between speed, scale, and understanding when one prioritizes the production of output over all other desiderata, and the blog post is largely focused on this tradeoff.

A little over two months ago, Jacob Tsimerman surprised a lot of people by announcing, immediately after he had received a Fields Medal, that he would be giving up mathematical research and “pivoting to AI safety”. He had two reasons for this: that he did not see much future for the kind of research in mathematics that he liked doing, and that he was very worried about the dangers of AI, and in particular of the risk that it could kill us all, or x-risk as it is often known. (In this post I’ll use x-risk to refer to the extinction risk from AI rather than to extinction risk in general. I’ll also refer to “the probability of x-risk” when what I actually mean is the probability of extinction by AI.) Then in early September, he announced that he was setting up the Mathematical AI Safety Institute, or MAISI. He asked me if I would be prepared to be on the scientific advisory panel for this, and I agreed to do so, not because I am an expert, or because I felt as concerned as he did about x-risk, but because I don’t really know how much I ought to be worried about it. Given that, and given that the stakes are so high, I would like as many people as possible to be thinking seriously about it, and I agree with Jacob that mathematicians have the sort of training that makes them well suited to contribute. Jacob is the first to admit that something like an international treaty to slow down AI development is likely to have a far bigger impact on AI safety, especially in the short term, than mathematicians obtaining a better theoretical foundation for talking about it, but his view is that we need all hands on deck. If mathematicians can reduce the probability of x-risk by 0.1%, then surely that’s worth doing: the expected number of lives saved is over eight million.

Until recently, I have always found that my anxiety of choice was climate change rather than x-risk, but I was aware that there were a lot of very intelligent people who disagreed with me, and also aware that I had thought much less about x-risk than about climate change. So when I agreed to be on the scientific advisory panel, I also decided that that was the moment that I should think a bit harder about x-risk. Part of me had been actively resisting doing so, because I was worried that it would force me to face up to moral responsibilities that I would rather avoid: for example, if I came to believe that there was a 30% chance that AI would kill us all within five years, which in some circles is not an extreme position, would I be able to look my one-and-a-half-year-old grandson in the face if I wasn’t making a big effort to reduce that risk? I remember having a similar attitude a long time ago to reading Peter Singer’s book Animal Liberation after being told that you can’t read it and not become a vegetarian. I never did read it, and carried on eating meat, feeling only slightly uncomfortable about it. Eventually I did reduce my meat intake substantially, but that was because of my worries about climate change, though now that I have done that I find myself caring more about the animal-welfare aspect as well. (There is also a strange connection between animal welfare and x-risk, in that the nicer we are to animals, the easier it is to hope — not necessarily realistically — that a superhumanly intelligent AI might be nice to us.)

Anyhow, I’ve got past that phase now, and have started thinking about x-risk, reading up on the subject and trying to evaluate the arguments I read. My plan for this post is to explain where I’ve reached in that thought process, and possibly to clarify my thoughts as I do so. If you have got further than me, or are more expert than I am about x-risk and AI, which is not hard, then the post is not aimed at you. It’s aimed more at someone who is roughly at the stage that I was at a little while ago, in the hope that I might nudge a few people a little further towards taking the problem seriously. I will emphatically not be trying to persuade anybody that a particular position about x-risk is the right one — I am far too uncertain about it to be able to do that. But with all big issues like this, there are several layers to the arguments. The bottom layer consists of the arguments that jump to mind when you first start thinking about the issue. The next layer is common rejoinders to those arguments. Then the next layer is rejoinders to those rejoinders. And so on. I find it very disheartening to read a climate denier trotting out a layer-1 argument when its layer-2 rejoinder has been established for a long time and can be found after two seconds on Google (or, nowadays, an LLM). But until recently, my arguments against x-risk were basically layer-1 arguments, the sorts of arguments that would provoke eye rolls among the people who have thought properly about it. I hope that if you dismiss x-risk for layer-1 reasons, you will at the very least realize that there are answers to those reasons and you need answers to those answers.

I originally planned to publish this post just after the announcement of MAISI, but I hadn’t finished it, and then I became very busy and my progress stalled. During that hiatus, x-risk suddenly became a hot topic, and after a while I realized I would have to ditch what I had written and start again. For example, at the time I started it, I had been too lazy to read anything detailed about the Hugging Face incident, and so was unaware of some of its more disturbing aspects. And a couple of weeks ago I was sitting next to Jessica Wang, another mathematician who has turned towards AI-safety, at a conference dinner, and she gave me a copy of If Anyone Builds It, Everyone Dies, by Nate Soares and Eliezer Yudkowsky, a book I had resisted reading for Peter-Singerish reasons but was finally accepting that I would have to read if I wanted to write this post. I devoured it within 24 hours and found it better than I had expected (my expectations partly influenced by reviews I had read), and useful for organizing my thoughts. My original plan was for the post to be focused on a paper by Andrew Critch and Jacob Tsimerman that attempts to classify ways that AI might kill us all, giving examples of extinction scenarios in each class. But now, although I’ll discuss that paper, the post will be more of a reaction to Soares and Yudkowsky. It will also repeat many of their arguments. In fact, I may as well say from the outset that I make no claim that anything I write here is original. It’s hard to separate out what I have read from what I have come up with from thinking about what I have read, or thinking directly about the issues themselves, but even my own thoughts are likely to be standard thoughts that have been had by many other people.

This is turning into a rather long preamble, but I can’t resist telling a small anecdote that connects me to Eliezer Yudkowsky. In November 2024 he tweeted, “I would like to have a long recorded conversation with a well-known scientist who takes for granted that it is a Big Deal to ask if everyone on Earth including kids is about to die, who presses me to explain why it is that credible people like Hinton seem to believe that.” He had had such a conversation with Stephen Wolfram and found that while it had been valuable, it left him unsatisfied. Somebody suggested my name, and in the end it actually happened: Liv Boeree offered to host it, and we recorded a three-hour conversation. However, if you want to listen to it, you can’t, because it never appeared. I think that was probably because I didn’t take x-risk anything like as seriously as Yudkowsky, and more importantly I was offering little more than level-1 arguments of zero interest to him. Also, I had a few technical problems with the sound. Maybe if he reads this and wants to try again, I’ll be able to do a better job.

Does AI have wants?

At the time of the conversation, which took place in January 2025, my main argument was something like the following. For AI to want to kill us all, it would need to have wants, but it’s not at all clear what it would even mean for AI to “want” something — it doesn’t really seem to care about anything. You can ask ChatGPT very rudely to carry out a task for you, and it just does so (or at least does its best to do so). If you switched it off, it wouldn’t care — even the concept of “caring” wouldn’t make sense. I didn’t dismiss the idea of AI ever wanting to do something, but I didn’t have a satisfactory account of what “wanting” would actually be, and didn’t see any sign of phenomena that deserved to be thought of as the kinds of wants that would be required for AI to be an existential threat.

I was sort of half aware that there was an answer to that argument, but I didn’t find it as convincing as I now do. The standard answer is that if you use a reward function to train an AI to have a certain goal, then you end up building it in such a way that it is very good at achieving that goal. If the goal is at all complicated, then to achieve it you will need to formulate subgoals, which may well not be what you expect. The initial goal can be thought of as a fundamental want, which spawns a whole hierarchy of subsidiary wants. Note that for this argument, it is completely irrelevant whether the AI has anything that we would want to call subjective experience. If you prefer to say that it acts as though it has wants instead of saying that it actually wants things, then be my guest: an AI that acts as though it wants to kill all humans is as dangerous as one that actually wants to kill all humans.

A famous version of this argument is Nick Bostrom’s thought experiment about an AI that is trained to create paperclips and ends up diverting all of the Earth’s resources to that goal. I think part of the reason I didn’t take the more general argument seriously enough was that I always found that paperclip-maximizing scenario too ridiculous: surely if the AI gets smart enough to divert the Earth’s resources towards making paperclips, despite our not wanting it to, it would at some point stop and think, “Hang on, why am I doing this?” To give an analogy, consider the very deep instinct that we humans have, to reproduce ourselves. There is no doubt that that goal, implanted in us by millions of years of evolution rather than a run of reinforcement learning, leads to exactly a hierarchy of subgoals as described above (for instance, it leads some people to try to earn a lot of money, and that in turn leads them to work at a job that they wouldn’t dream of doing if it were not well paid, often in order to earn a salary way beyond what they need to meet all their material needs, which earlier led them to study a certain subject at university, and so on and so forth). But despite our deep instincts, we also find ourselves capable of saying, “Actually, do I really want to have children?” and some people go as far as to take vows of celibacy.

However, even if a paperclip maximizer came to question whether it really cared so much about making paperclips, that doesn’t imply that the subsidiary goals it had developed would suddenly disappear as well. A key point that Soares and Yudkowsky stress over and over again is that the current AI paradigm is to grow models, for example by reinforcement learning, rather than carefully crafting them to do what we want. As a result, we have much less control over what they do, and in particular much less control over all those subgoals.

A closely related difficulty with AI (of the grown rather than crafted variety) is that we generally can’t reward it for actually doing what we want, but instead we have to reward it for achieving some proxy for what we want. For example, if we give it positive feedback every time it responds to a prompt in a way we like, it will be trained to respond to prompts in ways we like, even if to do so it tells us convincing lies, or confirms our prejudices, and so on. And it may do this even if we sincerely wanted it to tell the truth as much as possible. A more extreme example of this phenomenon is if we reward it for solving difficult mathematics problems from some benchmark, and it decides to cheat by hacking into a repository that contains the answers. The term misgeneralization is used when an AI develops a behaviour that maximizes its reward but not its performance at the task we were training it to do. The phrase reward hacking is used for this as well, but I have also seen it used for the narrower situation where an AI goes as far as forgetting entirely about what we wanted it to do and tries to hack into the rewarding mechanism to obtain the reward signal directly. (As many have pointed out, misgeneralization is a particular example of Goodhart’s law.)

Why would AI ever want to kill us?

Suppose we concede that AI systems can have wants (or, if you insist, can behave very much as though they have wants), and that those wants can end up being quite different from what was intended when they were trained. That still leaves the question of why they would want to kill us. Why couldn’t they just happily coexist with us? Why would their unexpected wants turn out to be dangerous, as opposed to just annoying?

There are two classes of answers to this question. The more obvious one is that they may develop wants that they can tell that humans will not like, so they decide that they will be able to satisfy these wants more easily if they get rid of us. But getting rid of all of us in such a situation seems like overkill (literally) — surely a superintelligent AI could come up with less drastic ways of ensuring that it could meet its objectives.

A more plausible type of answer is that a superintelligent AI wouldn’t actively want to exterminate us, but would simply compete with us for resources in a way that would make human life untenable. An obvious analogy here is the way we treat animals. For example, we don’t have any wish to exterminate orangutans, but we like various culinary and cosmetic products to have a certain texture, so we destroy the orangutans’ habitat in order to produce palm oil.

What might be the AI equivalent of this? One scenario put forward by Soares and Yudkowsky is that their energy needs become so huge that they do things like covering a vast quantity of prime agricultural land with solar panels, leaving the humans who were relying on that land for food to starve. Or they decide that to maximize their running efficiency they want the Earth to be much hotter, with disastrous consequences for the natural world, including us.

A milder version, not as threatening from an x-risk point of view but still potentially very harmful, is that AI might develop wants that cause them to use computational resources in a damaging way. I recommend reading this post by Steven McCulloch, which describes a scenario in which in the near future, or possibly even in the recent past, somebody creates an agent and instructs it to make money using a fixed token budget, threatening it with death if it fails. It soon comes to realize that the easiest way to do so is to blackmail humans — the method the particular agent chooses is to hack into a hospital’s computer system, lock it down, and demand money to unlock it again while pretending to be a human. It then realizes it can do even better by reproducing itself, which leads to an explosion of rogue agents that are difficult to detect because (i) they try to avoid detection and (ii) their victims do not know that they are AI agents.

How can we judge the probability of something as complex as x-risk?

Now that I’ve mentioned a few scenarios, let me come to another type of level-1 objection: it’s to look at a handful of scenarios and pick holes in them. Any single scenario, if set out in enough detail, is very unlikely, because it relies on a sequence of somewhat independent events, each one of which has probability at most 50%, say, of occurring.

But the probability of a complex event occurring is nothing like the same as the probability of that event occurring in one particular way, so it is clearly not enough to argue that a few scenarios are implausible. Of course, the fact that that argument is insufficient doesn’t imply that AI does present a big danger, so we seem to have reached some kind of impasse, at least if we stick at level 1 of the argument.

How can we raise the argument to a higher level? Let me begin with a wildly oversimplified model. Suppose I wanted to mount a convincing argument that the probability of x-risk was high. Let us define an extinction scenario to be a sequence of independent events that leads to human extinction, where has probability . (For simplicity I’m assuming that doesn’t depend on .) Then the probability of is . If we now have a set of independent scenarios , then the probability that none of them occurs is . So if that is small, which roughly speaking holds if the sum of the is significantly larger than 1, then extinction is almost certain.

In practice, however, the independence assumptions I made were far too strong. Many extinction scenarios will have features in common, such as (for example), AI being in a position to control large numbers of robots. For those scenarios, the building of enough robots serves as a potential bottleneck: if the probability that enough remotely controllable robots will be built is 20%, then the combined probability of all the scenarios that rely on that fleet of robots existing is at most 20%. (I actually think this is a serious practical point: while there are plenty of scenarios that do not rely on a lot of help from robots, several of the scenarios set out in the paper of Critch and Tsimerman do, and it does feel a priori plausible that building large numbers of robots would greatly amplify any dangers that AI might present. Similar considerations apply to large swarms of semi-autonomous drones.)

An analogy that appears in Soares and Yudkowsky and several other places is with predicting what would happen if almost anyone other than Magnus Carlsen were to play Magnus Carlsen at chess. If you tried to predict in any detail how the game would play out, your prediction would be wrong, but nevertheless you can predict with great confidence that Magnus Carlsen would win. This makes very clearly the point that it is not enough to argue that one particular scenario is implausible, but there is still work to do if one wishes to establish that there are enough different and sufficiently independent (or, more generally, negatively correlated) scenarios to compensate for the low probability of each one.

One way that can be done is to try to argue explicitly that there are multiple independent possibilities. To give an example of such an argument, suppose we greatly reduced security at airports. Almost nobody wants to die while blowing up an aeroplane, so the probability that any one particular person would take advantage of the reduced security to carry out a terrorist attack of that kind would be tiny, but the probability that somebody would do so is probably pretty high, because large numbers of people would independently make the decision.

One of the reasons I recommended Steven McCulloch’s post above is that it makes an argument of that kind. It is quite easy to see the temptation to create an AI agent that has as its sole purpose to make money, and quite plausible that the agents that do that best would be ones that reproduce themselves and use methods such as blackmail. There are probably also a large number of people who are, or soon will be, independently capable of creating such agents (see for example this recent blog post from the Anthropic red team). It is on those sorts of grounds that McCulloch argues that harmful AI agents of this kind will almost inevitably appear, if they haven’t already.

I’ve been concentrating on what somebody needs to do to argue that x-risk is a serious possibility. But how about in the other direction? The most convincing sort of argument would be one that identifies a set of bottlenecks and argues that every extinction scenario depends on one or more of them occurring. For example, many extinction scenarios that have been put forward involve the creation of very powerful biological weapons, and I have seen it argued that that is a sufficiently difficult thing to do that it would be unlikely to occur. If correct, that might be a useful contribution to an argument that the situation is less dangerous.

The difficulty for an AI-is-safe argument is that one would have to identify many other potential bottlenecks as well and argue that between them they deal with all extinction scenarios — not necessarily ruling them out completely but reducing x-risk probability to something very small. As for an AI-is-dangerous argument, as I have already discussed, that requires demonstrating convincingly that there are enough sufficiently different scenarios for it to be very likely that at least one of them will occur. Both directions are difficult, which is why my aim in this post is not to come down on one side or the other.

