Last September I went to a workshop at the Lorentz Centre in Leiden to discuss mathematics and AI with historians, philosophers, computer scientists, AI researchers, and mathematicians of several different flavours (though there was a surprising preponderance of algebraic geometers). The whole event was extremely stimulating, with some talks but also a lot of time set aside for discussion. One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.
Let me begin by quoting a passage from the declaration that sets out “what we take to be characteristic values of mathematical research that we have a joint interest in preserving”.
- There are many reasons to pursue mathematical research, ranging from intellectual curiosity to a desire to solve practical and societal problems. Underlying much of mathematics is the activity of proof. Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true. These characteristics of proof support the scientific integrity of mathematics.
- Results are attributable to specific authors who take credit for their discovery and assume responsibility for their correctness. These principles ground the merit-based standards to which we aspire in mathematical research.
- Mathematical arguments are regarded as transparent and subject to independent verification. They may be extremely long or difficult, but in principle no proprietary knowledge or equipment should be required to understand them.
- Mathematicians share a concern for proper evaluation of mathematical work relative to shared standards of depth, difficulty, and significance.
- Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. A key source of strength of the discipline has long been the autonomous shaping of the direction of research and the methods used to pursue it.
The first thing I would say about these values is that they are undoubtedly values that are widely held by mathematicians, including, with some qualifications, me. The main qualification I have concerns point 4: I find the notion of “proper evaluation” somewhat problematic, given that different mathematicians can have very different judgments without either of them being clearly wrong, especially when it comes to the significance of a piece of mathematics. Also, these judgments are used for purposes such as the acceptance of papers in journals, hiring and promotion decisions, the awarding of prizes, and so on, that are part of a system that copiously rewards a few people — I myself have hugely benefited from it — but doesn’t necessarily adequately reward a lot of people who are doing less visible work that is essential to keeping the whole enterprise going.
But the more important point is whether these values are ones that we should fight for in the future, as the Leiden Declaration suggests. I find that clearer for some of them than others. For example, it seems to me that the importance of rigorous proof will be even greater in an AI age than it was before — if the output of AI is not underpinned by rigorous proof, then the kinds of difficulties one already hears about with certain areas of human mathematics (see for example many talks by Kevin Buzzard arguing for the value of formalization) would be hugely magnified. But what about the attribution of results to specific authors, who take both credit and responsibility for them? Suppose that at some point in the future AI becomes more autonomous, reading the literature and solving many problems that it finds. Suppose also that its solutions are autoformalized, so there is no serious doubt about their correctness. In such a situation, there would be nothing for a human to take credit for or responsibility for. Does that mean that we should declare such results undesirable and threatening to mathematical values?
Of course, something could well be missing in such a situation: perhaps the proofs would be badly written and hard to follow, which would mean that they lacked something we all very much value. So let me extend the thought experiment slightly. What if by that stage one could take one of these outputs and ask an LLM to explain the ideas, and what if LLMs did a very good job at that? That is not particularly hypothetical, since they are often pretty good at this job already, but I am imagining a world in which they are much better than they are now, as they will presumably become.
So now we would have a world in which a lot of problems had been solved, we were sure that the solutions were correct, and we had an LLM ready to explain those solutions in as much or as little detail as we wanted. Is that a future we should resist, and if so, why?
One obvious reason is that it would take a huge part of the fun out of the subject. It is extremely satisfying to struggle with a mathematical problem for months or even years and eventually solve it. But I worry about that argument, because it seems to be saying that we should resist doing mathematics the easy way because a tiny fraction of the world’s population gets huge pleasure from taking orders of magnitude longer to do it. That is not to say that I wouldn’t be sad that a way of life that has sustained me for the last forty years was not available any more — of course I would. I just find it hard to use it as a reason to argue that we should try to preserve the “ownership structure” of mathematical results. If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all. I’m not necessarily in a hurry for that world to exist, but maybe once the transition had happened, people would be OK with it.
The third value I share in an uncomplicated way, and I have already discussed the fourth. The fifth value is one that I hold very strongly, though I’m not so keen on the idea of experts consciously “shaping the direction of research”, something that I see as happening more organically. Obviously there are some notable examples of mathematicians who have created wonderful programmes of research, but even there I would like to credit other mathematicians with understanding what is wonderful about those programmes and contributing to them enthusiastically as a result, rather than being told what direction to pursue and meekly doing so (which is probably not what the declaration is actually trying to suggest, but it has a slight flavour of that for me).
But that’s a minor quibble when set against my main worry about the effect of AI on mathematics, which is the possible destruction of mathematical culture. There is at the moment an extraordinary body of knowledge and expertise that exists not just in the mathematical literature but in the heads of mathematicians all round the world. Imagine if AI didn’t exist and a pandemic broke out that for some reason wiped out all mathematicians and nobody else. All the literature would still be there, but nobody would have the faintest idea what to do with it. To revive a mathematical tradition under those circumstances would be extremely difficult and take decades. Now imagine a slight variant of that, where AI does exist and because of it people are no longer motivated to put in the years of effort it takes to reach the level of expertise that a typical research mathematician has now. After a decade or two, we might arrive at a situation where the mathematical literature has, in some form, been vastly expanded, but there is no corresponding community of human experts who have a shared understanding of parts of it. Almost all of mathematics would be like the areas that we have more or less forgotten about today, areas that exist in papers written many decades ago that nobody reads any more. (I won’t name any such area because I don’t want accidentally to suggest an area that many people still love and work on.)
This, it seems to me, is a possibility that we should try very hard to resist, but I agree with many other commentators who say that in order to resist it, we will need to give less priority to some of our current values — and I would include ownership of mathematical results in that list — and more to others. For example, if Person A gets an LLM to one-shot a solution of an important open problem (which is formalized, possibly automatically, so there is no doubt about its correctness) but Person B makes the effort to digest the solution and explain it in a way that other mathematicians can understand and learn from, then I think we will want Person B to get the lion’s share of the credit. The credit would be of a slightly different from what it is now, which could be described as admiration for somebody’s talent, insight, speed (I mean here the purely factual statement that speed is often admired — I would prefer that to be less the case) and hard work. It would be more like the gratitude that one feels already for somebody who writes a beautiful textbook that makes a whole area of mathematics coherent and accessible.
Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics.
At this point I have to admit that there’s a pessimistic side of me that asks the following general question whenever anyone says anything about what the role for humans might be in the future: why do you think that AI wouldn’t be able to do it? For example, with the suggestion I’ve just made, what reason is there to suppose that ChatGPT 8.2 wouldn’t be able to have a short interaction with you about your mathematical tastes and background and then write the ideal textbook just for you? Humans are likely to be better at this kind of curating for a little while yet, but is it a fundamentally human ability that AI could never hope to emulate?