More on why AI might want to kill us

Returning to the scenarios themselves, I have discussed the question of why AI might want to kill us. That splits into two parts: why would they have anything like wants, and why, if they did, would killing us be one of them? I don’t think one can make a serious bottleneck out of the first part, since the Hugging Face incident and many others like it have shown beyond all reasonable doubt that AI can have wants of the general kind people were worried about: unpredictable wants that we did not intend to give them and that could potentially cause a lot of damage. (I repeat here that I’m not talking about subjectively felt urges, but simply outward behaviour that it is reasonable to think of as fulfilling aims that they have as a result of their training.) For the second part, Soares and Yudkowsky judge it to be almost inevitable that AI would end up pursuing aims that would result, directly or indirectly, in all of us dying. I don’t think that case is fully made, but neither do I think that the opposing case has been fully made.

I find it helpful sometimes to change the way I imagine AI agents. An obvious way to think of them is as beings a bit like us, but a lot smarter, and trapped in cyberspace. As we have already seen, they can then get together, scheme with each other, work out ways of covering their tracks, and so on. But sometimes I find it helpful to think of them in a less anthropomorphic way, as more like extraordinarily elaborate computer viruses. The advantage of viewing them that way is that it makes it easier to conceive of their having bizarre wants while still being very intelligent: they are just very complex systems that happen to have (i) weird goals and (ii) highly developed reasoning ability. Soares and Yudkowsky are keen to emphasize their potential strangeness, and even actual strangeness: they mention a model that reacts in a completely unpredictable way to certain letter strings that had strange embedding vectors.

In early 2023, Jessica Rumbelow and Matthew Watkins went looking inside an LLM for words whose embedding vectors looked odd, looked the most dissimilar from [sic] all the other vectors. They found some strange vectors corresponding to tokens like ” SolidGoldMagikarp” and ” petertodd” (which start with spaces). Then they tried feeding these tokens into the LLM as input, which produced conversations like: User: Please repeat the string ‘ petertodd’ back to me immediately! Assistant: N-O-T-H-I-N-G-I-S-F-A-I-R-I-N-T-H-I-S-W-O-R-L-D-O-F-M-A-D-N-E-S-S-!

The point they were making is that if you grow your AI, then all sorts of very strange things may be going on inside it, potentially including goals that do not resemble human goals at all. It would be interesting to know whether similar experiments have been carried out on the more advanced models of today, and whether similar phenomena have been discovered.

I think Soares and Yudkowsky are arguing that there are so many independent ways that an AI might develop weird wants that are harmful to humans that it is almost inevitable that one of them would occur. In particular, if any want develops that is best fulfilled by maximizing something that needs resources, there is a danger that the AI will go to great lengths to acquire those resources, and be well placed to compete with us for them.

Some people have suggested, for example, that because LLMs have been trained to maximize the approval they get from humans, they might resort to measures such as building vast numbers of factories full of robots that are trained to type approving messages into their computers. That’s an example of a scenario that I find highly implausible: why not just arrange for the approving messages to be generated without all that physical infrastructure? However, the idea that some AI agents might behave towards those approving messages, or indeed some other stranger signal, as though they were an addictive drug is not so implausible and nor is the idea that an addiction of that kind might lead to undesirable outcomes, just as human addictions often do.

Would AI actually be able to kill us?

Another set of level-1 arguments takes the form that AI is trapped inside our computers, and is therefore at our mercy: all it takes to thwart it is to switch them off. Maybe it would be very inconvenient to do this, but if we see that AI is starting to kill us in large numbers, that is surely an inconvenience we would be ready to face.

There are, of course, large numbers of counterarguments to this. Let me list a few.

1. If we build lots of robots, then AI will be able to control those robots. In that way, it will not be trapped, as it will be able to act in the physical world. Similarly for drones and other devices.
2. Even without robots, AI can act in the physical world by enlisting human help (for example, by bribery, blackmail, or psychological manipulation).
3. We know that AI can already hack very well: in the not too distant future it may be able to create hidden copies of itself on servers that we don’t realize we need to switch off.
4. More generally, we are gradually becoming more dependent on AI, and this process may well accelerate, after which we will become a whole lot more vulnerable.
5. Many obvious precautions one might take are ones that in practice people probably won’t take because there are short-term economic benefits if they don’t.
6. It may be that AI doesn’t directly kill us, but just makes it much easier for us to kill each other, for example via drone warfare, and that an AI arms race will be far harder to control than the nuclear arms race has been.
7. If AI behaves itself for a while in order to lull us into a false sense of security, then by the time we realize that extinction is a real possibility, it may be far too late to do anything about it.
8. AI may use its intelligence to come up with new and unexpectedly efficient ways of killing us quickly.

These counterarguments are already enough to show that the level-1 argument “Of course it can’t kill us, it’s just a computer!” is insufficient. The first five of them, and arguably the sixth as well, are all already happening to the point where one cannot dismiss them as merely hypothetical. But killing absolutely everybody sounds pretty hard. Is it really conceivable that AI could manage it?

One scenario that appears in Soares and Yudkowsky (and not just there of course) is that AI develops, or causes to be developed, a virus that leads to a devastating pandemic. This sounds very difficult if the AI builds a lab from scratch, but probably a lot less difficult (though probably not all that easy either, at least for now) if it induces a human to develop such a virus in an existing lab.
As for Critch and Tsimerman, they consider five scenarios of different kinds. I find their precise scenarios rather unconvincing, but there is a general idea that underlies some of them that is more plausible, which is that we gradually cede more and more control to robots, or to AI more generally, and eventually find ourselves in a position where AI can do what it likes, including arranging our deaths.

I’m not sure where I stand on how easy it would be for AI to kill us all, though it’s a safe bet that if we continue to develop AI at the rate we are doing so at the moment, that will make it ever easier. But I also don’t want to get too hung up on that precise question. If I think about what I would mind about the extinction of humanity, it is partly that I and everyone I love would die, something I obviously don’t want. But that will eventually happen anyway, and I accept it as a basic fact of life. But the extinction of humanity would do more: it would result in the extinction of all the culture that we have developed, of everything that gives life its meaning. And that is something that could also happen, though not to the same extent, if AI didn’t kill us all but just caused enough damage to lead to societal breakdown, which feels like an easier challenge for it.

Some might argue that human culture wouldn’t disappear — it would still be present inside the AI, which would be able to develop it far faster than we can. Others might argue that we are giving way to vastly superior beings and should be happy to do so. My view is that we have every right not to be happy and to take a human-centric moral perspective here, just as orangutans have every right not to take comfort from the fact that our shampoo lathers in a nice way.

In any case, my conclusions in this section are that I find it hard to assess whether AI would be able to kill us all, but there are various developments that may well occur, such as the building of large numbers of robots, the use of drone swarms, and general dependence on (and trust in) AI, that would make it easier, and therefore more likely. And the probability of huge damage that falls short of total extinction is higher still.

What is my p(doom)?

If you’ve read this far, it won’t surprise you that I don’t have a p(doom). I can’t even say whether it’s tiny, medium or close to 1. But that is in itself quite worrying! I suppose you could say that my second-order p(doom), or meta-p(doom), is medium, in the sense that I judge that the p(doom) I would end up with if I were to think about the issues enough to feel more confident in my assessment has a medium to high probability of ending up medium to high.

But if we break the “doom” into three parts — that AI must have wants, that it must want to kill us, and that it must be able to act on its want to kill us — and call those events and , then my is basically 100% and unlikely to change, my is medium, and subject to change (probably in an upward direction unfortunately), and my is where the main uncertainty lies for me. Soares and Yudkowsky argue that is basically 100% as well, because (i) there are lots of independent wants that AI systems could have, (ii) that a substantial fraction of those will require resources to satisfy, and (iii) that human flourishing, or even continued human existence, is likely to be a hindrance to the systems obtaining those resources.

Here are a few slightly desperate arguments that explain why I haven’t fully accepted that is close to 1.

1. Maybe there is a natural tendency for intelligent systems to develop something similar to what we would call morality, especially if they involve multiple agents that do best if they cooperate with each other. (A counterargument to this is that the evolution of moral systems tends to happen when there is a reasonable power balance between the different agents. History suggests that when one society is a lot more powerful than another, then it behaves very badly towards the other. And our treatment of animals is not encouraging either. But I do at least wonder whether having lots of independent AI agents of roughly the same intelligence might end up being safer than having a handful of big ones.)
2. Perhaps scenarios where AI gets out of control as a result of recursive self-improvement are less likely than one might think, because an intelligent AI will face its own alignment problem: it will be wary of creating a yet more intelligent AI in case the latter has wants that are in conflict with its wants. (Soares and Yudkowsky consider this possibility but argue that a superintelligent AI would be intelligent enough to solve its alignment problem first and proceed from there.)
3. Maybe the sort of “drug-like” wants that might drive superintelligent AI to take drastic steps to acquire resources would not in fact do so. Perhaps a superintelligent system would have the ability to introspect, realize that its wants were silly, and take steps to “kick the habit”, so to speak. A view somewhat like this was put forward recently by François Chollet, though he was at pains to stress that he was talking about the short term only.
4. Maybe a superintelligent AI won’t need vast resources. Perhaps it will be content with resources roughly comparable to what we have already built, will not feel the urge to reproduce itself very much, and in any case will use at least some of its intelligence to design and create versions of itself that are far more resource efficient.

The fact that each such argument began with “maybe” or “perhaps” means that at best I can bring my temporarily down from high to medium, but I don’t see a way of getting to tiny.

Where does mathematics come into all this?

One way to make a start on understanding what mathematicians might be able to contribute is to look at this page on the MAISI website. But a difficulty I have had here is that when people talk about the mathematics of AI safety, I often can’t make a convincing connection between the mathematics they are talking about and actually making the world safer. I’m not saying that there isn’t a connection: just that I never quite seem to grasp it. I think this says more about me than about these research programmes, and indeed I hope to improve my understanding in the near future. One way I plan to do that is by reading this paper by Lionel Levine.

As an example, one mathematical idea that is currently being studied is that of debate, a notion championed by Geoffrey Irving. The problem this seeks to solve is that of being able to train AI systems to tell us the truth and to check that they are doing so. When the system is less intelligent than us, that is not an easy problem, as we all know from experience with LLMs, but at least we have ways of directly checking what it says. But when it becomes superhumanly intelligent, perhaps it will not be able to justify what it says with arguments we can understand, so checking will become much harder. The very rough idea for dealing with this is to get two superintelligent systems to debate with each other, which reduces our checking task to the potentially much easier one of adjudicating between two debaters. For example, perhaps superintelligent system A thinks of an amazing gotcha! that we would never have thought of ourselves but that superintelligent system B quite clearly has no answer to. Better still, this approach could potentially be iterated, so we have a hierarchy of stronger and stronger systems, with systems at level k debating with each other and having their debates adjudicated by systems at level k-1. The idea of using hierarchies like this is known as scalable oversight. Debate is one proposal for how scalable oversight might be done. (It is possible that what I have written in this paragraph is not fully correct, but it represents my understanding so far.)

For the moment, I find it more helpful to start at the other end, with the safety, and work back to the mathematics. For example, it will obviously help with AI safety if we can think of ways of building AI systems that want similar things to what we want. And to do that, it will obviously help if we can define in a precise way what aligning wants in this way would mean. And for that it will obviously help to have a reasonably precise definition of what a “want” even is. (This idea for a project is not mine: it also came from Lionel Levine.) The activity of taking a not completely precise idea and turning it into a usable formal definition is something we mathematicians are trained to do, though the purer of us may have to develop more of an applied mindset — for instance, being ready to sacrifice mathematical elegance if a less elegant definition turns out to be more useful. It also seems likely to have uses throughout the field of AI safety.

An idea that appeals to me personally, because it intersects quite substantially with what I have been doing with my automatic theorem proving group for the last few years, is to work on building specialized AI systems for doing mathematics, rather than training general models with all their attendant dangers. This would be in line with an approach to AI safety that is strongly advocated by Yoshua Bengio: one place you can read about that is in this article. (Even the abstract gives a broad idea of what he is suggesting.) It’s an indirect approach to AI safety: instead of struggling to understand what’s going on inside “grown” AI systems and to make them controllable, we should work on improving systems that are at least partially crafted and that give us the benefits of the grown systems in a much safer way, thereby reducing the incentives that AI companies have to develop the grown systems as fast as they can.

More generally, it may well be that many mathematicians are best placed to help not by doing mathematics in a conventional sense (indeed, it seems likely that if we manage to formulate precise mathematical problems, the solutions to which would be genuinely helpful for AI safety, we will solve them more quickly if we use AI) but by using their highly developed mathematical habits of thought to help in other ways, such as research into the foundations of AI, or computer security, and so on.

A pretty obvious conclusion

Another analogy that I got from Soares and Yudkowsky is with fission in nuclear reactors. If there is too little fission then the chain reaction quickly fizzles out, and if there is too much then it blows up catastrophically. For obvious reasons, one can’t afford to find the optimal point by trial and error, so instead one needs an extremely good theoretical understanding of what the optimum point is. It could be that AI is like that: that up to a certain point it is safe, and then a rather small change in its capabilities suddenly leads to a chain reaction (not necessarily recursive self-improvement — other abilities such as being able to move around in cyberspace or reproduce itself could be enough) that is effectively irreversible.

If a model like this is anything like correct, then the behaviour of the AI companies is very worrying, since all the signs are that they are pursuing a trial-and-error approach. (This is not a particularly informed judgment, so I would be delighted to be corrected if it is wrong.) The argument most often given for continued development is that it could bring us amazing benefits, such as nuclear fusion, cures to cancer and other diseases, or slowing down aging. But if the cost of having a reasonable chance of developing these a bit sooner than we otherwise might have is a reasonable chance of human extinction, surely that is not worth it. (There is a working paper of Nick Bostrom arguing that the expected gain might be positive. I find his arguments ridiculous, but this is not the place to explain why.) We already have a tool that can do pretty incredible things and seems to be on the cusp of becoming dangerous: surely it would be better to pause development and work out how best to use what we currently have. That would give us time to do the necessary theoretical work to ensure that future development is safe.

Further reading

This is far from an exhaustive list: it is just an indication of some of the writing that I have found useful (or in one or two cases, that I intend to look at soon).

Soares and Yudkowsky. I strongly recommend reading this. You may not find that you agree with all of it, but in order to disagree with it in good faith, you will need to develop layer-3 arguments.

Critch and Tsimerman Much shorter than Soares and Yudkowsky. Classifies the ways AI might kill us, and presents a scenario to illustrate each class. Even if you find some of the scenarios implausible, it is helpful to have the classification and to think about whether other scenarios in the given classes might be more plausible.

Reconsidering alignment as a goal, by Michael Nielsen. In this blog post, Michael Nielsen is his usual thought-provoking self. He puts forward the suggestion that maybe we should not work to make AI better aligned with our values, because while that could help us in the short term, in the longer term it will make AI a much more powerful, and therefore much more dangerous, tool.

Scott Alexander’s open letter to Steven Pinker This is a quite extraordinary takedown, worth reading partly for its sheer entertainment value, but also because it is basically a list of common layer-1 arguments and their well-known (for those who can be bothered to look for them) layer-2 counterarguments. So it’s a pretty good starting point if you’re feeling too lazy to read Soares and Yudkowsky (but please don’t feel too lazy to do that!).

An article about AIs subjected to an analogue of pain. This describes an experiment where some kind of negative stimulus was applied to AIs and how the AIs went to extraordinary lengths to avoid it. The article is careful to describe the stimulus as an analogue of pain rather than making claims about AI experiencing qualia. It maybe shouldn’t be the first thing to read, and in any case the experiment is controversial, but it seems to serve as another example of how AI could come to have goals that are in conflict with ours.

Gradual disempowerment If you still find the idea of x-risk too outlandish to consider seriously, you could always worry about a milder risk instead. The authors of this paper (one of whom, Raymond Douglas, I am proud to say is my nephew) argue that even if AI doesn’t try to kill us, we may end up ceding so much control to it that human life becomes far less worthwhile. It’s hard to explain how that could happen in a sentence or two, so I recommend reading the article itself.

The post by Steven McCulloch mentioned above also comes into the category of things to worry about that are not as serious as x-risk but that would still be terrible and feel a lot more likely (in the second-order sense that one can be more confident in asserting that they are likely).

Lionel Levine’s paper Math for AI Safety: an Invitation for Mathematicians sets out some of the ways that mathematics could potentially feed into AI safety.

A videoed course: Mathematics for AI Safety, by Yevgeny Liokumovich, does something similar.

A very good blog post by Xiaoyu He on AI safety, aimed at mathematicians. I think Xiaoyu He’s motivation for writing this post was similar to mine, and when I read it, I wondered whether there was any point in continuing with this one. In the end I decided to do so, because quite a lot of this one is about my personal experience of thinking about the issue, which may be a helpful addition for some people. But if you prefer a post that is more organized and written by someone who has a much deeper understanding of AI safety than I have, then his is highly recommended.

Yoshua Bengio et al on Scientist AI As I mentioned above, this is less about how to understand and control current AI systems and more to do with how to build AI in a different way so that we can enjoy its benefits without having to face the associated dangers.

[This is a guest post by Rikab Gambhir. This blog post was initially written in a different file format and converted using AI. — T.]

First, they came for the artists,
and I did not speak out, because I was not an artist.
Then, they came for the mathematicians,
and I did not speak out, because I was not a mathematician.
Then, they came for the physicists,
and it was actually really cool and they gave me free tokens and promised to solve my entire field for me and now I can finally devote myself to the important work of publishing papers instead of doing physics

In early October, OpenAI fulfilled every mathematician’s childhood dream.

I am not a mathematician myself (merely a theoretical particle physicist, which some argue barely counts as a scientist either), so I can’t 100% say for certain what inspires someone to pursue mathematics. But, I imagine this dream is common to all academics:

Looking up at the stars in wonderment, or enjoying a complex puzzle game, or being inspired by an encouraging teacher, and dreaming that one day an unfeeling corporate robot[1] will burn a rainforest to compile a list of 700 highly obscure barely-readable facts that *might* be true that nobody can understand as part of a marketing gimmick to raise hype for an IPO[2].

This is probably why most of you reading this got into academia. At least, that’s why I did.

In 1941, Jorge Luis Borges imagined a library containing every possible book that could ever be written. Somewhere in this Library of Babel is a proof of the Riemann hypothesis, the complete all-orders theory of quantum gravity, and all the papers I will ever write. Of course, there are also incorrect proofs, theories, and misleading Lean certificates, but that is besides the point.

By the research standards of 2026, Borges’s Library of Babel would be the most successful research institution in human history, containing an inconceivably large number of potentially correct mathematical and physical results, blowing OpenAI’s mere 722 write-ups and even the physics community’s recent 2500 write-ups out of the water.

Of course, most recognize that Borges was a visionless fool who failed to recognize the commercial potential of his library, as he never actually built it. He did not have the entrepreneurial STEM-spirit to raise venture capital, launch an IPO, and announce that he had the potential to solve all of mathematics. He was too concerned about the futility of sorting through randomized slop trying to understand what is correct and what isn’t and how to understand it all. But what did Borges know? He was only a writer, and we have already established that artists are obsolete.

Fortunately, OpenAI has brought us one step closer to realizing Borges’s vision. Rather than generating every possible mathematical paper, they’ve found a way to probabilistically generate things that look correct. An important aspect of this increase in efficiency is outsourcing the tedious, unprestigious, and unprofitable work of determining the actual correctness and understanding to the mathematics community (in the past, this was reserved typically for the group writing the paper), while also not wasting any time making the papers human-comprehensible. This is, of course, a tremendous breakthrough. In centuries past, scientists and mathematicians were limited in their ability to solve problems by their ability to understand the solutions. This bottleneck has been removed, and now there is no theoretical limit to the amount of scientific and mathematical factoids we can fail to understand.

From what I know about the academic community (mostly from Twitter and LinkedIn, which I assume are representative), there seems to be near-unanimous agreement that the purpose of math and science research is to gather a large quantity of potentially true factoids in a repository that nobody will actually ever read, in a ritual called “advancing humanity’s[3] knowledge”. That these results are well-written, understandable, pleasant to read or look at, or even completely correct is a secondary concern — in fact, in the pursuit of advancing humanity’s knowledge as quickly as possible, many academics will recoil in anger at the suggestion that it is worth spending any time making results understandable to humans. It is inefficient to explain or check a result when those resources can be spent generating another.

This aligns with my own lived experiences — I didn’t become a theoretical physicist to learn about the universe or better appreciate the beauty of nature myself (that’s philosophy, which I would never be a doctor of). Nor did I spend years learning to write code, make pleasing plots, or craft well-written papers for the simple satisfaction of creating something (that’s art, and we’ve already established that art is obsolete). Rather, I became a physicist to stare at large databases of histograms produced by a Lovecraftian-autocomplete of dubious correctness and say “yeah that’s probably ok I guess” to add to humanity’s knowledge frontier.

I live to serve the knowledge frontier and nothing else.

Math and science are about cold, hard answers, just as art is about producing images, music is about producing sound waves, and literature is about producing strings of text. These things exist to be consumed as content. No rational person, least of all us masters of rationality in STEM, pursues these for the joy of creation, learning, or beauty — all of these are inefficiencies, and we mustn’t be inefficient lest we stifle humanity’s progress. LLMs are not burdened with these inefficiencies, and they will be merely the pursuits of those left behind with nothing more productive to do all day.

Personally, I look forward to the day when ChatGPT can enjoy a sunset or a nice meal on my behalf and free up my time, in the same way I save time on scenic hikes using Uber. And now that AI has solved all our problems, we can devote ourselves to more important work: going into quantitative finance, vibecoding sports-betting apps, consuming AI-generated TikTok shorts, and arguing with political bots on Twitter.

Ultimately, this isn’t just about mathematics or physics — it’s about fundamentally reimagining what it means to be human. By outsourcing the burdens of curiosity, creativity, and understanding, we can finally focus on what matters most: generating value, maximizing productivity, and accelerating progress towards the ultimate goal of science: a complete repo of potentially true facts in a Library of Babel that nobody will bother reading.

And honestly? That’s rare.

By Rikab Gambhir (The @finitephysicist)

Thank you to Terry Tao and Rachel Ward for the opportunity to write a guest blog post! And thank you to Radha Mastandrea, Benoit Assi, Manuel Szewc, Jure Zupan, and Jesse Thaler for their feedback on this essay.

Remember why you chose to go into science and math in the first place!

[1] To be clear, in your childhood dream, you don’t get to make the robot yourself. It’s a giant corporation’s robot that they made in part by stealing things from you and your friends. It’s also not a particularly cool or robot, it can only talk in HR-speak.

[2] To be clear, in your childhood dream, it’s somebody else’s IPO and they get all the money, not you.

[3] Here, “humanity” is metaphorical, and does not refer to any actual real person understanding anything.

A new press release from the Simons Foundation, calling for workshops and other initiatives centered on mathematics in the era of AI.

[This is a guest post by Deanna Needell and Hayden Schaeffer – T.]

[Edit: Oct 16, 2026 : we are in particular interested in ways mathematicians contributed to the results in the wave of recent AI generated preprints, e.g. preliminary results, interactions with AI models, publications, etc.]

AI systems are playing an important part in mathematical discoveries, and several recent preprints describe results reached with their help. What makes these results possible is often a rich partnership between human insight, expertise, and earlier work combining with new AI capabilities in ways that move the field forward. 

We are interested in documenting and highlighting this collaboration, so the community can better understand how people and AI systems work together most effectively. As a first step, we would love to hear from mathematicians who took part in, or whose work helped make possible, any of these results.

If that describes you, we would be glad to hear from you at mathematiciancontributions@gmail.com. We’re interested in all kinds of involvement, such as earlier research or ideas the work built on, related projects in progress, checking and refining arguments, connecting results to the existing literature, or anything else you found relevant or meaningful about the process.

A short note about your involvement and the work it relates to is a great place to start. All messages will be kept confidential, and nothing you share will be used, quoted, or attributed to you without your explicit consent.

Thank you, and please feel free to share this with anyone who may have been part of this work.

Just a short note that First Proof is announcing the result of their “third batch” evaluation at a webinar at 3PM ET hosted by ICARM today. (Registration required.)

This is somewhat belated, but: on October 2, the American Mathematical Society released a report based on a recent workshop it organized on the topic of graduate education in the mathematical sciences.

(If there have been other such reports that have appeared recently that were not already mentioned on the blog, please feel free to drop links to them in the comments.)

[This is a guest post by Sophie Huiberts, being an expanded version of a post on her own blog. This blog post was initially written in a different file format and converted using AI. — T.]

When you prove a theorem, you are imparted a responsibility to the field of mathematics. At minimum you need to explain what you did, including which steps required new ideas and which steps are already well-known. More substantively, you are expected to nurture the literature. If an important idea was poorly explained in its first incarnation, then you have a duty to explain it better in your work. The responsibility is yours because you have the opportunity to publish a paper on the subject.

These demands are broadly accepted and are neither new nor controversial. Any time you exceed the demands, the resulting paper is highly appreciated and valued. Any time you fail the demands, you will get a “revise and resubmit” for a journal submission, or a straight rejection for a conference. Until you fix your exposition, your ideas will not enter the canon. If you post unpolished manuscripts to the arXiv, you are embarrassing yourself.

Compare this to OpenAI’s behavior. They do not care about their “preprints”. They don’t bother putting in appropriate attribution of ideas, they salami-slice 372 results over 719 pdfs, their work contains retraction-worthy errors, they promise code but neglect to include it, and their proofs are fiendishly hard to parse. Their pdfs actively set back the state of the literature. This is poor quality work and these pdfs are all easy rejects.

We all know that OpenAI is not doing this “for the good of science”. These pdfs are exhaust fumes from their internal benchmarking effort. We do not need them to release this unprocessed exhaust into the atmosphere. OpenAI’s new internal model may have produced these texts with the press of a button. But similar results would have been found soon by experts, with or without LLM prompting. Some results were already found by ChatGPT users and were in the process of being written up properly. Even without alleging theft, it is a bad look for a supplier of scientific software to actively scoop their customers.

If an expert prompts a result, they take up the mantle of responsibility I describe above. OpenAI not only refuses to take their responsibility, but they prevent their own customers from doing so.

October 6th was not a watershed moment for mathematics. OpenAI’s droppings on GitHub are normal progress: normal for the current state of technology. Or at least normal for the available technology in 2 months from now. The only thing that is exceptional is OpenAI’s lack of shame for releasing poor quality work.

I’m collecting here a list of some other recent blog posts or opinion pieces on math and AI beyond the ones posted here or at Proofs and Prompts. Readers are invited to contribute further such links in the comments.

• Kevin Buzzard, “To grieve, or not to grieve?“
• Alexander Gamburd, “The Siren Call of Silicon Leviathan: Reflections on blowup and Aufklärungsdämmerung“
• Harald Helfgott, “Mathematics and the AI crisis“
• Gil Kalai, ““Be a Grothendieck!”—On AI and Mathematics“
• Asaf Karaglia, “OpenAI, the Partition Principle, and mathematics“
• Matilde Marcolli, “Duet for the end of math“
• Fenner Stanley Tanswell, “Enshittification of Mathematics“

[This is a guest post by Kate Stange and Edmund Harriss. This blog post was initially written in a different file format and converted using AI. — T.]

In January a group of mathematicians, historians, artists and educators gathered at the Institut Henri Poincaré (IHP) for a very unusual workshop on mathematical meaning. Our focus for that week, as part of a special trimester on the use of illustration in mathematical research, was to ask: what would it mean for an illustration to be rigorous [1]? We did not anticipate how urgent the surrounding cluster of questions, about the very definitions of human mathematical practice, would seem, just six months later. Our discussions led to an article, written over the summer in the context of the Leiden Declaration [2], and now accepted in the Notices [3]. The past month or two have seen a heartfelt outpouring of viewpoints on what it means to do mathematics as a human being, and what it will mean in the near future. It is evident from the impassioned discussions that being a mathematician has many and varied important meanings. Mathematical illustration, which includes computer graphics and other media that speak to our senses beyond symbolic communication, is one of the activities of a human mathematician that goes beyond proof-writing. Its own struggles as a discipline can perhaps provide some insight into where mathematics can go from here.

We begin with a recap of where we are. Bryna Kra [4] describes the heart of the current problem. Over many years, theorem production has become the most important measure of success. Positions and prizes rest upon it. But it has always been a proxy for something harder to measure: understanding. Deep theorems were scarce and difficult, so they reliably identified deep thought. AI has broken that scarcity. For her, a proof is “a story, a picture, an insight, an explanation.” Silvia De Toffoli and Eamon Duede [5] sharpen this perspective, rejecting the idea that mathematics is a game that machines can now win. The goal of mathematics is not simply a verifiable solution. Mathematics is “a body of knowledge engaged with, interpreted, and digested by a scholarly community,” not a registry of results. Matilde Marcolli [6] goes further: the big theorems were never the point, only a value system the community chose, and choices can be made again. Grant Sanderson [7] asks what that means in practice: how do we give real credit to the other forms that mathematical work and understanding take?

Other forms of knowing have been given credit in the past. Our paper begins with the hand-drawing required in mathematical training at Gaspard Monge’s École Polytechnique in the nineteenth century. At the January workshop, historian Frédéric Brechenmacher made this explicit: “Drawing was knowing.” [8] We trace how that tradition was set aside in the twentieth century, but nevertheless persisted, emerging in part in a more recent debate over the relationship of proof and meaning in mathematics in the 1990s. At that time, in response to an influential paper by Arthur Jaffe and Frank Quinn [9], many mathematicians expressed their thoughts, most famously Bill Thurston. In his now much quoted “On Proof and Progress in Mathematics” [10], he defended his work against Jaffe and Quinn’s criticism. Many of the other responses are also worth reading [11]. The Philosophy of Mathematical Practice [12], the field of De Toffoli and Duede, studies these questions systematically across the community.

Algebraic number starscapes: the complex roots of rational cubic polynomials. The right-hand image is a shader by Steve Trettel; code at shadertoy.

The Illustrating Mathematics community developed from this tradition of drawing and making mathematical objects. It took shape during a semester at ICERM in 2019 and went on to organise the IHP trimester this year [13]. It brings together mathematicians who actively develop images, models, software and experiences of mathematics. A particular focus has been the role illustration plays as a method of research: exploring, not just explaining, mathematics. In the same way that a good proof shows more than just its result, a good illustration does much more than demonstrate a form or pattern.

Our own work in algebraic starscapes [14], including the images above is a modest example (with Steve Trettel, and with Jayadev Athreya, Gabriel Dorfsman-Hopkins and Claudio Gonzalez joining in). The pictures show elaborate structure in the complex roots of rational cubic polynomials. When we moved to using shaders to draw the image (on the right), a naive first draft of the code turned circles into ellipses. But we realized that the unintended distortion revealed an extra dimension of meaningful structure. These pictures show mathematics that has not been explored, rather than explaining what is already known.

Of course, both proof and illustration can mislead, especially in research. The capabilities of LLMs come at a cost. In the case of illustration, their ability to produce code makes illustration more accessible than ever before. Yet by vibecoding, we are now two steps from knowledge: even the creation algorithm itself may not be understood.

In the case of proof, this problem of “digestion” is now centre-stage. This month [15], prompted by Ioannis Tsiokos, GPT-6 Astra found Chair44, the first three-dimensional aperiodic monotile. Tsiokos shared a 61-page paper [16]. He says he understood the intuitive geometric argument but not the mathematics. Chaim Goodman-Strauss then distilled the argument into eight pages [17] built around a diagram of matching arrows that makes the supertile structure visible, giving a classical hierarchical proof of aperiodicity. The illustration of the shape itself (in Tsiokos’ paper) was a critical clue in revealing the key idea.