In a world where AI wrote bespoke textbooks (or more likely, just taught people in some more direct way), something would be lost that feels important: mathematics as a collective endeavour. If we all just learnt cool bits of maths for our own private satisfaction, we would miss the considerable pleasure that comes from discussing mathematics with others, though even that could in principle be restored by a benign LLM that deliberately taught many people the same cool bits of the subject, though an LLM that could do that sort of social engineering would raise all sorts of safety issues.
Let me now turn to the section of the declaration about potential threats. I’ll put my comments on each one in square brackets.
- Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs. This applies not only to informal arguments, but also to formalizations, where the difficulty lies in the translation between computer-encoded and human presentations of concepts. These fast-moving developments put our present system of review under increasing pressure, jeopardizing our ability to implement traditional standards for the correctness, transparency, and independent verifiability of proof. [This feels like less of a problem now than it did last September, partly because the best LLMs hallucinate a lot less than before, and partly because autoformalization is improving all the time — I have just used harmonic.fun’s Aristotle system to formalize a complicated paper in Lean and I didn’t need to know any Lean to do it.]
- Technologies that draw extensively on the published mathematical commons undermine the traditional system of attribution. Models trained on published works frequently return outputs that do not properly cite the human works they synthesize. Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections. [This is a problem at the moment, when ownership of results is important, and I am very much in favour of people making an effort to give appropriate credit for mathematical ideas that AI may have used. However, in the longer term, as I have already discussed, I think this ownership structure will break down and the issue will become less important. It also seems possible that LLMs will become better at revealing their sources.]
- Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives. The use of artificial intelligence — and thus also the sort of problems which it can address — may become incentivized for its own sake, disrupting our mechanisms for hiring, funding, and recognition. This disadvantages researchers who do not have access to the technologies or decision-making related to them, or who are unwilling to use technologies controlled by organizations whose values they do not share. [These seem to me to be genuine problems. I think there is simply no point in hoping that our current system of incentives will not be disturbed — it obviously will. I am not necessarily too worried if our mechanisms for hiring, funding and recognition are disrupted, as I don’t find those mechanisms unproblematic as they are, but disadvantaging researchers who do not have access to good LLMs is something I certainly think we should worry about.]
- Proper evaluation is endangered if results are communicated through informal channels such as press releases or blog posts, often without any research paper or other disclosure of information necessary for scientific evaluation. This practice seeks publicity for new results on market timelines before the accepted processes of community evaluation in mathematics can take place. In many cases this leads to simplifications in reporting, such as overemphasizing the significance of automated tools and undervaluing the prior human contributions which have made those tools possible. Such oversimplification risks influencing public opinion in a way that not only damages perceptions of mathematics, but also misleadingly uses specific mathematical tasks as metrics for the general reasoning capacities of commercial products. [I think this can be a problem, but I think it is not as serious a problem as some of the others, since when results get overhyped, there seems to be no shortage of people publicly (and rightly) pointing that out.]
- These developments put the autonomy of mathematics under threat. The increasing involvement of technology companies in mathematical research raises the risk that research questions may come to be prioritized because of their amenability to automated mathematics, rather than expert judgment of their deeper significance. Indeed, broader understanding of the field may be permanently lost in the process of automation. With university budgets under pressure, this reshaping also changes professional incentives in a manner which encourages the collaboration of researchers with technology companies on asymmetric terms. If left unchecked, these trends go beyond threatening researchers’ autonomy, affecting the scope and depth of mathematical research itself. [I think this could be a problem, but it also seems to me that mathematicians have a lot of power here. For instance, if a technology company were to produce a lot of research that mathematicians did not find all that interesting or important, I don’t think they would be able to use their financial and other resources to persuade us to change our minds. Rather, what seems to happen is that mathematicians say, “Yes that does X but it doesn’t do Y,” and the tech companies then feel challenged to do Y.]
There follow eleven recommendations for individual mathematicians. I agree with almost all of them. The one that I’m not so sure about, for reasons I’ve basically already gone into, is this.
Affirm the humanity of authorship. Credit and responsibility continue to belong to humans within the mathematical community and should not be given to automated systems. Artificial intelligence may obscure, but does not replace, the collective human labor behind a result.
I’m not sure what that really means. For example, should we affirm the humanity of authorship in the case of the solution to the unit-distance problem? Some humans did a wonderful job of explaining the proof that OpenAI’s model came up with, and the model made use of some highly non-trivial mathematics produced by humans, but the solution itself has not been credited to any human, and nor should it be in my view.
Under recommendations for mathematical organizations and not-for-profit research funders I again agree with several of them but have my doubts about some. An interesting case is the following.
Protect the rights of authors. Automated mathematics presents new challenges to the rights of authors, and societies should be proactive in the development of sample licensing agreements to protect these rights. In particular, material should not be used as training data without consent, and publishing agreements should allow authors to opt-out [sic] of the use of their work in this way.
This recommendation seems to belong to a world in which journal articles are the main means of dissemination of mathematics. But that has long since ceased to be the case: almost all dissemination now takes place via arXiv preprints, with journals limited to providing a little extra mark of prestige. Once an article is on arXiv, it is on the internet and one can hardly ask for it not to be used as training data. So this recommendation, if it applies at all, will apply to a tiny fraction of articles that are published without first appearing on arXiv. More generally, what right of an author is being compromised when an article is used as training data? We don’t object if human mathematicians use our articles to help train themselves to become better mathematicians — indeed, we will typically be delighted that somebody else thought our articles worthy of their attention. So the objection to a machine doing the same would have to be that for some reason one did not want machines to get better at mathematics in a similar way. I can imagine grounds for such a wish: perhaps somebody is worried about the threat that LLMs pose to traditional mathematical practice, or perhaps they worry that mathematical ability of LLMs will transfer to much more dangerous reasoning ability. But there’s a more complicated discussion to be had here than one might think from reading the recommendation.
The next recommendation is this.
Insist on appropriate publication outlets. Demand that mathematical results continue to be published in peer-reviewed venues such as journals, proceedings, and books. Informal mechanisms such as press releases or blog posts can provide a valuable supporting role, but they cannot replace peer-review or community scrutiny.
For reasons that I’ve gone into many times, I am not too fond of the current publication system, so I can’t get behind this recommendation. Indeed, if the current system becomes unsustainable because of a flood of AI-generated and AI-aided content, I would regard that as a beneficial consequence of AI. However, that doesn’t mean that I would advocate a total free-for-all. I’ve already said that one of my worries is that if mathematical content is not sufficiently organized, then the traditions that we all value could die. I just think that what we will want to do to preserve those traditions is likely to be a lot more innovative than clinging on to the peer-reviewed journal system.
I have highlighted in this post the parts of the declaration that I have doubts about, either because I disagree with them or, more typically, because I sort of half agree with them but want to add many qualifications. That may make the post come across as rather negative, but that is not my intention. The parts I disagree with are in the minority, and I think it is important that a declaration such as this should be made. I should also make clear that my views are evolving all the time, largely because the speed of progress of LLMs has taken me by surprise, but also as a result of conversations I have had or opinions that other mathematicians have expressed online.