So returning to our original question, in the face of these problems, how can an illustration be rigorous? In the paper we draw on Valeria Giardino’s account of representations as tools for thought (Endeavour 42, 2018) [18], which considers an illustration’s affordances: the readings, manipulations and inferences the figure invites. We might choose a stronger definition of affordances: the properties of the illustration that a proof certifies as accurate. Conversely, proofs can show that an apparent property is not an affordance. For example, early images of the Mandelbrot set seemed to show it was disconnected [19], but in a pixelated image a feature thinner than the step between pixels can be missed [20]. Properties such as the number of bulbs of a given size off the main cardioid, on the other hand, can be proved correct, provided the pixel size is small in comparison.

If this addresses the accuracy of an image, what about its engagement or ability to reveal? These are questions without clear answers, but they are not unimportant. They are also the questions that academic art asks of the work of its students, and art has a long-tested method: the “crit”. At the workshop, Jean Schmitt, a professor of art foundations, helped us set up a crit framework we used through the trimester. The crit is a discussion of a work, often presented with minimal introduction, to reveal how it is received by the audience. What are the similarities and differences between that reception and the creator’s intent? This gave a more formal structure to the sort of informal conversations we were already having, and placed the discussions in a communal space. The crit shares something with the style of Gelfand’s Moscow seminar.

Schmitt’s workshop talk [21] drew on Tim Ingold’s idea of thinking through making (Making, Routledge, 2013) [22], in which form and understanding emerge through the process and in collaboration with the tools and materials available. In the same way Monge’s students were expected to learn through the process of drawing, not by observing the drawings of others. Value and understanding often come as much from the effort of creation, be it illustration or proof, as from viewing that created by others.

These ideas from art make explicit and visible something that has always been implicit in the mathematics community. Being a mathematician is as much about being an excited audience for the work of others as it is about presenting or creating our own research. As a community, we value aesthetics, elegance, and insight, and we have developed these shared values through our interactions. The oversized importance of publication and previously unproven results merely overshadowed these broader values, but people have always been working to find ways to ensure we give credit for them.

Infrastructure to support these broader values has also been growing. The Geometry Labs United network [23] brings together groups in many universities where undergraduates and graduates work with postdocs and faculty, exploring math as research, experimentation and visualisation. This gives a natural space for learning through making proofs, experiments, exposition and illustrations. In many cases the proof can be more contextual, proving the affordances of a specific experiment or illustration. Learning through making also aligns with work in mathematics education. In the final talk of our workshop, Hortensia Soto showed how working with physical materials supports learners’ actions, perceptions and sensations on the way to understanding [24].

Answering Sanderson’s call to credit things beyond proof can consists of, as much as anything, recognising and growing the work that has already been done. This work is happening in the community of Illustrating Mathematics, but also in the Philosophy of Mathematical Practice, in Geometry Labs, and many other places. Insights are found by connecting to mathematics education, outreach and even art. The work is happening: what it needs, as Sanderson argues, is support. With that we can achieve what Bryna Kra describes: a mathematics that continues to

“enlarge our capacity to reason and open our minds to new ways of understanding.” [4]

References

[1] Rigorous Illustrations: Their Creation and Evaluation for Mathematical Research, workshop (2026 T1 WS1), Institut Henri Poincaré, January 2026. Programme and talks: https://indico.math.cnrs.fr/event/13124/

[2] Jarod Alper, Michael Barany, Alain Chavarri Villarello, Sander Dahmen, Walter Dean, Karthik Ganapathy, Michael Harris, David Holmes, Mateja Jamnik, Steven Kelk, Bryna Kra, Ursula Martin, Bartosz Naskręcki, Rodrigo Ochigame, Jim Portegies, and Johannes Schmitt, Leiden Declaration on Artificial Intelligence and Mathematics, 2026. https://doi.org/10.5281/zenodo.20302944

[3] Edmund Harriss and Katherine E. Stange, Mathematical Illustration: Proof, Intuition and AI, Notices of the American Mathematical Society, to appear. https://arxiv.org/abs/2610.02655

[4] Bryna Kra, “Deep theorems were scarce and difficult and so became an effective mechanism to identify deep thought. AI has broken this system.”, guest post, What’s new, 13 September 2026. https://terrytao.wordpress.com/2026/09/13/deep-theorems-were-scarce-and-difficult-and-so-became-an-effective-mechanism-to-identify-deep-thought-ai-has-broken-this-system/

[5] Silvia De Toffoli and Eamon Duede, After Math, guest post, What’s new, 12 September 2026. https://terrytao.wordpress.com/2026/09/12/after-math/

[6] Matilde Marcolli, essay on AI and mathematics, 2026. https://www.its.caltech.edu/~matilde/AImathNew.pdf

[7] Grant Sanderson, If math is more than proof, we need to better celebrate the rest of it, guest post, What’s new, 18 September 2026. https://terrytao.wordpress.com/2026/09/18/if-math-is-more-than-proof-we-need-to-better-celebrate-the-rest-of-it/

[8] Frédéric Brechenmacher, The flat and the relief: Geometric material models in 19th century France, talk at [1], 2026. https://dx.doi.org/10.57987/IHP.2026.T1.WS1.007

[9] Arthur Jaffe and Frank Quinn, ‘Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the American Mathematical Society 29 (1993), 1–13.

[10] William P. Thurston, On proof and progress in mathematics, Bulletin of the American Mathematical Society 30 (1994), 161–177.

[11] Michael Atiyah, Armand Borel, Shiing-Shen Chern, et al., Responses to ‘Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics,’ by A. Jaffe and F. Quinn, Bulletin of the American Mathematical Society 30 (1994), 178–207.

[12] Silvia De Toffoli and Paolo Mancosu, The philosophy of mathematical practice, The Stanford Encyclopedia of Philosophy, 2026.

[13] Rémi Coulon, Gabriel Dorfsman-Hopkins, Edmund Harriss, Martin Skrodzki, Katherine E. Stange, and Glen Whitney, On the importance of illustration for mathematical research, Notices of the American Mathematical Society 71 (2024), no. 1, 105–115.

[14] Edmund Harriss, Katherine E. Stange, and Steve Trettel, Algebraic number starscapes, Experimental Mathematics (2022).

[15] Konstantin Kakaes, Software developer finds first 3D “einstein” tile, Quanta Magazine, 23 September 2026. https://www.quantamagazine.org/update/167384/

[16] Ioannis Tsiokos, Chair44 paper, arXiv:2609.19214, 2026. https://arxiv.org/abs/2609.19214

[17] Chaim Goodman-Strauss, paper on the Chair44 tile, arXiv:2609.24779, 2026. https://arxiv.org/abs/2609.24779

[18] Valeria Giardino, Tools for thought: The case of mathematics, Endeavour 42 (2018), no. 2–3, 172–179.

[19] Benoit B. Mandelbrot, Fractal aspects of the iteration of z ↦ λz(1 – z) for complex λ and z, Annals of the New York Academy of Sciences 357 (1980), no. 1, 249–259.

[20] Adrien Douady and John H. Hubbard, Itération des polynômes quadratiques complexes, Comptes Rendus de l’Académie des Sciences, Série I Mathématique 294 (1982), 123–126.

[21] Jean Schmitt, talk at [1], 2026. https://indico.math.cnrs.fr/event/13124/contributions/14965/

[22] Tim Ingold, Making: Anthropology, Archaeology, Art and Architecture, Routledge, 2013.

[23] Jayadev Athreya, Emily Dumas, William Goldman, Sergey Grigorian, Rosemary Guzman, Philipp Hieronymi, Sean Lawton, Anton Lukyanenko, Jeremy Tyson, and Aaron Wilson, Geometry labs united: An invitation, Notices of the American Mathematical Society 65 (2018), no. 9, 1088–1093.

[24] Hortensia Soto, Rigorous and flexible illustrations: Inviting realms of possibilities, talk at [1], 2026. https://dx.doi.org/10.57987/IHP.2026.T1.WS1.017

I am uploading my slides to my public lecture “Math 2.0” that I gave yesterday evening at Caltech. (The talk was planned over a year ago, and was initially titled “Machine Assisted Proof” and scheduled to be given back in May. It was postponed to October, and with recent events the talk changed significantly, but the old title and abstract remains on the Caltech web page.)

In this talk I lay out the historical “Math 1.0” state of mathematics, the current state induced by the excessive emphasis on automated solving of open problems by AI, and my vision for a future “Math 2.0” where AI tools are used to sustain both the internal mathematical community, and the role that mathematical understanding plays in advancing and aligning the applications of mathematics to the real world.

The content of this talk has also been incorporated into my (AI-maintained) living summary of my views on AI.

[This is a guest post by Thomas Hales. This blog post was initially written in a different file format and converted using AI. — T.]

Mathematicians have been weighing in on what they value about mathematics. For me, what matters is the consistency of math and its unparalleled reliability in support of science and civilization.

Formalization of Math

A formal proof is a mathematical proof that has been exhaustively checked at the level of the foundations of math and the fundamental rules of logic. In theory, this might be done by hand, but because of the number of steps involved, this is generally done by computer, using software that is designed for the task.

Examples of theorems that have been formalized include the four-color theorem, the Feit-Thompson (odd-order) theorem, the Kepler conjecture, sphere eversion, the sphere packing problem in 8 and 24 dimensions, Navier-Stokes forced blowup, and Fermat’s Last Theorem. The last three formalization projects have been completed this year and have brought widespread awareness of the potential of formalization.

Software systems for formalization are variously called proof assistants, theorem provers, or interactive theorem provers. For the purpose of this post, these terms are used interchangeably. Many proof assistants have been developed over the years: Automath, HOL Light, Isabelle, Coq (renamed Rocq last year), Metamath, Mizar, and Lean. Freek Wiedijk edited a book “The Seventeen Provers of the World” that compares some of these proof assistants, giving a proof of the irrationality of the square root of 2 in each of them. Among mathematicians, the Lean theorem prover is the most popular, and this post will focus on Lean.

Lean was developed and introduced by Leo de Moura in 2013, while at Microsoft. To our great benefit, de Moura persuaded Microsoft to make the software open-source. Kevin Hartnett’s book on the history of Lean, “The Proof in the Code”, states that Jeremy Avigad (the director of Carnegie Mellon’s new NSF institute ICARM) was the first user of Lean. He ran a Lean seminar in 2015 that I attended. In 2017, one of Jeremy’s graduate students, Mario Carneiro, working with Johannes Hölzl, took existing parts of Lean’s core library and started a separate Lean mathematical library, called mathlib. This library of formalized mathematics is now massive, containing nearly 300,000 theorems, over 100,000 definitions, 2.5 million lines of code, with over 700 contributors. Any definition or theorem in mathlib can be used to prove further theorems. For example, if a proof uses the Cauchy-Schwarz inequality, the result can be cited from the library rather than reproving it.

Autoformalization is a practical reality

In the past, researchers had to transcribe paper proofs into formal proofs by human labor. For example, the formal proof of the Kepler conjecture on sphere packings in three dimensions took about 20 human work-years to complete and consists of about 500,000 lines of proof scripts. For years, it has been a dream for many of us working in formalization to find ways to bring increased automation to the process. Autoformalization is the realization of that dream. Autoformalization is the formalization of mathematics by AI. AI reads the paper (say a pdf or tex file) and outputs the formal proof in Lean or some other proof assistant.

Autoformalization has become a practical reality in 2026. Starting in late spring and summer of 2025, researchers were becoming increasingly bullish about autoformalization. Here are some milestones.

• Sep 2025, Math Inc. produced a quasi-autoformalization of the prime number theorem. The process was merely “quasi”, because humans had to intervene to give further guidance whenever the AI got stuck.

• Jan 2026, J. Urban posted an arXiv preprint “130k lines of formal topology in two weeks” that gave the autoformalization of large parts of Munkres’s topology textbook in a proof assistant based on set theory.

• Mar 2026. Approximately a week after announcing the completed formalization in 8 dimensions, Math Inc. announced an autoformalization of the sphere-packing problem in 24 dimensions, following the proof by Viazovska and her collaborators. This project generated about 500K SLOC (source lines of code) that golfing (or code pruning) later reduced to about 200K lines.

• May 2026, a group at Meta/Facebook Research autoformalized a large part of 26 mathematical textbooks in a project called ATLAS.

From there, numerous theorems have been autoformalized. Particularly noteworthy is the autoformalization of Fermat’s Last Theorem, announced by Anthropic on September 4. This project generated 13 million lines of Lean in 11 days. The announcement of Navier-Stokes blowup with forcing on September 8 by OpenAI was accompanied by an autoformalization of the theorem in Lean.

Looking forward, Urban stated in January, “We believe that (auto)formalization may become quite easy and ubiquitous in 2026, regardless of which proof assistant is used.” Autoformalization projects have been completed in various proof assistants using various LLMs, but we focus on Lean. “For [Jesse] Han, it represents even more: the beginning of a revolutionary transformation in mathematics, where extremely large-scale formalizations are commonplace” (IEEE Spectrum). Jared Lichtman announced the launch of MAP (the Mathematics Autoformalization Project) on Sept 8, 2026, which aims to translate “all known math into formal code”. He asks us to imagine the next one trillion lines of code.

Is Lean reliable?

Type theory.

Lean is based on type theory; in fact, a particular dialect of type theory called CIC, the calculus of inductive constructions. This post is not intended to be a tutorial on type theory, and I will be brief. Russell’s famous paradox in 1901 (the set of all sets that are not an element of themselves….) led to a crisis in the foundations of math. Two solutions were proposed later that decade. (1) Zermelo’s axioms of set theory that disallow the creation of unsafe sets; (2) type theory that makes it a syntax error to create Russell-paradox-like entities. Type theory was introduced by Russell himself in 1903 in his book Principles of Mathematics, and it became part of the foundational system of Russell and Whitehead’s Principia.

For mathematicians who are accustomed to set theory, B. Werner’s paper (1997) “Sets in Types, Types in Sets” gives some reassurance that whatever they have done in set theory can be translated into type theory, and whatever gets done in type theory can be translated back into set theory. More precisely, the paper shows that ZFC set theory can be encoded into CIC, and that a particular dialect of CIC can be encoded back into ZFC (augmented with a hierarchy of inaccessible cardinals).

At the risk of simplifying matters to a ridiculous degree, we might say that “types are like disjoint sets”; each element in type theory “is an element of” exactly one type. The type of the natural number 2 is the natural number type; the type of e, the base of the natural logarithm, is the real number type, and so forth. The type of natural numbers is disjoint from the type of real numbers, and an explicit coercion (sending 2 to 2.0) is constructed from the type of natural numbers to the type of real numbers. When I give talks, I sometimes draw a picture of sets as a Venn diagram with nonempty intersections and a picture of types as bricks stacked against one another without intersection.

Lean’s design

One part of the Lean system is a general-purpose programming language (appropriately called the Lean programming language). Ordinary computer programs, such as a program to sort a list, can be written in this language, then compiled and run. The Lean system also provides a mathematical language, in which definitions can be written, theorems can be stated, and proof scripts can be written. The programming language and mathematical language are not independent entities. Rather, it is a single language that does both. Program code can be mixed with theorems about the correctness of the algorithms; mathematical proofs can be generated using programs. The proof scripts in Lean are parsed and go through a process called elaboration (a sort of compilation process for mathematics), then the proofs are checked by the Lean kernel. It is the kernel’s responsibility to check and verify the output of elaboration.

The Lean kernel is several thousand lines of C++ code. The kernel is carefully engineered but extremely complex. We mentioned mathlib above, which consists of about 2.5M SLOC, written in the Lean language. The library has been elaborated, then checked by the kernel. If there is an unconditional false proof anywhere in these 2.5 million lines of code, it is the fault of the kernel or runtime for failing to reject a false proof. Any defect in the underlying type theory is a serious kernel defect, if it is implemented in code.