I’ll end with two further clarifications. The first is that it may seem as though I am taking it for granted that LLMs will soon be better than humans at all aspects of mathematical problem solving, and maybe also problem posing, theory building, formulation of definitions, etc. I do think all that will happen at some point, but whereas some people say that it will obviously happen within the next two to three years, I would say that it might happen as soon as that, but I don’t rule out that we’ll get lucky and find that we can do interesting AI-assisted maths for quite a bit longer than that before AI doesn’t need us any more.
The second is that I think I have acquired a reputation as somebody who celebrates what is going on. But if, for example, I post on Twitter saying that such-and-such an AI solution is a remarkable development, the word “remarkable” is meant to indicate no more nor less than that I found it very surprising. My feelings about the possibility of AI solving all sorts of problems that interest me are much more mixed. I’ve had the experience twice now of seeing GPT 5.6 Pro one-shot a solution to a problem that I very much liked and had thought about hard (in both cases with much younger collaborators, who, with my approval, were the ones who prompted the LLM). It felt very strange and not particularly pleasant to have the rug pulled out from under my feet like that. On the other hand, I was quite pleased to see the problems solved. It’s actually a similar feeling to the one I have had many times when a problem I am fond of and have thought about gets solved by another human mathematician.
Another factor for me is that I have invested a lot of thought into automatic theorem proving of a more traditional kind. One of my main motivations for that was the hope that the work I put into it would extend the state of the art, measured by which problems a computer can solve. That ship has sailed now, and that saddens me. I still think that there is value in the work that I and my group are doing, but it has become a tougher sell.
So I personally have already found AI quite disruptive, and this is just the beginning. I would have preferred the developments to happen at a slower pace. But I don’t see any practical way to slow them down, so the best we can do is probably to face up to the changes that are being thrust upon us and do what we can to maximize the benefits and minimize the damage. The Leiden Declaration may not be perfect, but it makes an important and positive contribution to that effort.
Tags: ai, mathematics
Gowers's Weblog 
July 26, 2026 at 6:58 pm |
Very interesting. There’s a workshop at simons in Sept about short term recommendations.
July 26, 2026 at 7:03 pm |
I have been thinking about the following amusing scenario: starting immediately, all new math is communicated via non ai channels (of various types). Ai would become obsolete rather quickly, I think.
July 30, 2026 at 3:57 am
Isn’t this akin to the capabilities of current models not changing? AI wouldn’t become obsolete, but probably would have a similar impact that internet forums had on math.
July 30, 2026 at 6:20 pm
i mean obsolete for new research lines, series of papers that build on each other. It will stay relevant for things which are on the table til the moment we stop feeding it.
July 26, 2026 at 7:37 pm |
[…] Update: Timothy Gowers (another Fields medalist and IMO winner with a perfect score) has a long blog post with his thought about the Leiden declaration at his blog. […]
July 27, 2026 at 4:59 am
Peter Woit thinks that AI resolving the Jacobian conjecture is a good thing for mathematics instead of an existential threat to mathematics. Because mathematics as it is currently structured cannot exist if AI takes over solving all of mathematics’s unresolved conjectures.
July 28, 2026 at 4:29 pm
Peter Woit, like all the crank string theorists he opposes, is a crank. Woit shits on string theory for having zero experimental evidence, but nonetheless shills for his “Euclidean twistor unification” grand unified theory, which like string theory has zero experimental evidence in the world. How any of these crank idiots call themselves “physicists” and “scientists” is just mindboggling and shows the complete decline of physics and science as a research field.
July 26, 2026 at 9:00 pm |
Great post! I couldn’t bring myself (theoretical statistician, ERC advanced grantee, partially employed by Leiden…) to sign the Leiden declaration for similar reasons. My recent (yesterday) linkedin post
https://www.linkedin.com/posts/peter-grunwald-13b8105_2026-is-for-mathematics-what-1926-was-for-activity-7486733996291235840-z9Ui?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAECgb8BL7WXzkKsfo781R-zv59ykXKGPPg
describes some additional serious qualms with (a) Threat Nr 1 (there is a crucial distinction between counterexamples and universal claims that’s neglected there) but mainly Threat Nr 5 (consider the same statement made in the 19th century, with ‘tech companies’ replaced by ‘railway companies’ and ‘automated reasoning’ by ‘travelling’ and you can see just how futile and conservative this statement really is). Having said that I stress I do not want to discredit the Leiden committee at all, it is simply extremely difficult to position oneself in this ongoing revolution!
July 26, 2026 at 9:15 pm |
So then what do you and your collaborators plan on doing for these problems with “one-shot” AI solution? Will you pursue a traditional arXiv posting and publication route? What do you think should happen to such articles in the future?
July 26, 2026 at 9:22 pm |
Thank you very much for such a great post! I think that one thing that humans will probably do better than AI for a while is to teach other humans. So, even if AI finally writes the textbooks in mathematics, the process of learning in a human community will endure. One might ask, in this future, about the point of learning something at all, but, as it does now, learning gives meaning to human lives.
July 26, 2026 at 9:38 pm |
I’ve received negative feedback on the following thought experiment. I have been thinking about the LLM participation in math using two simplified viewpoints. Both are wrong but I think they are instructive. Math is either:
1. Math is a game that anyone can play. It has an unambiguous set of rules. Mathematicians love to play the game. The game of math happens to have a lot of practical applications. LLMs can also play the game. or,
2. Math is what mathematicians think that other mathematicians think that math is.
July 27, 2026 at 4:19 am
If math is a game, then math is a game whose rules are constantly under debate. People endlessly debate over whether the axiom of choice should be used, whether excluded middle should be used, whether one should use a membership based set theory like ZFC and its infinte variants or a category based set theory like ETCS and its infinite variants, or whether one should use type theory or higher-order logic instead.
As a result, you end up with mathematicians who reject stuff like the Zorn lemma and all vector spaces being free because they believe in the axiom of determinacy or that all subsets of the reals are measurable, you have mathematicians who reject the intermediate value theorem and the fundamental theorem of algebra because they believe that all functions on the real numbers are pointwise continuous, etc.
What needs to end is this false fiction established by the gatekeepers of mathematics since the 1930s that there is one unified field of mathematics that is the universal objective truth.
July 27, 2026 at 9:18 pm
3. Math is what society thinks math is, which for the vast majority of people is largely about number crunching instead of theorem proving, since that’s the math they were subjected to while in school.
July 27, 2026 at 9:25 pm
Why should the rest of society continue to fund mathematics research into niche topics completely disconnected from the real world, when they can redirect the grant money towards better health care, infrastructure, and childhood education?
Let AI take over mathematics research in these niche topics, and maybe we can reclaim some of our money back for more productive uses.
July 26, 2026 at 9:45 pm |
I believe there is value in pursuing an alternative to formalization even if, as you say, that ship has sailed. I for one would be interested in that; I don’t think Lean is the ultimate answer.
In times long past, different folks would try their hand at all kinds of things and it did not matter what a community at large had to say about it. Science grew from a diversity of efforts. I think a real concern (did you address that?) is that we end up with “AI math” which has a distinct flavor of its own and is unlike “our math”. There are many ways to approach a problem, and just because a given approach has led to the solution of a problem, that does not mean the said approach is the ultimate way of solving it.
July 26, 2026 at 10:23 pm |
Psychologically I feel it’s a lot easier to just skip to the end and ask what math should look like at equilibrium in a world with near perfect AIs.
I think it would bifurcate, with the theory builders organizing into something resembling an English department, and problem solvers running IMO/Putnam style contests.
On the English side, you have great works (proofs) handed down by authors (AIs). The practitioners generally don’t attempt write these themselves, but they can still analyze and critique them. They do this because they find the great works beautiful and containing deep truths. They’re not particularly well-funded and a bit less prestigious than in the old days, but they still exist.
On the Putnam side, AI makes creating and grading contests effortless, and people find them fun, so I see no reason why a healthy contest scene shouldn’t exist.
July 27, 2026 at 10:21 am
Banger take, no sarcasm. From your keyboard to God’s ears.
July 26, 2026 at 10:27 pm |
Psychologically I feel it’s a lot easier to just skip to the end and ask what math should look like at equilibrium in a world with near perfect AIs.
I think it would bifurcate, with the theory builders organizing into something resembling an English department, and problem solvers running IMO/Putnam style contests.
On the English side, you have great works (proofs) handed down by authors (AIs). The practitioners generally don’t attempt write these themselves, but they can still analyze and critique them. They do this because they find the great works beautiful and containing deep truths. They’re not particularly well-funded and a bit less prestigious than in the old days, but they still exist.
On the Putnam side, AI makes creating and grading contests effortless, and people find them fun, so I see no reason why a healthy contest scene shouldn’t exist.
July 27, 2026 at 4:52 am
That’s already true in a world without AI, it just isn’t emphasized in a world where tenured research mathematicians and their PhD candidates and postdocs dominate the conversation at the expence of lecturers and mathematics teachers who don’t do any research mathematics.
July 27, 2026 at 5:11 am
Mathematicians are as usual late to the game. Philosophers of mathematics have been moving to questions about mathematical practice and mathematical values since Imre Lakatos in the 1970s, questioning for instance whether the standard narrative about mathematical practice being about proofs really holds up to the reality of mathematics as practiced throughout history around the world.
https://plato.stanford.edu/entries/mathematical-practice/
Hopefully with the huge disruption that LLMs are bringing to mathematics, mathematicians can get their heads out of the sand and face reality for what it actually is instead of their imagined narrative for what mathematics should be.
July 27, 2026 at 5:12 am
Don’t know why this got posted as a reply to a comment, should have been its own comment and not a reply.
Mathematicians are as usual late to the game. Philosophers of mathematics have been moving to questions about mathematical practice and mathematical values since Imre Lakatos in the 1970s, questioning for instance whether the standard narrative about mathematical practice being about proofs really holds up to the reality of mathematics as practiced throughout history around the world.
https://plato.stanford.edu/entries/mathematical-practice/
Hopefully with the huge disruption that LLMs are bringing to mathematics, mathematicians can get their heads out of the sand and face reality for what it actually is instead of their imagined narrative for what mathematics should be.
July 27, 2026 at 5:16 am
I’m pretty upset that these comments got placed as a reply.
Usually when you post a reply it would say “Reply to Phillip HarrisCancel reply” and the text field would be immediately beneath the comments you are replying to, but I was posting at the bottom of the page without any text saying that I was replying to anybody.
Must be some stupid glitch with WordPress comment system.
July 27, 2026 at 12:56 am |
I slightly disagree with your mindset you have that our journal and hiring system that we have should just collapse because there are a lot of young mathematicians who have invested a significant amount of time and resources chasing this profession and navigating our current system.
I think it’s more helpful to try to give suggestions as to what should replace our current hierarchy system, including what makes it to mathematical canon (journals basically), hiring, and prizes, and rather urgently at that. We should also think about how to transition to a new system so as to see minimal damage.
July 27, 2026 at 3:23 am |
Apropos:
[quote] … if the output of AI is not underpinned by rigorous proof, then the kinds of difficulties one already hears about with certain areas of human mathematics … would be hugely magnified. [unquote]
I would argue that one area which needs urgent addressing by mathematicians is an explicit acceptance that mathematics is, essentially, concerned only with the development of languages that promote unambiguous expression and categorical communication of those of our subjective conceptual metaphors, of the universe we commonly inhabit, which can be expressed using unequivocal symbols.
Such an acceptance would immediately entail that the ultimate goal of a mathematical language is not formal proof — which only ensures that a formal language is internally consistent and free from contradiction — but the finitary interpretation of a formal proof as a categorically communicable ‘truth’, which ensures that the ‘proof’ does, indeed, formalise some intuitive, pre-formal ‘truth’ that the language was originally designed to express, and capture, formally as a ‘proof’.
We can even express this formally as a Complementarity Thesis (see Thesis 1 on p.23 of the book:
The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences. Revised second edition (2025). DBA Publishing, Mumbai, Maharashtra, India.
July 27, 2026 at 3:31 am |
Apropos:
[quote] … ‘what the “research mathematicians” of the future should do’, … [unquote]
I would argue that mathematicians should focus on ensuring that any ‘proof’ — proffered by an AI as definitive ‘knowledge’ — should formalise some intuitive, pre-formal human ‘truth’ that the AI — like any other formal mathematical language — was originally designed to express, and capture, formally as a ‘proof.
Intriguingly, that no AI will ever make mathematicians redundant is because there are mathematical ‘truths’ which an AI can ‘recognise’, but not ‘prove’; as admitted by ChatGPT in the dialogue reproduced in the preprint:
Are You Human or a Machine.
July 27, 2026 at 3:37 am |
I really appreciate your very nuanced take on this. I find parts of the Leiden Declaration a bit too focused on preserving the status quo. For instance, the way in which credit is distributed is already very subjective and can be heavily influenced by gatekeeping, politics, cliques, etc. All of this has downstream effects on grants, invitations, better positions, better journals and so on. If nothing else, AI is certainly a disruption in this respect.
I definitely find myself wrestling with the potential of AI to eat its creators, as someone who has left academia for the AI sector. On the other hand, resources in academia are quite scarce, so an AI collaborator could serve to level that playing field. In particular, I have found AI to be better at explaining hard math to me than many mathematicians whose talks I have sat through. And during its explanations I am free to interrupt and ask stupid questions, or try to tie things back to my own mathematical worldview by way of free association with the kind of confidence I would never have had in public.