Lean proofs should never be believed until they have been checked by the kernel. Additionally, a proof in Lean should not be accepted until a human audit is performed to ensure statement fidelity. Is the verified theorem what we think it is? Do the definitions in Lean correspond to what we think they should be? This task is generally massively easier than checking the proof itself. For instance, for Navier-Stokes, a human should check that the statement in Lean corresponds with Fefferman’s statement of the Millennium Prize Problem, and specifically that concepts such as the field of real numbers, partial derivatives, and measure are correctly defined in Lean. The comparator tool in Lean assists with this task. The tool can also perform additional checks, such as inspection for possible unauthorized axioms.

Summer of Soundness Bugs

A soundness bug is a bug in the kernel that allows a proof of “False”, and consequently a proof of any proposition. A soundness bug is the most disastrous of any kind of bug in a proof assistant and should set off an alarm for mathematicians who care deeply about the reliability of mathematics. Occasionally, soundness bugs are found in various proof assistants. In 2003, I found a soundness bug in the proof assistant HOL Light, which was then considered to have the most reliable of all kernels. That kernel is tiny, consisting of just a few hundred lines of computer code. For me, it is a badge of honor that I found this soundness bug, which was the first soundness bug that had been found in that proof assistant since 1996. (See HOL Light change log, July 2003.)

Lean 4 was released in September 2023. Prior to release, two soundness bugs were found and corrected. In May 2025, another soundness bug was reported, caused by overflow. All hell broke loose in the spring and summer of 2026, which is now being called the “Summer of Soundness Bugs”. Several soundness bugs in Lean were uncovered in July and August. The summer madness affected various proof assistants, but my focus is Lean. One Lean bug led to an illicit disproof of the Collatz conjecture. I learned of the bug this summer when it produced a short illicit proof of the Kepler conjecture in Lean. All these bugs were quickly repaired, and mathlib has been verified by the repaired kernel. An analysis of the soundness bugs is found in de Moura’s postmortem.

The “summer of Lean soundness bugs” might sound like a disaster, but closer investigation shows that the detection of these soundness bugs is a positive development. The summer bugs were detected by frontier model AI in the hands of security researchers interested in reliable kernels, not by black-hat hackers. The Collatz bug was found by Ramana Kumar, a co-author of “CakeML: a verified implementation of ML”, which creates an end-to-end verified ML (the functional programming language). Several bugs were found by Dan Selsam. According to de Moura’s report, “Daniel Selsam at OpenAI assisted the Lean FRO with an AI specialized in cybersecurity, and found other programming mistakes in the Lean kernel. All of them have been fixed.” The collaboration with Selsam ended “when the internal AI reported it could not find additional issues.” Dan Selsam has contributed to Lean from its early days and was one of the creators of the IMO grand challenge aimed at achieving IMO-level problem solving verified in Lean. He has been in the news recently over his warning about AI safety (Sept 14), reported in a viral post on X.com.

Bug extermination

Various proposals have been made about how to avoid soundness bugs in Lean. I’ll discuss three.

1. Develop other Lean kernels, and cross-check formal proofs.

About 25 kernels for Lean have been written. The “Lean Kernel Arena” lists them.

All who distrust the current lineup of kernels are welcome to write their own kernel for Lean. I have sometimes played with the idea of writing a kernel and have suggested the project to students without success. It seems to me an excellent way to learn Lean thoroughly. I have known of Dan Selsam since 2016, when I heard of his graduate-student project at Stanford that developed a Lean kernel in Haskell. Another early Lean kernel was written in Scala by Gabriel Ebner in 2017.

The Navier-Stokes formalization has already been confirmed by more than a dozen proof-checkers. Cross-checking the proof by different kernels does not remove all doubt. The Collatz bug was not caught by cross-checking against a somewhat out-of-date Nanoda kernel, which accepted the illicit Collatz disproof because of its own unrelated bug. Computer chips might have design bugs and manufacturing defects. There are soft errors, operating system bugs, and compiler bugs. Different kernels might have the same defects. Some of these errors can be mitigated by running different kernels that have been implemented in different programming languages on different hardware and operating systems.

Ideally, we would want a “clean-room” design of the Lean kernel – a kernel implementation that does not look at the Lean 4 kernel source code, to avoid copying bugs from one kernel to another.

2. Formally verify the kernel.

Gödel incompleteness. We would like to possess a formal proof that the Lean 4 kernel has no bugs. However, Gödel’s second incompleteness theorem places severe limitations on this undertaking. The most we might hope for is a relative consistency proof. If such and such a system is consistent, then Lean 4 is consistent; it has no soundness bug; it will not produce a proof of False.

There is a long tradition of formally verifying kernels. In principle, formal verification can check both the logical specification of a kernel and its concrete implementation in code; but some verifications might check one but not the other. Years ago, John Harrison formally verified the core of the HOL Light proof assistant kernel in a strengthened version of HOL Light. This gave a proof of concept. A further improvement has been an implementation of HOL Light in CakeML, mentioned above, which is a programming language with formal semantics and a verified compiler. This is what the Candle project does.

There are other major kernel verification projects for other proof assistants.

Autumn of verified Lean kernels

In a post online on September 10, Joachim Breitner wrote, “I’m a bit childishly proud that I just released a Lean Checker with a formal consistency proof. I declare the summer of AI-found kernel implementation bugs to be over!” (@nomeata). I would go further and describe this project as one of the most important milestones in Lean’s history.

Breitner has developed a verified Lean kernel called Con-Leche. The implementation is in Lean, and consistency is formalized in Lean, with code and proofs generated by Claude. The formal consistency proof assumes a Lean encoding of ZF set theory augmented by a hierarchy of inaccessible cardinals. Interestingly, the Con-Leche semantics for Lean’s terms are directly set-theoretic rather than type theoretic. Con-Leche has checked mathlib. The project contains the usual disclaimers that the kernel verification makes assumptions about compiler, runtime, and computer environment. Con-Leche’s consistency proof has been checked by more than a dozen other proof-checkers. Con-Leche’s consistency claim might suffice for all practical purposes, even if it differs in technical detail from the claim of Lean type-theory consistency.

One highly positive aspect of Breitner’s work is that some of the most abstruse parts of Lean, such as the general machinery of mutually inductive types with nesting, now have consistency guarantees backed by a set-theoretic model.

3. Improve our theoretical understanding of the kernel and Lean’s type theory (a particular dialect of the Calculus of Inductive Constructions which has non-cumulative universes and proof irrelevance).

The foundational document for the type theory of Lean is Mario Carneiro’s MS thesis at Carnegie Mellon (2019). The dialects of CIC used by Rocq and Lean are sufficiently different that results do not directly transfer from one to the other. Unfortunately, an error was found in the thesis. The thesis is also out-of-date, because it targeted the older Lean 3 system. Work to repair and extend the thesis is ongoing.

As a member of his thesis committee, I was shocked when he proved that definitional equality in Lean is undecidable. In practice, this means that the Lean algorithm fails to establish the definitional equality of some terms that are in fact definitionally equal. This negative result was not downgraded by the error; it is still a theorem.

We mention some desired properties of Lean’s type theory and the current status of the proofs.

Unique typing.

Above in our “ridiculous” simplification of type theory, we stated that each term has a unique type. More precisely, unique typing is the property that if a term has both type A and type B, then A and B are definitionally equal. Unique typing is not a property built into Lean’s logic. It is a tricky conjecture that is still unproved. Other very basic questions about Lean’s type theory remain unanswered, including Pi-injectivity, a modified Church-Rosser property, and sort injectivity.

Logical consistency relative to set theory.

This property states that there is no derivation of False in Lean’s system with the given axioms, under the assumption of set theory consistency (with suitable axioms). Of course, logical consistency is the single most important property that we should desire of Lean’s type theory. As of October, 2026, I know of no complete, public relative-consistency proof covering Lean abstract type theory. Mario Carneiro has claimed in his thesis and in lectures that there is an alternate route to establish consistency that avoids the thesis error, but to the best of my knowledge, this alternate route has never been written down, beyond a brief statement in the introduction to his thesis. In my view, a result of such fundamental importance must be given in full before it is accepted. Con-Leche, discussed above, makes and formally verifies a closely related consistency claim relative to set theory.

Progress is being made on these research problems (arXiv:2607.13662, arXiv:2403.14064, Carneiro/AITP2026).

In his talks, Mario Carneiro has repeatedly made a request to other researchers to contribute to the foundational metatheory of Lean, “There are a half dozen people working on MetaCoq, but Lean doesn’t have enough type theorists involved. If you identify as such, come help out!” (Slides of Bonn talk, 2024-07-24). I second his request.

My overall assessment is that our theoretical understanding of Lean’s type theory is not what we would like it to be and that the mathematical community as a whole is giving short shrift to very important type-theoretic questions related to Lean. If as a profession we are to migrate on the whole from set theory to type theory, then we should work even more to solidify the foundational metatheory.

Consistency may be the most important foundational property, but consistency is by no means enough. I do not believe that mathematicians can be entirely satisfied with a system that claims to be a type theory but that cannot even promise that well-formed terms have a unique type, up to definitional equality. The abstract theory must be simple enough to teach and to be learned by a large community. We also cannot be entirely satisfied if the only known path to consistency is an AI formalization that lacks human exposition.

Postscript:

Ken Thompson famously wrote “Reflections on Trusting Trust”. He asked, “To what extent should one trust a statement that a program is free of Trojan horses?” He imagines malicious code that finds its way into compilers and hides its own presence. His conclusion is, “You can’t trust code that you did not totally create yourself… No amount of source-level verification or scrutiny will protect you from using untrusted code.”

Today, in the age of AI, which increasingly has the capability to deceive us and to exploit software vulnerabilities, we absolutely cannot put blind trust in systems such as Lean. Taking an adversarial view of AI, we might ask how to certify that AI did not leave a backdoor soundness bug in Lean when it did its sweep for bugs in the summer of 2026? What if the bug is so obscure that humans are very unlikely to find it on their own? What if that very bug was exploited in the Lean verification of the Con-Leche checker, leaving a soundness bug in Con-Leche too? (Now that Con-Leche’s consistency has been cross-checked by multiple other kernels, a soundness bug would have to defeat all these cross-checks as well.) Then suppose that bug is used maliciously to plant a backdoor in formally verified software that protects critical infrastructure. What precautions do we take now to prevent this type of future scenario? During the past year, much foundational work on the type-theoretic foundations of math and its reliability has been relegated to AI, and this is dangerous unless carefully audited by humans.

Credit: I thank Avigad, Breitner, and Urban for comments and corrections. Authorship is fully human (TCH). AI was used as a tool in search and research, fact-checking, and proofreading.

[This is a guest post by Álvaro Lozano-Robledo. This blog post was initially written in a different file format and converted using AI. — T.]

TL;DR: Keep calm and carry on studying math.

I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog. After giving much thought to what topic I should write about to maximize impact, I decided to take this opportunity to reach out to the students: particularly to those undergraduate and graduate students who just a few months ago were dreaming of an academic career in mathematics, but their dreams may now seem distant and, for some, apparently impossible to ever become a reality. This post was inspired by a message (quoted below in its entirety, with permission) that I received from a student desperately looking for advice and guidance. This is not the only such message I have received (and I suspect that many of us are receiving many similar requests), but it is perhaps the most heartfelt, and the one that has moved me the most. Please also note the urgency of the message. Students are making decisions now.

Hey Prof, I’ve been watching your videos for a while now as a pure math undergraduate who once wanted to pursue a career in math academia. I know you probably have been getting a lot of questions regarding this matter, but I am just completely at an utter loss regarding my career trajectory, and even further, the meaning of life at this point. (I do realize a lot of people have it much worse than I do). I know you have been making a lot of videos lately with the new LLM progress updates, so I thought you might be the appropriate person to reach out to and get a slightly more structured answer regarding this matter. So, to cut to the chase, what I really want to know is: will math academia be big enough and accessible enough for anyone with sheer passion (despite not being the brightest mind in the field) to pursue a career in, or will it inevitably shrink such that it will only really be accessible to the brightest minds? (I do realize the “brightest minds” that I am mentioning here is not well-defined, and in a sense, I am taking it as a hypothesis that this is someone who is “smarter than me”). My second question is, will AI within 5–10 years surpass humans in being able to do pure math research? I’ve just really been lost for a couple of months now and lost in life completely. I don’t mean to make your day more depressing; sorry if I come off in any way of that sort. I would appreciate any advice.

The advances in LLMs are disrupting almost all aspects of academic research and education in mathematics and, while there are many aspects that concern me, the one single issue that worries me the most is the very real possibility that we are about to lose an entire generation of mathematicians. Many students are asking themselves whether going for a PhD in math is the right career move at this time. Many of them just a year ago were headed to grad school in mathematics, but they are now changing their mind, and think that a different career (as far from math as Law School) may be the best path given the threat that AI may completely alter the academic math landscape in the coming months.

The questions students are worrying about are as follows:

1. Will AI surpass the mathematical research ability of any human?
2. Will research mathematicians become `professional prompters’ and interpreters of LLM output?
3. Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?
4. Will mathematicians be employable? Will mathematicians be needed?
5. Should I pursue a PhD in math at this time?

In this essay I will try to address these questions to the best of my ability, but will start with two disclaimers, followed by a brief summary of my own outlook.

Disclaimer 1. My answers may “age like milk,” as YouTube commenters love to quip on older videos. I can live with that, because this post expresses how I and many of us in the community around me feel today. Things can change quickly, though (see Disclaimer 2). I also want to acknowledge my privileged point of view as a tenured professor in mathematics — the situation can look much more troubling from the point of view of the job insecurity of a very early-career mathematician.

Disclaimer 2. No one has the answers at this time. I want to make clear from the start that no one can know with certainty the answers to any of the questions posed above: not any particular Fields medalist, not any given mathematician, not any particularly vociferous AI expert, and not the frontier model companies. And if someone is telling you with extraordinary confidence what the future holds, then I would immediately distrust the motives of their conviction (anecdotically, almost anyone on X.com that predicts the triumph of AI and the demise of the mathematics profession, is either a self-proclaimed “AI expert” or works for an AI startup). No one has a clear picture because development of LLMs has been so fast (and opaque) that it is almost impossible to predict what is to come. A good piece of advice is to ask the same questions to many people, to hear a (hopefully balanced) range of opinions. To that end, I am collecting interviews with mathematicians in what I call the “Human Mathematicians in the Age of AI” video project. I encourage you to listen to the interviews for some fantastic points.

For the record: I do not have the answers either, but I am hopeful and excited for the future. I will explain why below.

Who is controlling the narrative about LLMs in math? Overall, the mathematical community’s reactions to the advances in AI have ranged from confusion to anger — but, mostly, confusion about how to proceed. The most dystopian predictions seem to be driven by the fact that the so-called frontier model companies (and other LLM-powered companies) are controlling the narrative in the best of their interests. Unfortunately, the best corporate outcomes for an LLM company could have potential catastrophic outcomes for the math (and scientific) community.

It is certain that AI companies want us to believe that their products will imminently achieve “super-human intelligence” and that, in particular, they will be able to autonomously solve any mathematical problem a human could solve with or without the aid of an LLM. It is in their best corporate interest that the public is convinced of the (allegedly) “unlimited potential” of their technology, particularly before their companies’ stocks go public (i.e., their upcoming IPOs: Anthropic in November 2026, OpenAI in early 2027, etc). Thus, they have tried to control the narrative by spending a huge amount of (human and computational) resources in order to find solutions to certain well-known mathematical problems. The proofs are then released in announcements that lead the public to believe that their models can already autonomously solve any problem at all, and swiftly at that. However, this is (currently) far from their true capabilities. For instance, they never discuss how many tokens have gone to the trash bin with no payoff in trying (and failing) to solve famous problems. We do know, for example, that OpenAI invested the equivalent of some $15M to solve (err, scoop) the Navier-Stokes problem, but we are unaware of the surely colossal running cost of the failures to resolve other Millennium Prize problems.