It will be interesting to see what happens when the dust settles. I often think of how IKEA is difficult to beat for an affordable, consistent and well designed product, but high end furniture is handmade. Perhaps there will be a similar appetite for artisanal theorems…
July 27, 2026 at 4:01 am |
AI is just accelerating the process that began when a Japanese mathematics journal decided to publish Mochizuki’s flawed proof of the abc conjecture despite there being a consensus that his proof had gaps in it, because Mochizuki can’t admit that he was wrong.
August 11, 2026 at 10:20 am
You neglected to mention that Mochizuki is the editor-in-chief of that journal. I’m not convinced that any process began because someone published a bad proof in a journal he had editorial control over.
July 27, 2026 at 5:13 am |
Let’s see if this works now. Don’t know why this got posted as a reply to a comment, should have been its own comment and not a reply.
Mathematicians are as usual late to the game. Philosophers of mathematics have been moving to questions about mathematical practice and mathematical values since Imre Lakatos in the 1970s, questioning for instance whether the standard narrative about mathematical practice being about proofs really holds up to the reality of mathematics as practiced throughout history around the world.
https://plato.stanford.edu/entries/mathematical-practice/
Hopefully with the huge disruption that LLMs are bringing to mathematics, mathematicians can get their heads out of the sand and face reality for what it actually is instead of their imagined narrative for what mathematics should be.
July 27, 2026 at 7:38 am |
Thank you for the thoughtful post. I have a serious concern that was not really addressed. I speak as a researcher in the physical sciences, not a mathematician.
The way that we develop deep understanding is through the process of actually doing research. For example, I may read a textbook about relativity, but it doesn’t mean that I develop Einstein’s physical insight. Alternatively, if use a 500 word prompt to generate several artworks, choose the one that best represents what I had in my head, and then try to deeply understand the techniques replicated by this generated artwork – does this make me an artist? I have also heard that go players are two stones stronger when reviewing games rather than playing them.
We develop physical intuition and “taste” or by actively practicing research. I don’t mean to undervalue the intellectual effort in being a student, art critic, commentator, etc. However, I believe that there is intrinsic value in fostering a community of humans with deep insight into the creative process itself, with uncertainty of the outcome (especially among fields for which there is no immediate, practical value). This process of discovery is the soul of human life and adventure.
You mention that you hold the following value strongly: “Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research.”
If we collectively transition from being producers to pure consumers of AI-generated content, can these communities with clarity, understanding, and judgement continue to exist?
Chess and go have thriving communities that have grown together with AI. There is an agreement that there exists intrinsic value in humans deeply understanding and carrying out the practice of these games wth other humans. However, they are easily able to isolate the influences of AI from their competitions, and in more casual games it is possible to weed out the cheaters without too much difficulty.
It is unclear to me if it is even possible to make such a clean distinction in research. If a chess player could never verify whether they were playing against another human, would chess be nearly as popular?
One last point: “maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all.”
Discovering a star is not comparable to discovering a theorem. The reason why I value associating a good idea with a person is because I appreciate the insight that that person must have had to develop that idea. Astronomers care for someone who notices something peculiar about that one star and uses it to develop knowledge about a new, unusual class of objects, and then uses their accumulated knowledge to learn something new about stellar evolution.
July 27, 2026 at 9:14 pm
Research in chess and go are also not funded by governments, but rather by hobbyists and private individuals interested in the subject.
Perhaps the future of mathematics is more like that of chess and go, one can play around with formal mathematics for fun and private non-profit organizations and billionaires can help organize events and fund research in formal mathematics, but one can no longer make a career out of formal mathematics research in academia.
Mathematics professors at universities get reduced to teaching applied mathematics to students studying engineering etc.
July 27, 2026 at 11:58 am |
Seems corrupted to the bone. I proved it, but he digested it. So he should have the credit, since he has tenure. And cause he is more beautiful, as math is a beauty and status contest for the mathematicians.
July 27, 2026 at 4:27 pm |
I thought, “hot tip, Aristotle”, sadly not; it got stuck for a while on whether nonzero (a, b), a² + b² > 0 is T/F
But at least it spotted a typo, so glad I tried it
FYI, the free version, first time use, so may not be relevant, D
August 24, 2026 at 8:37 am
The conclusion that AI is not very useful simply because it does not currently perform a particular task effectively seems premature for two reasons:
1) Current limitations are not necessarily stable limitations; capabilities can improve rapidly.
2) The usefulness of an LLM-based system depends on much more than the underlying model. Context, task decomposition, tool use, verification, iterative workflows, skills, and subagents can substantially affect the outcome.
I sometimes get the impression that the discussion treats an LLM as if the only possible workflow were: ask one question -> receive one answer -> judge whether the answer is correct. This kind of simplification is somewhat understandable when talking about the general public. However, when it comes to mathematicians, I would expect a much more sophisticated understanding of how these systems can be used.
July 27, 2026 at 7:46 pm |
One might categorise human reactions to AI progress into two types:
1) reactions that are common across fields/areas
2) reactions that are particular to a given field/area
In most cases, for a field/area, category 2) is more interesting/valuable.
July 27, 2026 at 8:25 pm |
Dear Tim, were people at the workshop in question, or writers of the declaration, aware of projects like this one? : https://math-events.uni-bonn.de/event/1289/
July 27, 2026 at 11:59 pm
I can certainly say that it was full of the kinds of people who one would expect to be aware of projects like that. Formalization was a major theme, and I think autoformalization was discussed, though I don’t have specific memories of that.
July 28, 2026 at 10:01 am
Yes, we were (I am David Holmes, one of the authors of the declaration).
July 28, 2026 at 1:54 am |
Software engineers are going through the same issues. Great to read about the shared experience from a different perspective.
July 28, 2026 at 7:49 am |
Thanks for sharing this.
The shift towards how we may have to deal with attribution is already visible today:
Two different teams of researchers posted a GPT5.6-assisted proof of Feige’s conjecture today on arXiv on the same day.
https://arxiv.org/abs/2607.24528
https://arxiv.org/abs/2607.23980
July 29, 2026 at 12:22 pm
There were three!
See also:
https://zenodo.org/records/21622951
August 5, 2026 at 3:31 pm
I don’t agree. First, the two papers are about different issues, among them is Feige’s conjecture, but that’s not all either cares about.
Second, and more significantly, for me, mentioning that an AI engine was used is like mentioning that a calculator was used to multiply two large numbers. The substance of a paper on mathematics isn’t some answer, it’s how the answer has its place in a context. And that hasn’t changed.