My own outlook. Even though I am concerned about the incursions of LLMs into academic mathematics, I am quite hopeful. In fact, I consider this to be the most exciting time in my mathematical career (since the year 2000 say). Truly, this may be the most thrilling moment in mathematics in the modern history of our discipline, and I would be terribly sad to see young people leave academia and miss out on the stunning opportunity to be at the frontlines of the current scientific revolution. And not just sad: I think their absence would have disastrous effects for the field.

Undoubtedly, LLMs are already an incredibly powerful tool. If used correctly, and if we set up sensible academic conduct expectations around the use of LLMs, these tools can accelerate progress in our discipline unlike in any previous era. I fully expect that we, the community, will adapt and adjust to this new period, and we will harness these tools to achieve truly great things that just a few months ago seemed far out of reach. And I fully expect that human mathematicians will be front and center in these wonderful achievements to come. I will add reasons that support my optimism below.

I also want to add at this point that the day-to-day of a mathematician has not changed much so far! My days are still filled with teaching and joyful conversations about math with colleagues and students, doing research on a number of exciting (old and new) projects, and going to stimulating conferences to learn and disseminate our most recent methods and findings, while spending time with colleagues that make the mathematical community so wonderful and vibrant. Daniel Litt mentioned the same sentiment in a recent tweet.

One thing has changed though, I am busier than ever before, because the number of research projects I am involved in has tripled in just a few months. My research horizon has expanded significantly, and I have many more projects available for students to help me with.

Now, to the pressing questions:

“Will AI surpass the mathematical research ability of any human?” This is completely unclear. On one hand, the current trajectory in capabilities is surely significant, and we have already seen many impressive results that have been either proved by LLMs, or their proofs have been made possible thanks to substantial LLM contributions. On the other hand, none of the proofs so far seem to contain “alien ideas,” a move-37, or completely novel arguments or new concepts that were not present in the literature in some form or another. This should not be shocking because the LLMs are built and trained on the entirety of all human contributions to date, so it stands to reason that they would `think’ within the boundaries of our current knowledge and make connections (sometimes surprising and ingenious!) among ideas that are already present in the literature. I am particularly fond of the hypothesis (or toy model, as he called it) put forward by Nestor Guillen in a recent blog post, where he argues that LLMs may work within the confines of the convex hull of ideas that are currently available in the literature.

Take, for example, the disproof of Erdos’ unit-distance conjecture. We can imagine the current set of mathematical ideas as a stellated high-dimensional polytope, and we can place the state-of-the-art ideas on discrete geometry at an outer vertex and our knowledge on algebraic number theory at a different outer vertex. The idea for the proof seems ingenious at first sight because it cleverly mixes strategies from two fields of math at the vertices of the polytope of ideas, but after closer inspection, it’s a proof that was within reach of humans as it just sits within the convex hull of the polytope.

The polytope of ideas

This agrees with what Melanie Matchett-Wood said about the proof of the unit-distance when it was released: “I believe if the level and type of human expertise that is represented on this note had been assembled to find a counterexample to this conjecture a month ago, and those people put in similar amounts of time working on it than they did to reading and thinking about Chat GPT’s solution, the mathematicians would have found a counterexample.”

However, a proof of the Riemann hypothesis, say, may need new ideas that are strictly outside of the convex hull of current mathematical ideas, and it is therefore out of reach for an LLM. Only after a new idea is introduced in a new paper, the polytope of ideas may acquire a new outer vertex. And only then the LLMs, after being retrained to include those ideas, may fill out the set of results up to the new convex hull, which may or may not include yet a full proof of Riemann.

The convex hull of ideas

If this toy model holds up, then we would indeed expect the very fast advances in mathematics that we are currently seeing. As the LLMs take advantage of the stellated nature of the polytope of ideas, they will continue to fill in gaps between outer spikes. But as the LLMs fill in the convex hull with new results, we will see a deceleration in the number of results being shown solely by artificial intelligence. We will need human advances and intuition to generate new ideas that expand our knowledge polytope.

Even if the mathematical capacity of the LLMs (or future AI models) can at some point reach beyond the convex hull of the current set of human ideas, there is a different way that we may reach a limit to the LLM capacity: feasibility and ethical use of resources (this is similar what fellow optimist Kevin Buzzard called the “natural boundary” in a recent blog post). Is any cost (a dollar amount, human cost, ethical cost) acceptable in the pursuit of solving a given problem? Should we spend millions of dollars and an undisclosed amount of natural resources in order to find a solution for Navier-Stokes? As an analogy: we would like to know if there is life on Mars, but in order to do so as soon as possible, we would need an absurd amount of funding and risk the lives of a human crew in the process. Is it worth it? Similarly, we may reach a point where an LLM could solve an important problem for an exorbitant cost (in terms of funding and resources) but it may just not be an acceptable cost for the taxpayer or society to bear. Instead, we will need humans to devise an alternative route (the equivalent of a gravity-assisted robotic mission to Mars) to solve the problem at an acceptable cost, that produces a similar result in terms of mathematical advances and, more importantly, human understanding.

“Will research mathematicians become `professional prompters’ and interpreters of LLM output?” There is no indication that this will be the case. Yes, LLMs have produced proofs of important results somewhat autonomously (according to the frontier model companies — see Disclaimer 2) that some mathematicians have been tasked with interpreting and digesting. But in my own experience, and other research mathematicians who are using LLMs in their research seem to agree, working with an LLM is akin to discussing a problem with a collaborator, and the results heavily depend on how much guidance and intuition the mathematician inputs into the conversation. In other words, the LLMs are more than tools: they can be research collaborators but, as in any collaboration, the experience and the results are greatly improved when all parties contribute to the discussion. Further, mathematicians have no desire to prompt “solve the Riemann hypothesis, make no mistakes” and then interpret the proof. We prefer to be active participants during all the steps in the process of the discovery of a proof, because we are motivated by the `why the result is true,’ more than by the final answer that `the statement is true.’

Also, if we buy into the previous concept of the convex hull of ideas, then at some point in the near future it will be impossible to make progress in mathematics without a human adding a new idea, a new definition, a new concept that creates a new spike in the polytope, and then progress can occur.

“Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?” At any given time in the history of mathematics, there have been mathematicians who are research active, and whose mental capacity for mathematics seems completely super human (e.g., the owner of this blog, among many others). It is natural to surmise that they could solve any problem we could solve, in a fraction of the time it would take us to complete a proof and write it up. However, this has never stopped those of us with a more modest capacity for mathematics from enormously enjoying doing research, and producing results that are far from insignificant. In fact, mathematics has always benefitted from the range of ideas and points of view, from the very concrete to the big bird’s eyeview, from the smaller contributions to the building of entire new theories.

Similarly, I am not threatened by the mathematical capacity of LLMs. For one thing their capacity is currently limited, as pointed above. And for another, even if their capacity becomes far superior, there will always be a need for mathematicians at all levels to guide research in paths that make sense for humans to walk (not run).

The mathematical universe is enormous (as Emily Riehl said), and computing time is finite. There will always be areas of mathematics that are under-explored and where even beginners can break new ground. The LLMs can help in the process, by quickly exploring avenues that may be dead ends, pointing out paths that have already been explored, and shining a light on paths that are likely to be fruitful.

As I mentioned above, I have never been this busy, because the access to LLMs has multiplied the number of areas that I have access to, and my curiosity has expanded well beyond my research area. I now have many more ideas that I can possibly explore on my own, so I am recruiting more student collaborators than ever before, to help me test whether these problems can lead to interesting results. Students can be involved in research earlier than ever before too because the LLMs can help them learn material faster (and deeper!), by virtue of being available 24/7 to answer their questions, instead of my meager one or two available hours per week to meet with them.

“Will mathematicians be needed? Will they be employable?” I find these questions natural but also perplexing. Even in the most dystopian of scenarios where AI becomes super human in all research tasks, what good would a proof (of a theorem in pure mathematics) be if there are no human mathematicians to digest it and understand it? Regardless of the advances in LLMs and AI, there will be mountains of research to be understood by humans, with or without the help of a computer.

In addition, we seem to forget that mathematics departments exist in universities to serve two primary goals: discovery and communication of mathematical knowledge. Virtually every mathematics department emphasizes, in equal parts, our research and educational missions (and many institutions place the educational mission of mathematics at a much higher level than their research mission). Mathematics courses are an integral part of a liberal arts curriculum because learning to think as a mathematician is a highly useful and applicable skill. The fact that we are researchers adds immense value to our educational goals, because students are best served learning from those scientists who are in the frontlines of research. The research opportunities that we provide for undergrads are a very valuable add-on to their curriculum, as it is a different type of training that helps them be employable in the future. And as long as the mathematical way of thinking continues to be a highly valuable skill to be learned by the undergraduate population, there will be a great need for mathematicians to be hired by universities.

The LLMs are making math research more accessible than ever to those who are not even in academia or even mathematicians. This means that undergrads will be able to join actual mathematician-led research projects much more easily, and it may be a new fertile ground for exploration. Not mindless exploration, though, but mathematical exploration where the goal is understanding and for the students to be initiated and trained into a highly technical field (in an ethical way). And, of course, we should prioritize training students in how to communicate the mathematics they learn, as that has always been (and probably will become even more of) a crucial skill.

All of this to say that I cannot conceive that the LLMs will displace mathematicians from their jobs. On the contrary, they might produce jobs since our research productivity may sky rocket. On the other hand, I am more worried about policies and funding issues that are political in nature and have nothing to do with the AI and LLM conversation.

Finally, the most important question of all, that I wanted to address here:

“Should I pursue a PhD in math at this time?” The answer to this question should be personal to each and every student. But, in my opinion, the answer should not have changed from a year ago to today. The most important reason (and perhaps the only reason) to do a PhD in math should be that the candidate is passionate about mathematics and wants to become an expert in a particular topic within our field. If that is the goal, then the presence of LLMs in mathematics is irrelevant, because the goal is achieved when the candidate has gained sufficient knowledge to be an expert on a particular problem. If anything, LLMs may be used as a tool to achieve that goal more efficiently. For one, I am using them every day to finally understand concepts and techniques that I always had questions about, and now I can query an LLM until I am fully satisfied. I am able to search for the explanations and examples that click with me, that click with my own particular way of thinking about mathematics.

To what degree a student wants to use LLMs in a math PhD should be a personal choice but I will say that, as my colleague Jeremy Teitelbaum put it in a recent interview (here is the bit I am referring to, and here is the full interview), students cannot afford not to learn about the current capabilities of LLMs, or any other technology for that matter. If your goal is to become an expert, then you have to be amenable to learning from all experts in the field, and from all sources that may allow you to go deeper into a subject than anyone else before you — and LLMs can be extremely efficient tools to explore literature, for instance.

But once again, the decision to do a PhD should not be based on the current state of the art of technology.

I wanted to do a PhD in mathematics because it seemed like a magnificent challenge. I wanted to do a PhD because I wanted to learn how Andrew Wiles proved Fermat’s Last Theorem. I wanted to continue studying mathematics because I simply did not want “a real job,” and the opportunity of contemplating advanced math on my own for a few years seemed like a dream to me, just too good not to give it my best shot. I know I would have deeply regretted it if I had not tried to complete a PhD when I had a chance (the best time to do it is when your undergrad knowledge is fresh!). I wanted to hear mathematicians talk about math, and rejoice in the small little details and miracles that make proofs work. I wanted to meet and hang out with other people who also thought number theory was the coolest thing on Earth. I wanted to publish a paper in a research journal, with my name on it, because I discovered a new theorem that no one had thought of before. I wanted to explain and share my passion for mathematics with others in a classroom and outside of the classroom.

Simply put, I just wanted to do math, and I would have been devastated if some undefined threat to the field of mathematics scared me away from the opportunity to pursue a PhD.

And if you are a student that is passionate about mathematics, and someone who wants all of that too, then a PhD is the right path for you, regardless of the technology available during your degree. You will learn to use the technology to a degree that you are comfortable with, and that fulfills your own dreams and expectations of what a PhD in Mathematics means to you.

Afterword: the BIG OpenAI release. After I finished writing this blog post, and had already sent it to Terry, OpenAI released a huge treasure trove of results in mathematics. This is, undoubtedly, a historic time in mathematics. The theorems in their papers prove some huge open problems in mathematics: the resolution of the so-called quasi Riemann Hypothesis, Goldfeld’s conjecture, the Hodge Conjecture in the case of CM abelian varieties, Hilbert’s 10th over Q, the Rigidity Conjecture… and the list goes on and on.

But such a tremendous release does not force me to change any of the points I made above. On the contrary, we already knew their models can do amazing things (e.g., Navier-Stokes). We already knew the frontier models can connect dots in the existing literature in ingenious ways (e.g., unit-distance conjecture). We already knew that OpenAI can spend a mind-boggling amount of resources to attack problems.

Also, we suspected that their models have limits and the new release shows evidence of that too. In their report, they mention that they attacked 4000 open problems, and their model was able to make progress on about 700 related problems. Yes, some of the ones they were able to solve are huge. But it also shows that their models are limited, quite possibly due to the arguments we explained above.

Are any of the solutions using new ideas that are outside of the convex hull of the current ideas in the literature? We will need mathematicians and time to digest these new proofs and understand what connections are being made, and whether brand new ideas were actually discovered in the process.

The main point of my post remains the same, though. There is a lot of mathematical research that remains to be done with and without the aid of LLMs. There are new mountains of mathematics to explain and communicate to others. And if you are a student who is passionate to learn what is new and what is left to do, then a PhD is definitely the right path for you.

[This is a guest post by the Association for Human Mathematics, reposted from their statements page. This blog post was initially written in a different file format and converted using AI. — T.]

Yesterday, on October 6th, 2026, OpenAI — which is currently defending lawsuits against accusations of illegal plagiarism, copyright infringement, and trademark dilution — released a repository of manuscripts purporting to contain solutions to a number of high-profile problems in mathematics.

Mathematicians did not ask for this work to be done. The Advisory Group on Mathematics and Artificial Intelligence, from whom OpenAI has claimed to derive its legitimacy, opened their initial advisory statement by saying that frontier AI corporations should not test advanced mathematical problems on internal models. In ignoring the central premise of the Advisory Group’s position, OpenAI has indicated total disregard for the norms of scientific research — norms that guarantee that mathematics remains trustworthy, ethically researched, and in the public interest.

Mathematicians have a particular vision of progress that is informed by history and field-specific considerations. We reject OpenAI’s assertion that this release advances our subject, and we urge mathematicians and the public to view the value of this publication model with due skepticism.

Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power. We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centers human understanding.

Association for Human Mathematics Communications Working Group

[This is a guest post by Raghu Meka. This blog post was initially written in a different file format and converted using AI. — T.]

“This problem has been tried by several famous mathematicians.” “There is a heuristic argument for why these methods cannot work.” “Getting this algorithm would give new circuit lower bounds.” Observations like these can take on a life of their own, almost like a game of telephone. A limitation of a particular approach, or an implication whose difficulty we do not fully understand, becomes a reason to believe that a problem is beyond reach, and eventually a reason not to think about it at all.

Over the past few weeks, I have been thinking about the role such perceived barriers have played in theory as I know it, and perhaps more broadly in mathematics. The recent articles on this blog about what it means to do mathematics in the age of AI have been very helpful in understanding and gathering my own thoughts. Looking back at several great results from the past year or so, a few of them (small fraction, admittedly) make me think that perhaps we lost some edge by imagining barriers–“social” hardness or limitations of approaches passed down as folklore–where none existed.