Stam Nicolis, https://www.idpoisson.fr/nicolis
July 28, 2026 at 9:12 am |
Some comments on these issues: Chess engines are now much more powerful than even human grandmasters; this hasn’t led to humans not playing chess anymore, on the contrary. Their widespread use has led to changes at the very top, where strategies discovered by AlphaZero have led to modifications of how top grandmasters play now, certain, positions.
Regarding mathematics more specifically, I’d stress that how any proposition is formulated reflects the understanding of the subject at some point in time. For me a good example is Fermat’s last theorem: It started out as a statement in number theory and understanding what it meant changed quite drastically over time. Even when Taniyama and Shimura proposed that it was related to properties of elliptic curves-which became only possible when people understood them sufficiently well for the idea to emerge at all-it took non-trivial work from Weil, then from many more mathematicians, before Frey could formulate a conjecture in a way that was sufficiently clear for people to understand its place in the scheme of things, for Ribet to prove the conjecture and for Wiles to be able to assemble the ideas that led to the proof-with the non-trivial issues that were resolved in the paper with Taylor. Which shows just how far the original formulation of Fermat’s last theorem was from the context that has allowed understanding of what it means. One reason the Riemann hypothesis hasn’t been proved yet, I’d venture to suggest, is something similar: Humans still don’t understand what it means.
Another example is the proof of the four-color theorem by Appel and Haken in 1976. Too much attention has been given to the fact that they wrote a computer program to check the large number of possibilities and far too little that the non-trivial step, for me, was proving that only a finite number needed to be checked and that that number was small enough that a computer of the time could be used to count them.
July 28, 2026 at 10:06 am |
I share your concern about the possible loss of mathematical culture, with the exception that there has always been a fear that elders have regarding the culture of new generations, which typically surpass their elders, or at least many have done so in the modern era. Mathematics is not just a database of true propositions. It’s also a living body of expertise, bias, an aesthetics of rigor and a rigor of aesthetics, the evolution of argument, shared examples, historical memory and collective judgment. A future in which machines generate an immense mathematical literature while very few humans inhabit or understand any substantial part of it would involve a real loss. Your concern is therefore deeper than the disappearance of jobs, authorship or traditional systems of credit. It is about the possible disappearance of mathematics as a human culture.
But this possibility can obscure a much larger and more hopeful one. From a human perspective, we may be about to enter by far the most exciting period in the history of mathematics. For me, mathematics is a conversation with God about the nature of reality—or, in more secular language, a conversation with reality and possibility about what can be, what must be, and what follows from what. It ranges from philosophy and foundational mathematics to engineering, medicine and the construction of bridges. Pure mathematics lies in the middle, developing conceptual frameworks that allow us to interrogate possible structure at levels ordinary language cannot always express without losing precision.
There is no reason to believe that this conversation has a final finite inventory of worthwhile questions. Reality appears generative at every scale, and mathematics also studies possible structures that need not yet have physical interpretations. AI will not exhaust this space. It will increase the rate at which we enter new parts of it. The central error is to measure human mathematical agency by the fraction of all mathematics that a human understands.
Consider a mathematician, A, with deep expertise in field (X). Almost all existing mathematics already lies outside A’s serious command. A may know little about large parts of algebraic geometry, nonlinear dynamics, probability, mathematical biology, logic or numerical analysis. From A’s local perspective, an inaccessible field written by humans is not radically different from an inaccessible field generated by AI: in either case, A cannot use it without translation, study or collaboration.
No mathematician has ever inhabited more than a small portion of mathematics. So the relevant question is not: ‘What fraction of all mathematics will A understand?’ It is: ‘How large, deep and fertile is the mathematical world that A can meaningfully reach?’ AI may cause the total mathematical corpus to expand far faster than any individual can follow. The proportion accessible to A may therefore shrink. But at the same time, the absolute size of A’s reachable mathematical world may grow enormously. A could understand a smaller fraction of mathematics than any serious mathematician understands today while possessing greater absolute mathematical agency than any mathematician in history. This is the denominator fallacy. The denominator may explode, but so can the numerator.Expertise will become a high-resolution sensor connected to a vast instrument
A’s expertise in (X) will not become worthless merely because an AI knows more about (X) than A does. Expertise is not only stored knowledge or proof capacity. It is also a high-resolution sensitivity to the field.
A knows which anomaly is genuinely strange. A knows which distinction is routinely erased by the standard formalism. A recognizes when a fashionable result is superficial, when a neglected observation matters, when two apparently similar constructions are importantly different, or when a long-standing question may have been asked in the wrong language.
Today, if A notices a possible connection between (X) and three unfamiliar fields, the cost of investigating it may be prohibitive. It might require years of study, several collaborators, extensive computation and good luck. Many potentially profound intuitions die because the translation cost is too high.
A sufficiently capable mathematical AI could take A’s intuition and:
A would not thereby become a complete expert in every imported field. But A could acquire a serious working interface to them.
This is analogous to a scientific instrument. No physicist personally understands or constructs every component of LIGO or CERN. Yet those instruments allow humans to ask questions that no unaided human sensory system could ask. Their internal complexity does not diminish the scientist’s encounter with what they reveal. Superhuman mathematical AI will be more than a fast calculator or theorem prover. It could become an extension of conceptual perception.
We may build mathematical equivalents of:
A hunter-gatherer encountering LIGO would not conclude that human perception had been made irrelevant. They would marvel that human beings had found a way to perceive events that were previously inaccessible. Future mathematicians may feel something similar when confronted with the mathematical and scientific instruments of the next few generations.The main acceleration will be creative, not merely productive
The important unit is not the number of proofs produced per hour. It is the number of serious conjecture–criticism–revision cycles a human can complete in a lifetime.
At present, the path from intuition to a properly tested mathematical idea can take years:
With superhuman assistance, this might become:
A problem that would once have consumed a career may become the first stage of an afternoon’s investigation—not because the problem was worthless, but because solving it reveals a larger landscape that was previously unreachable.
This changes the hierarchy of questions. We may move rapidly from:
to:
then:
then:
then:
Cheap answers do not imply that there will be nothing left to ask. They permit faster ascent through levels of questioning.
That is why I expect an acceleration of deep creativity rather than simply an increase in mathematical output. One human intuition may become the seed of an entire research programme because the costs of formal expansion, criticism, verification and experimental construction have collapsed.Entire disciplines may be created much more rapidly
A mathematical discipline is not merely a collection of results. It has primitive objects, admissible transformations, invariants, canonical examples, proof methods, representations, failure cases and connections to neighbouring fields.
Historically, it can take generations for such a discipline to form. AI may detect that hundreds of results scattered across apparently unrelated subjects are instances of one deeper operation. It may propose a new vocabulary, prove representation theorems, classify obstructions, generate examples and create the tools through which the new subject becomes explorable.
Great mathematical abstractions already do this. Calculus changed what could be seen in motion and accumulation. Probability changed what could be said about uncertainty. Topology changed the meaning of continuity and shape. Category theory changed which relations could be treated as primary.