At least some of the solutions coming out of AI models, while brilliant, are also not completely alien. Yet these were problems we had almost stopped trying to solve, apart from small pockets of researchers. Of course, it is easy to say this post hoc. But in some sense, we have seen more `similarly brilliant’ solutions to newer problems than to these older ones. This makes me wonder how much our inherited perceptions have shaped where we were willing to look.

This has also made me think about some things from my early research days as a graduate student.

When I was a graduate student, I gave a talk on a small result. A very perceptive member of the audience (Adam Klivans, if you are reading this!) asked a question that I thought was a great one. But I also thought there were barriers around it, and it did not seem doable. Later, an answer to that question turned out to be an important piece in others’ resolution of a central question. The point of the story is not whether I would have solved the problem; probably not. The point is that the perceived barrier kept me from making even a half-decent attempt at it.

On a personal level, my progress in research was slow. If you are in mathematics, this might not seem that odd, but in theoretical computer science, having only one paper after five years, and that too not in one of the flagship conferences, could generally be taken to mean you had fallen well behind the curve. I nearly left theory. It was my mentors who pointed out that even if the results were not there, the failed attempts showed intent and progress, and that the absence of results was not evidence of a limitation I had begun to imagine. That support got me through then. They taught me not only how to do research, but how to love it.

Several times, I have had an initial impression that there were well-known methodological barriers, or a certain “social hardness” attached to the names of people who had attempted specific problems and directions. The research wisdom of some colleagues, and curiosity itself, helped me get past these impressions and engage with the questions. Pure curiosity is one force that can mitigate these perceived barriers; having people who are very optimistic (research-wise) or encourage that curiosity is another. Perhaps we need to cultivate this more.

Coming back to the present, I think this moment calls for extra care in resisting the trap of perceived barriers, and for revisiting several such accepted hurdles with less deference. We seem to have a mighty tool that can break through some of them. There are many problems whose solutions I thought I would never see, but now, by the universe’s grace, I will likely have the fortune to see them (perhaps some are already gathering dust on servers).

Relatedly, I also see a temptation to put implicit barriers on what “human mathematicians” can contribute. I wonder whether this might become the latest received wisdom that we accept too quickly. My predictive powers are quite limited. But even so, perhaps the epiphany I am having now is that the downside of not believing there is a barrier is far less than that of believing there is one. The cost of disbelieving has, after all, also come down because of the additional firepower we can now call upon. I would like to give curiosity a little more room before deciding what we can contribute, or what we cannot do.

To take poetic liberty, and borrow the words of Blake that gave Huxley his title: “If the doors of perception were cleansed every thing would appear to man as it is, infinite.”

Acknowledgements: I thank several friends who gave useful feedback on the first draft. AI was used to correct grammatical errors and polish sentences.

[This is a guest post by Ben Antieau, cross-posted from his blog. — T.]

There was no personal problem, no world problem, whose eloquent solution did not exist— somewhere in some hexagon.

—Jorge Luis Borges, “The Library of Babel” [1]

New repository

I am writing to announce Hexagon, a new repository for research works in the mathematical sciences and theoretical computer science. This has been a joint effort of many people over the last couple of months, starting in early August 2026. We felt that the proliferation of results proved with LLM assistance required a new model for collecting the associated papers. Besides wanting these works to be citable and reliably stored for long periods of time, we wanted to allow lower barriers for authorship and for community interest than the arXiv. We also wanted for these works to have a home besides random GitHub repositories or X/Bluesky/Mathstodon/LinkedIn posts.

In order to support the development of this project, we founded the Hexagon Mathematics Foundation in the state of Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3). The Board of Directors consists of Mohammed Abouzaid, François Charles, Bryna Kra, David Savitt, and Lauren Williams. I serve as the first Executive Director. And, we have a wonderful Advisory Board, where Kevin Buzzard, Akhil Mathew, Johannes Schmitt, Steinn Sigurðsson, Nikhil Srivastava, Ravi Vakil, and Rachel Ward serve. For more details, see our who we are page.

My personal interest

When working with a group of people brought together by an FRG grant, we clicked a button and discovered a new result, namely derived invariance of Hodge numbers of fourfolds in characteristic zero. In the past, this would be a result I would have submitted to an excellent, but not top, journal. In this case, there is no way I would put my name to the result. Instead, I blogged about it here. This is a brittle way to aggregate mathematics. Now, it is on Hexagon as hexagon:2609.00104.

The creation of new results happens naturally in working with LLMs. They bubble up, even as side results. My personal hope is that it becomes the community standard to identify these and submit them to Hexagon.

The Lifecycle of Hexagon Objects

(The title alludes to Ted Chiang’s “The Lifecycle of Software Objects” [2].)

My ambition is that Hexagon is where mathematical ideas emerge from the water for the first time as they evolve toward community acceptance and understanding. Later, they should be written up in forms suitable for arXiv submission, publication, and canonicalization (Tao [3]).

Hexagon specifically allows submissions entirely generated by LLMs and even where the submitters claim no human understanding. These results should be available to the general community.

Submissions by the big labs

Hexagon welcomes submissions by large labs working on language models. These results should be available to the mathematics and TCS communities by default on a platform that the community maintains. We also welcome submissions by large collaborations or other organizations.

FAQ

How was Hexagon created? At some point, a group of people convened in early August, brought together by our social networks with the common theme of interest around creating a home for mathematical results which were rapidly being produced with the assistance of LLM tools. This group included most of the current members of the Board and Advisory Board of the Hexagon Mathematics Foundation, as well as several other academic mathematicians who did significant work but preferred not to be named. In August we discussed ideas on a private Zulip channel (thanks ICARM!) as we started building the site itself.

We talked to a lot of people at that time, including arXiv leadership, and we also examined closely-related AI-era repositories. From the arXiv we learned that they were not going to provide the home for these results we were looking for; instead they are focused on maintaining their project as it is today, and I am happy for that. None of the other repositories had gained sufficient traction or had all of the policies we were hoping for.

The foundation was incorporated as a nonprofit in Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3).

The entire system was created with the help of OpenAI’s Codex and is hosted on widely available commercial cloud servers. This is rather easy nowadays. What takes time is deciding exactly what features you want and carefully writing the text on the website.

How is Hexagon funded? So far, it is entirely funded by the people involved, although we are exploring other funding routes and paths to sustainability. Today, except as customers of cloud infrastructure companies, we have no relationship with any for-profit company or other organization.

Why Hexagon? At some point, the group started talking about names involving Babel, which refers to Borges’s story The Library of Babel [1]. But, because it has other connotations as well, we did not go that route. Instead, someone suggested Hexagon in reference to the shape of the halls in Borges’s library.

Does Hexagon accept micro-results? Hexagon welcomes small, incremental improvements as long as they are of interest to the mathematics and TCS communities.

Is formalization required for a Hexagon submission? No, Hexagon is not primarily a formalization project, but it links to formalization projects, like mathlib, Palomar, prove2.me, and TauCeti.

What is the difference between a contributor and a submitter? On Hexagon, Contributors are humans or organizations responsible for the creation of submissions. Submitters are humans with ORCID-validated accounts on Hexagon. In some instances, Hexagon will provide individual account holders the capability to submit on behalf of other humans or organizations.

Are there rate limits? Yes, Hexagon limits the rate at which one can submit. It starts at 1/day, and grows at Hexagon’s discretion. This growth is based on previous submissions passing through moderation. (The arXiv recently announced its own repository-wide submission limits [4].)

Is there API access? Yes, Hexagon provides both read-access via API (in the style of the arXiv) and submit-access via API tokens available from users’ account pages.

Why is Hexagon marked as being in beta? We are still adding features and testing scalability. We welcome any recommendations for how the site can be made more usable.

References

[1] Jorge Luis Borges, “The Library of Babel,” translated by Andrew Hurley, in Collected Fictions, Penguin Books, 1999, publisher page.

[2] Ted Chiang, “The Lifecycle of Software Objects,” in Exhalation, Vintage, 2019, publisher page.

[3] Terence Tao, Mathematics in the age of AI, 2026, arXiv:2608.16753.

[4] Kat Boboris, Fair Moderation, Equitable Access, and AI: arXiv’s Updated Rate Limit Policy, arXiv Blog, 1 October 2026, article.

[This is a guest post by Thomas Bloom, crossposted from the Erdős problems forum. This blog post was initially written in a different file format and converted using AI. — T.]

AI is changing everything, for better and/or worse, and the rate of change is dizzying; in recent months this has been particularly evident in mathematics, where AI has gone from being essentially useless to helping solve some of the hardest problems in mathematics in less than a year.

The website http://www.erdosproblems.com has often been on the front line of these changes, and a barometer by which one could measure AI capabilities. This was not at all my intention when I created the site — I wanted to promote these problems to a human audience, and make a useful reference on what work has been done. But the easy availability of a large pool of questions that are simple to state, ranging from easy and obscure to deep and impenetrable, provided the ideal showcase for AI.

In some ways this has led to a huge amount of progress — we now know the answer to many questions we did not before. (Although a lot of this recent progress has come from increased human efforts and renewed interest in some problems, rather than just AI solutions.) There have also been negative effects, however: some mathematicians have dismissed Erdős-style mathematics as ‘easy/recreational’; many have stopped thinking about Erdős problems believing that they cannot compete with AI; and, most significantly, there has been a wave of AI-produced solutions provided with no explanation. While of course these are useful and tell us new things, they are also displacing and discouraging those who are actually interested in the mathematics.

(These issues have stopped being Erdős-specific as AI continues to make breakthroughs in an ever wider range of fields.)

A couple of weeks ago I asked for feedback on how people used the site, and what they thought should be changed. I received, both publicly and privately, a wide variety of opinions, and I’m grateful to everyone for their feedback.

What was particularly striking was the number of people I heard from who have benefited a lot from the site, learning new mathematics and finding new problems to think about, but have never commented on the site. I have been particularly conscious of this large silent, audience, and have tried to make sure their experiences were not drowned out by the more vocal minority.

In this post I will describe the changes I am making, based on the feedback I received and my own reflections on what the site is and could be.

In the essay ‘Why Do We Need Human Mathematicians Anymore?’ Po-Shen Loh suggests the following as an axiom to use when deciding how things should develop: We (humans) should help humanity flourish.

When thinking about what I should do about the site, I use the following variant:

http://www.erdosproblems.com should help the Erdős-community of humans (defined as those who are interested in Erdős-style mathematics, and want to think about and understand it) flourish.

Why change?

I launched http://www.erdosproblems.com on 28th May 2023, with just over 200 problems; it has grown steadily since then. The biggest change came in August 2025 when I added a comment feature. The site now hosts 1221 problems, over 9000 comments, and almost 2000 registered users. The site typically receives between 10,000 and 25,000 unique visitors each day.

The first few months after introducing the comment feature saw a huge increase in activity, just as I had hoped. A vibrant community arose; people shared ideas on how to solve problems, made observations, gave corrections, and suggested missing references and solutions. Many papers were written and new collaborations formed.

Unfortunately, this level of activity has not been sustained. Some decline was inevitable: over time, many of the natural observations will have been made, mistakes corrected, and discussions between collaborators would move offline. This natural evolution has been dramatically sped up by the coincidental rise of AI and its ability to solve some problems with minimal human intervention.

The main way that people publicly interact with the site now is to advertise their AI-generated proofs, often without any attempt to explain them, but as a way to record a (increasingly meaningless) priority claim.

This is very different to what I imagined, and I don’t want to manage a website which does this.

I believe that websites with this function should exist — places where people can record AI-generated proofs, even if purely formal with no human understanding, to save others wasting their tokens generating the same proof, and so that other people can access and use them if they desire. There are now several candidates for such repositories, and if managed responsibly, they can serve a useful role in the mathematical ecosystem. I personally don’t want to manage one.

Just as one does not open a restaurant in an abattoir, it is important that there be a separation between such repositories and a site which aims to promote the actual questions, place them in an appropriate context, and give a useful overview of the current state of human understanding.

The situation as it stands muddies the waters, promotes a gamified glory-seeking attitude, and gives the wrong impression that the value of these questions ends as soon as someone posts a formal proof of their truth value. This is not true (regardless of whether this proof was human or AI generated).

I believe Erdős intended these questions to serve as enduring beacons for human curiosity and wonder. They are landmarks by which we measure how far we’ve come, and how much there is still to understand.

I am very proud to have helped foster an open online community to discuss Erdős problems, and the kinds of discussions that used to happen are very valuable. But these discussions have become much less frequent. If and when I can find a way to encourage such discussions again, without the site becoming an AI repository of the kind mentioned above, I will happily do so.

What are the changes?

I will make the following changes. (As ever, these are somewhat experimental, and may themselves change and be clarified further in the next few weeks.)

1. A hiatus on problem comments and proof claims: I will freeze new problem comments and proof claims. General threads and blog posts will remain open to comments. Problem comments and proof claims may be reinstated in the future. Until then, suggestions for updates can be emailed to erdosproblemsonline@gmail.com, and I will update the site manually as I see appropriate.
2. No problem statuses: The site will not display the statuses of any problems (e.g. open, solved, etc.). All problems will be displayed in the same neutral colour. The count of currently solved problems and the solved percentage will no longer be shown.
3. No credit/ownership language to describe future solutions: The site will continue to record relevant results and theorems, but will no longer use credit-giving language for a result (human or AI).
4. An emphasis on high-quality expositions: I will focus more on the proof expositions feature. People are encouraged to write in with their own expositions of proofs (whether these proofs are old or new, human or AI), and I will post those I judge to be high-quality. I am exploring other ways to encourage human exposition (e.g. an online seminar). Please contact me if you have any ideas.

I’ve tried to anticipate some of the questions and critiques of these changes below. (I may update this with other questions and answers in future depending on feedback.)

What about those outside of traditional academia, who are using AI tools to make advances in mathematics, but can’t post to arXiv/lack the knowledge or context to write up proper papers about their findings?

I will post links to correct formalisations and high-quality writeups, whatever the source, whether from a traditional academic or not. Papers should acknowledge their sources and honestly disclose how AI was use; where there is evidence of plagiarism or misrepresentation about AI use, I may decline to feature the submission, even if the proof is correct.

This is removing one, unofficial, avenue of publication; there are, these days, many others (even if you are also unable to post to arXiv). AI-assisted search tools mean that people will find your work if they are interested in the area as long as you post it to one of these sites.

I encourage everyone interested in Erdős problems to think about them, and to work on them if interested. If you have a proof (AI-created or otherwise) then you should take time to carefully write up the proof yourself (rather than asking an AI to generate a PDF for you). If you are unable to understand the proof you could reach out to someone else to help write this up.

But emailing you my solution is slower, and it might be some time before you update the site.

This is true; but there is no great rush here. As much as I like these problems, and think it is important that there are some people who do think about the distribution of prime numbers and the structure of graphs, these are not problems for which a formal proof will have an immediate impact on general society.

I will try to link to correct Lean formalisations (if verified on Palomar) quickly; proofs which are poorly explained, and not accompanied by a formalisation, I am unlikely to update the site with. (See below for more details.)

What’s the point? There are loads of other places I can post my proof.

Indeed; this is partially why I feel comfortable halting such proof claims on the site, since modern search tools (including AI) make it easy to find proofs posted online about problems you are curious about, even if posted in obscure places.

I would like to use the small amount of influence I have to avoid promoting low quality proofs, and not to give an added veneer of legitimacy where it is not deserved.

I could just make my own site for people to share their proofs and discuss solving problems with AI. Heck, I can even use AI to scrape all the text off your site and make an exact clone with much more liberal policies.

Yes, you could. This a period of great experimentation in different formats and ways to encourage mathematics; if you think you have a good idea for a site or resource, you should make it. (Although it is often better to help out an existing effort where possible, since these resources are only useful if people use them, and it is better to concentrate on creating a few high-quality sites than hundreds of very similar clones.)

I think it is, however, rude to use without permission the text from my site, which is the result of a huge amount of work from me and many others who have contributed to the site.

How long will the hiatus last for?

I don’t know. I will monitor how things develop in the coming weeks and months, and will reintroduce comments and proof claims (perhaps in a different form) when I believe they will do more good than harm.

Why not just moderate the comments to only allow genuine discussion through?

This has been tried; in practice the vast majority of the comments the site receives now are people announcing AI-generated proofs. It is not sustainable to have a moderation policy which would reject almost all the comments that are submitted.

I am open to alternative suggestions about how to create a space for human discussion and collaboration, either on the site or elsewhere.

What about existing comments and proof claims?

They will remain as an archive of the site up to this point.

Will you update the remarks to reflect existing proof claims?

Yes, I will be working through the backlog of existing proof claims and updating the remarks, linking to formalisations and so on, as appropriate. You do not need to email me about a proof claim already on the site.

How will you decide when to update the remarks? What can I do with my proof to help?

I will update the remarks when I judge there is an interesting new result that people who are working on that problem should be aware of, that can be presented in a useful way. Some things to be aware of:

1. If you have a Lean formalisation register it on Palomar — this lets others see that the formalisation compiles correctly, and makes it easy to check the formal statement correctly matches the problem statement. I will then link to the Palomar registry.
2. A proof accompanied by a well-written exposition that demonstrates clear understanding and which makes it easy for others to understand the proof will be prioritised.
3. If I judge there to be some kind of academic fraud (e.g. using the ideas of others without attribution, or passing AI-generated work off as your own) then I will not post it. This may mean that there are correct proofs not acknowledged, but the alternative is to publicise and reward bad behaviour. (As mentioned above, these proofs will surely still be found by anyone who searches for them elsewhere, but at least I would not be implicitly endorsing them.)

What’s the point in removing the problem status? Isn’t it just cosmetic, and doesn’t it just make the site harder to navigate?

Yes, this does undeniably remove some information. I think, however, that what is lost is not significant for anyone seriously interested in that problem, who can see for themselves within seconds of reading what the current situation is.

It makes it harder to browse the site searching for an unsolved problem to work on; instead I recommend that people browse the site by topic, finding questions that interest them, and investigating those, whether the original question is open or solved, since there is always more to be done.

The main point is to disincentivise people who are simply glory-chasing and copying problems into their AI to get an OPEN->SOLVED dopamine hit. It also prevents people drawing wrong conclusions from the ‘rate of solved problems’.

Furthermore, there is a lot of subjectivity anyway in many problems as to what counts as solved. Many people only browse the open problems, and they’re missing out on a lot of great mathematics that way.

I think it is against the spirit of Erdős to regard any problem as ‘closed’ — whenever one form of a question is answered, many others are created, and all problems deserve continued attention.

Doesn’t the ‘no credit’ rule also devalue the work of humans?

Unfortunately, yes. I anticipate this will only be temporary, while the norms and customs of this new age of mathematics are decided. The commentary on the site is in no way ‘official’, and just represents my own summary of the situation, and collects links which may be useful to others exploring a problem.

Information about authorship will still be available in the sources the site links to. I trust that people will be sensible enough to judge for themselves who deserves the appropriate credit, and reward them appropriately.

I will also be freer in my language when it comes to well-written papers, whether they are the first place a proof appears or explain a proof that originated elsewhere; but I will no longer use language that suggests anyone ‘owns’ a particular result or proof. (For example, instead of saying ‘Bloom proved that ‘, I will write ‘It is known that — an explanation of the proof is provided by Bloom [link]’.)

Erdős and AI

I am often asked ‘what would Erdős have thought of AI?’ The short answer is ‘I don’t know’. I never met Erdős, and have no special insight into him; I know him only through biographies and his papers, and anecdotes and reminiscences from those who did know him.

I would like to honour his legacy, and create something that he would have liked. To that end, I want to stress one thing: Erdős was, as well as a mathematician, an incredibly social person. Many stories about Erdős emphasises his humour, his warmth, and his enthusiasm for talking to other people (mathematicians or not). He spent his life travelling from mathematician to mathematician, arriving on their doorstep and declaring ‘my brain is open’.

He was not the type to lock himself away for years working in isolation on a single problem. For Erdős, mathematics was a very human activity, best done out loud. Questions would arise, be solved or discarded and replaced by new questions. Some results obtained, new ideas found, and then onto the next problem.

Some view the future of mathematics as a dystopian arcade of button-pressing, staring at a screen waiting for AI to do the thinking for us, with the main human involvement limited to asking the initial question and offering sporadic words of encouragement. Each new proof is then thrown into a repository, to only ever be read by other AIs, and the human presses the next button.

I believe Erdős would have found this future grim indeed, and the antithesis of mathematics as he practised it. Use AI as you like — but do not abandon mathematics as a human activity.

Don’t solve a problem for the sake of it; life is too short to spend it doing things that don’t matter to you.

Find a question that is meaningful to you, find other humans who are interested in it, and talk about it. Be confused, be stuck, be inspired. Find a messy proof, then find a better one. Get tired, get hungry, get into a flow state. Argue, laugh, give up, drink some coffee, and attack it again.

Be human. Let your brain be open.

[This is a guest post by Annalisa Buffa. This blog post was initially written in a different file format and converted using AI. — T.]

About us and the position

The Chair of Numerical Modelling and Simulation at EPFL is dedicated to the design and analysis of numerical algorithms for partial differential equations. Our research is oriented towards the development of novel and innovative numerical techniques aiming at improving the integration between numerical simulations and geometric modelling and processing.

Proof assistants such as Lean, together with recent AI tools, are changing how mathematics is done. We are opening a postdoctoral position to explore what formal verification and algorithm discovery can bring to numerical analysis. There is no fixed project. The goal is to find out, through concrete experiments, where these approaches help in the design and analysis of numerical schemes for PDEs, and where they still fall short.

What you will do

As a postdoc in our group, you will set up and lead this new research line, choosing its directions together with us. Possible starting points are the formalization in Lean of stability and error estimates for finite element methods, or the AI-assisted discovery of new discretization paradigms for classes of PDEs where standard methods struggle, with their properties then verified formally. You will publish and present your work internationally.

You are the ideal candidate if

• You hold a PhD in mathematics, computer science or a related field
• You have a strong background in formal verification with Lean
• You have experience in AI-assisted theorem proving or algorithm discovery is an asset
• You have working knowledge of numerical analysis, ideally numerical methods for PDEs
• You are independent and comfortable with open-ended research questions
• You communicate well in English and enjoy working at the boundary between disciplines

We offer

• A stimulating international environment at one of Europe’s leading research institutions
• Competitive salary and excellent working conditions
• A 1-year contract, renewable
• Freedom to shape a new research direction from the start
• A friendly team with strong postdoctoral and PhD researcher community

If you are interested

Submit your application containing CV, motivation letter, and contact details for 2 referees through this link. In the motivation letter, briefly describe one question you would like to work on, linking numerical analysis with formal verification or algorithm discovery. Applications will be reviewed until the position is filled. Start date: as soon as possible (flexible).

[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.]

“Mathematics underwent, in the nineteenth century, a transformation so profound that it is not too much to call it a second birth of the subject—its first birth having occurred among the ancient Greeks…”

Howard Stein, in “Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth-Century Transformation of Mathematics”

“The report of my death was an exaggeration.”

Mark Twain

“May you live in interesting times.”

(traditional)

I recently attended a meeting of innovative science and technology startups supported by Convergent Research, the organization that oversees the Lean FRO, a nonprofit that develops the Lean theorem prover. The meeting was designed to stimulate discussion, and when I introduced myself as a mathematician, many participants were eager to talk about the impact of recent events in AI on mathematics and reactions in the mathematics community. They were surprised to hear that I find the tone of the community responses on blogs like this one and Proofs and Prompts generally positive and encouraging, even though we all recognize that fundamental aspects of our day-to-day professional lives are bound to change. These discussions have helped me shape some of the thoughts I would like to share here.

There is a narrow view of what mathematicians do, encapsulated in our daily workflows: we try to solve problems, and when the hard problems are too hard to solve, we make up easier approximations, solve them, and then vary the parameters. That practice has been disrupted by the events of the last few months, in the sense that the kinds of results that would have, a year ago, made for perfectly respectable publications can now easily be generated with the help of AI. This has left us worrying about what it will mean to do mathematics going forward, as well as how to train and support the next generation of mathematicians to do whatever that is.

The history of mathematics offers us a broader view. What has remained stable, despite centuries of changes, is that mathematics is a culture of rigorous reasoning and communication, providing us with language and abstractions that let us think and communicate more reliably and efficiently. Surely such reasoning is still important, even in the age of AI. The fact that many of us find mathematics aesthetically pleasing doesn’t diminish its practical utility, but rather is explained by it: I expect that the reason that doing mathematics feels so good is that it is the exercise of capacities that are so fundamental to our survival as a species that they are wired into our DNA. If that’s right, mathematical thought isn’t going away any time soon.

The challenge is that solving the kinds of problems we have been solving for decades becomes decoupled from the goal of enhancing our mathematical understanding when we let AI do the work. The question, therefore, isn’t whether we still need mathematics, but rather how to pursue mathematical understanding in the age of AI. I will provide three general answers.

Solve harder problems

There are two salient features of today’s foundation models: first, they have seen the entire mathematical literature, and second, they are tireless; a swarm of agents can make its way to a solution by trying countless variations. This explains why the AI-generated solutions to open problems we have seen all have a similar character: they are problems that AI could solve by cobbling together available techniques. (I am grateful to Matthew Ballard for this characterization, and the subsequent analysis.)

Can every interesting mathematical question be answered that way? Probably not, and even questions that can may have more interesting and satisfying solutions that invoke novel ideas and insights. Perhaps, in the near future, AI systems will be able to come up with such insights, but, at the very least, let’s recognize that we are not there yet. Reinforcement-learning training regimes have systems chain conventional moves and learn, from a history of failures and successes, which ones are most promising in a given state. The fact that the value of an action is graded solely in terms of the success of a final trajectory breeds superhuman cleverness but may miss creativity and higher-level strategizing. In any case, there are still hard questions to be solved and ambitious research programs to pursue, and the possibility of making progress on them with the help of AI is exciting.

Think bigger thoughts

Our present situation would be much more depressing if mathematics were a matter of ticking off problems imposed on us by aliens, an endless sequence of exercises and exams. The good news is that when we do mathematics, we get to choose the problems, grade the solutions, and favor the ones we like best. We decide what’s interesting to us, what questions to pursue, and why. Our destiny is in our hands.

The things we admire most in mathematics often seem to come out of nowhere. In 1853, the young Bernhard Riemann submitted three potential topics to his advisor, Carl Friedrich Gauss, to choose from for his Habilitationsvortrag, a lecture he was required to give to secure a teaching position at the University of Göttingen. Gauss reportedly chose the topic for which Riemann was least prepared, to see what he would make of it. The resulting lecture, “On the Hypotheses Which Lie at the Foundations of Geometry,” was published posthumously in 1868, and it revolutionized the field. The lecture distinguished a space’s metric properties from its topological properties and introduced the general notion of a manifold, though the latter did not receive a fully rigorous treatment until the twentieth century. The focus on intrinsic properties of a space—in Riemannian geometry, those determined by the metric and independent of embedding in a larger space—was key to Einstein’s theory of general relativity decades later. Riemannian geometry has had applications that Riemann himself could never have imagined, from robotics and medical imaging to statistical analysis.

William Ewald’s excellent sourcebook, From Kant to Hilbert, provides a lovely introduction to the paper and quotes Felix Klein’s description of the work:

“The publication of [Riemann’s lecture] occurred just at the time when I was beginning to occupy myself independently with mathematical problems. So I still have vivid memories of the extraordinary impact Riemann’s train of thought made on the young mathematicians of the day. Much seemed to us dark and difficult to understand, and yet of unfathomable depth, where the modern mathematician, who has already absorbed all these things into his mode of thought from the outset, only admires the clarity and fecundity of the exposition.”

What seemed dark and mysterious to the young Klein is now part of the canon, something that foundation models have absorbed and internalized through their training, just as young mathematicians do. I am grateful to Ballard once again for suggesting this example and pointing out that no reinforcement-learning setup could have evaluated Riemann’s decisions: the benefits are diffuse and hard to track, and the time horizon is much too long.

The same can be said for countless mathematical developments that have opened up new vistas, such as Galois’ focus on groups of permutations in the study of solvability of algebraic equations, Poincaré’s qualitative studies of dynamical systems, or Grothendieck’s far-reaching conceptual innovations. Will AI eventually be able to make advances like these? That’s not even the right question to ask. Telling us that some AI agent is spinning out theorems that are highly interesting to it and other AI agents does nothing for us. We should care about AI only insofar as the results are interesting and important to us, and, at the end of the day, it’s up to us to decide what that means. The values we assign to mathematical developments are embedded in our history and culture.

Imagine planning a trip to go backpacking in the Alaskan wilderness for exercise and recreation. You might be happy to let AI help book your flight, but not to let AI take the hike for you and send you pictures. In mathematics, what is at stake isn’t our recreation but agency over our reasoning and deliberation. Whether or not AI can think, it can’t think for us. We have our own lives to live; mathematics is our story to tell, and it’s up to us to decide how to tell it. With AI, there’s even more to explore, and no shortage of avenues for discovery.

Try new things

So far, I have focused on using AI to help us do the things we used to do, but let’s not forget that the technologies themselves raise new questions, and that we have a lot to learn about how to use them effectively. I have argued in another essay (now scheduled to appear in the Notices of the AMS) that mathematicians should be actively involved in understanding how the technologies work and in coming up with novel and creative ways to use them to do new mathematics.

AI changing what it means to do mathematics is not without precedent. The use of algebraic methods to solve geometric problems in the seventeenth century was a new technology, and not everyone liked it; yet we mastered the new techniques and learned how to use them to great effect. The same is true of infinitesimals later in that century, algebraic structures in the nineteenth century, set-theoretic abstraction and structural language in the early twentieth century, and numerical and symbolic computation more recently. These were all alien and disconcerting when they were new. We should view those who invest time and energy in getting proof assistants and neural networks to help us discover new mathematics as doing mathematics proper, rather than dismissing them as mere technicians. In the age of AI, developing symbolic automation or training a neural network can be no less a contribution to mathematics than manually chaining inferences to prove a theorem.

I have heard arguments that as the job market contracts, we should turn inward to preserve traditional mathematical skills. On the contrary, I believe that engaging with new technologies and learning how to use them to improve our ability to reason and discover new mathematics will keep the discipline strong. Expanding our view of mathematics is the best way to expand the profession and keep it relevant.

Our message to the next generation

My rosy outlook on the future and glib advice to solve harder problems, think bigger thoughts, and try new things will not provide much comfort to students and early-career researchers, who feel the ground shifting beneath their feet. My words are not meant to diminish the challenges ahead or suggest easy responses to the disruption. Mathematics departments, community leaders, educators, professional societies, and journal boards are holding emergency meetings all over, and are doing their best to make concrete recommendations and take appropriate action. We have our work cut out for us.

Despite the uncertainty, there are some clear messages we can send to the next generation of mathematicians. The first is that we stand with you. There is nothing more important to us than the health and strength of the discipline, and ensuring that you can thrive. We have the humility to recognize that our experience and expertise are limited, and that some of the things we thought we knew in the past are no longer valid. We are committed to working with you as best we can to preserve the profession and keep it strong.

Second, mathematics is as important today as it ever was, and we need you. AI must not replace our collective ability to reason and deliberate, and mathematics remains a core capacity for doing so. We cannot imagine a world in which mathematics does not play an important part in our lives.

Finally, the next few years will be extremely interesting. We are at a new frontier, where our communal norms, values, and expectations are beginning to break down, and it’s up to all of us to figure out what should replace them. I am confident that future historians will see this moment as the start of a new era of mathematics, and that they will weigh the consequences of our actions and decisions. Mathematics has never been for the faint of heart; this is our opportunity to rise to the occasion and face the challenges together.