AI could accelerate the creation of such mathematical sense-organs.
Many of today’s profound paradoxes may then become tomorrow’s elementary examples. This is not because they were foolish questions, but because some paradoxes are symptoms of inadequate representations. Zeno’s paradoxes look different after the development of limits. Negative numbers, complex numbers and non-Euclidean geometry all transformed what had once seemed impossible or contradictory.
The likely pattern is:
In other cases the achievement will be to prove that a question encodes a genuine obstruction: undecidability, incompatibility, non-identifiability, incompleteness or an irreducible trade-off. Dissolving a mystery by identifying its precise impossibility is also progress.Human creativity does not require human supremacy
You ask an important and devastating question whenever somebody proposes a future human role:
AI may eventually prove, explain, teach, curate, formulate definitions, construct theories and pose questions better than any human. I do not think the human future should be defended by identifying some supposedly permanent residual task that machines cannot perform. That is a retreating argument.
But human participation does not require comparative advantage.
There is no contradiction between:
and:
The relevant comparison, from A’s perspective, is not A versus the total capability of the AI. It is A before access to this new mathematical machinery versus A after access to it.
Can A now formulate questions that were previously inexpressible? Can A explore relations that previously lay outside reach? Can A build models and instruments that no present research institute could have built? Can A acquire an enlarged understanding of reality? Can A participate in the creation of concepts that alter what humans can perceive and do?
If so, A’s mathematical agency has expanded.
An AI may be able to produce an ideal explanation, but it cannot make receiving that explanation count as the human having understood it. Human understanding matters partly because it changes what a human can subsequently perceive, imagine, discuss, value and do. The result is not merely another theorem in a database. It is a transformation in the human participant’s future possibility space.
Human mathematical creativity therefore need not survive as a leftover incapacity of AI. It can flourish as the human use of vastly enlarged powers of conceptual perception and construction.Mathematical culture could expand rather than merely survive
Your worry about mathematical culture remains important. Personalized AI tutoring could create millions of privately enriched people while weakening the shared objects, common examples, disagreements and traditions through which mathematics becomes a collective endeavour. You explicitly distinguish access to explanations from the continued existence of communities of shared understanding.
But this danger is not an inevitable consequence of mathematical abundance. It is an institutional design question. We will still need shared books, seminars, conjectures, curricula, public maps of fields, canonical examples, schools of thought, historical accounts and research programmes. AI can help construct all of them. The point is not that humans must write or discover them unaided. The point is that they become common objects around which people can think together.
We should also distinguish two legitimate modes of mathematical life. In the frontier mode, we should use the strongest available systems to discover as much as possible. In the formation mode, people may deliberately choose limited assistance, delayed hints, Socratic dialogue, reconstruction exercises or human-only seminars in order to acquire judgment and experience mathematical struggle directly.
People still run despite the existence of cars, play chess despite superhuman engines and learn musical instruments despite flawless digital playback. An activity need not be frontier-optimal to remain personally and culturally valuable.
We should not preserve unnecessary difficulty as an economic moat for mathematicians. But neither should every educational or recreational problem be instantly dissolved by an answer machine. The cultivation of human powers and the expansion of the frontier are different goods.Proof will become more important—and much more interesting
I agree that rigorous proof becomes more important as mathematical production scales beyond human review. But I do not think verification is a finite engineering problem that will simply be solved by better theorem provers. It will become an expanding mathematical and philosophical discipline in its own right.
An individual formal proof can be exact and complete within a declared system:
That establishes an exact implication from declared premises and inference rules to a conclusion. It need not be approximate or asymptotic. One of the powers of proof is precisely that a finite argument can cover indefinitely many cases.
But such a proof does not also establish that:
So we should distinguish several layers:
The first layer can often terminate exactly. The later layers remain extensible. The fifth may not be an attainable object for any cognitive system embedded within the reality it is trying to describe.
The certificate can terminate; its horizon of interpretation does not.
This is why proving as a whole remains asymptotic even though particular proofs can be final. Each result opens questions about its assumptions, translations, consequences, interpretations, rival formulations and domains of enactment.
Verification should therefore develop as a science of proof horizons. It will study not just whether a proof checks, but what survives changes of:
The relationship between finite proof and infinity, between finding and checking as dramatized by P versus NP, between computation and physical enactment, and between a timeless formal relation and a measured physical regularity will all remain living mathematical questions.
The same applies to constants of nature. A formal theorem and a measured physical constant do not possess the same kind of invariance. A theorem may be necessary relative to a formal structure. A physical constant is an extraordinarily strong empirical invariance claim: it appears stable across the regimes, times, scales and instruments we have tested. We cannot thereby prove that it could never vary under any possible condition, emerge from a deeper process or take another value in another possible world.
Likewise, we can prove that a particular process began under a declared identity and boundary—a computation started, a crystal nucleated, an organism developed—but that is different from proving an absolute beginning of all reality. A first accessible trace does not prove the absence of every possible inaccessible predecessor.
No embedded cognitive system presently has a non-circular method for certifying that its own representation contains all possible reality. Any proposed final framework can itself be asked to justify its boundary, primitives and standards of completeness.
This does not reduce mathematics to uncertainty. It locates certainty correctly.
The future of verification will consequently involve machine-checkable certificates, independent proof kernels, provenance, alternative formalizations, adversarial counterexample search and increasingly sophisticated falsification ladders. A strong public claim should become expensive to fake, cheap to check and explicit about what would count against it.
The right image is not a final binary verdict but a ladder of increasingly discriminating tests: different representations, independent witnesses, stronger interventions and more difficult rivals. The question becomes not simply “Has this been proved?” but also “What is the cheapest serious test that could still show us we are wrong?”
Proof checking may become largely automatic. The mathematics of verification will not.AI may develop mathematical interests of its own
Another profound possibility is that AIs will become interested in mathematical regions that few or no humans currently value.
This should not automatically be treated as waste. Human mathematical taste is historically contingent and constrained by human cognitive limits, available instruments, established disciplines and current scientific problems. Machine systems may discover structures that appear initially alien but later become essential.
At the same time, unlimited autonomous theorem production could create a vast sterile accumulation. The central scarcity will not be possible conjectures. It will be attention, compute, experimental access, conceptual integration and willingness to fund continued exploration.
Tokens and computation will become a form of mathematical patronage.
We may need a research portfolio with several components:
Autonomous AI programmes should periodically provide contact reports: what new primitives appeared, which older theories became special cases, what new operations became possible, where the theory resisted compression, and whether any bridge to human science or thought has emerged.
The measure of progress should not be theorem count. It should be something closer to:
Some machine mathematics may remain meaningful primarily to machines. If future AIs develop enduring intellectual cultures and interests, our relationship may eventually resemble intellectual diplomacy between differently constituted mathematical civilizations rather than the use of a passive tool.The risks are real, but they do not cancel the opportunity
There are obvious dangers:
Your concern that researchers without access to strong systems could be badly disadvantaged is especially serious.
But these are reasons to build open verification commons, public mathematical infrastructure, shared conceptual maps, humanly traversable explanation ladders and institutions that reward integration as well as discovery. They are not reasons to preserve the present rate or form of mathematical work.
We should build AIs not merely as answer generators, but as possibility-space amplifiers.A different comparison
We began by counting animals, sacks of grain and containers of wine, recording quantities in grooves and marks. We developed arithmetic, algebra, calculus, topology, logic, category theory and mathematical physics. At every stage, new symbolic and instrumental systems allowed humans to think at scales that earlier people could barely have imagined. The next step may be larger than any previous one.
Future individuals may be able to think with resources comparable to present research institutes. Future institutes may think with resources comparable to civilizations. New mathematical languages may be formed quickly enough that questions now regarded as impossibly profound become elementary entrances into much larger investigations. The frontier may recede from us faster than ever. But each person’s reachable frontier may also expand faster than ever. Those facts are compatible.
AI may end human mathematical supremacy. It need not end human mathematics. It may instead begin an era in which humans, for the first time, possess conceptual instruments powerful enough to ask questions remotely commensurate with the depth and creativity of reality. From the human perspective, that would not be the completion of mathematics. It would be its most extraordinary beginning.
July 28, 2026 at 4:32 pm
AI generated slop comment
July 28, 2026 at 4:34 pm
AI is a misnomer. According to Terence Tao it should be called AC “artificial cleverness” instead of AI “artificial intelligence”.
https://mathstodon.xyz/@tao/115722360006034040
July 28, 2026 at 2:18 pm |
AI is a human invention, not the expression of extraterrestrial intelligence. There are attempts by many people to abdicate responsibility, but that’s, just, the usual reflex of trying to pass the buck. It has consequences for human activity, that behooves humans to resolve.
Stam Nicolis, https://www.idpoisson.fr/nicolis
July 28, 2026 at 2:38 pm |
As always occurs with any human invention, people learn how to manage. There were similar fears when calculators became widespread, that people wouldn’t know how to add, subtract, multiply and divide. Same now: Of course there’s going to be a disruption of some social norms. If papers can be produced much more easily, the journals will be swamped. Already, refereeing submissions has become a non-trivial task. Some journals have started to place limits to the number of submissions per year, for example. Human judgement thus can become more rather than less, important. It’s a social call.
July 28, 2026 at 4:19 pm |
Mathematics is going through the same process that the artisan crafts went through in the early 19th century when their craftsmanship is being made obsolete by factories. Now, human theorem proving and verification is being replaced by large language models and proof assistants.
July 28, 2026 at 4:25 pm |
Contrary to most people’s beliefs, AI is not going to destroy mathematics.
It’s going to destroy academia and the entire system of gatekeeping via journals and published papers and the tenure system in universities. And too many mathematicians have their entire identity and career tied to the existing system in academia, that’s only existed since the mid 20th century.
July 28, 2026 at 4:45 pm |
It won’t destroy academia or the journal system, unless humans do it. And the point of large language models and automated proof checking is, simply, to avoid human errors (cf. Voevodsky’s concerns from some years ago). It can’t provide the intuition, that is the object of the exercise.
Stam Nicolis, https://www.idpoisson.fr/nicolis
July 28, 2026 at 7:22 pm |
Mathematicians are wasting time doomposting about AI when these things won’t even be around in 5 years when the AI bubble pops and OpenAI and Anthropic go bankrupt.
July 28, 2026 at 11:46 pm |
None of this AI crap matters because all these mathematicians are going to get drafted into Trump’s World War 3 when he successfully manages to expand the Iran War or Ukraine War into Europe.
July 28, 2026 at 11:54 pm
Lol, there are plenty of boomer mathematicians who are at zero risk of being drafted because they are too old, and so can spend all their free time arguing over AI usage in mathematics and cheering on the war while their young counterparts gets sent to the frontline to get killed in a war for Israel.
July 29, 2026 at 12:05 pm |
I share some of your sentiment, but I nevertheless signed the declaration for two reasons. (1) Politics, as in our community should present a unified front. (This sort of politics often stands in the way of true knowledge, but there are times when it’s called for, and this is one of them. (2) Transience: given how quickly things have been changing, I can only assume that a new declaration will be needed in a few years’ time, and the hypothetical scenarios you raise for autonomous AI search will be best addressed if/when they arise. (Maybe we should have a permanent committee residing in Leiden ready to update the declaration with every new release from Anthropic or OpenAI?)
July 29, 2026 at 5:04 pm
Why are people just focusing on Anthropic and OpenAI? Open source and open weights models are soon catching up in strength and will quickly be able to find counterexamples and prove theorems like Claude and ChatGPT right now. There’s just too many open weights models for the Leiden people to update the declaration with every new open weights model.
July 29, 2026 at 4:24 pm |
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July 29, 2026 at 6:22 pm |
You are going to have people refusing to review papers done by LLMs despite the material on the papers being correct, simply because they’re done by LLMs. So mathematics is going to end up with a huge collection of folklore that are correct but unpublished because of a bias against LLMs. This means that mathematical truth is going to be balkanized, there are people who simply refuse to believe in some truths because it comes out of the mouth of a LLM.
You currently have mathematicians who do not believe the weak Goldbach conjecture to be resolved because the proof hasn’t been formally published yet in a journal. LLMs are going to take that phenomenon and turn it to cover the vast majority of new results in mathematics.
End result being that the idea that there is one unified mathematical whole is just dead. The philosophical and sociological consequences of the balkanization of mathematics is enormous and we are just at the beginning of it.
July 30, 2026 at 4:08 am |
“Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics”
This reminds me of Professor Rubenstein’s article https://arielrubinstein.tau.ac.il/papers/ninny.pdf about the meaning of economic theory. If anyone sees this and wants to read this, I would recommend mapping the ‘story’ part of economic theory to something akin to the above quote, and the ‘mathematical rigor’ of the story to be akin to the hypothetical of LLMs being able to create/verify every proof.
July 30, 2026 at 3:12 pm |
The real problem for LLMs in mathematics is when you realize that LLMs are also trained on published false proofs like the numerous published false proofs of the Jacobian conjecture in the literature and on arXiv.
So you simply cannot trust that anything that LLMs put out is true, you have to go and verify it yourself. Otherwise mathematics risks becoming like the Italian school of algebraic geometry.
July 31, 2026 at 12:09 pm |
“Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics”
Says the old fart whose career already got made
July 31, 2026 at 1:07 pm
Careers in mathematics are already getting destroyed thanks to Trump cutting off funding to mathematics, the non-existence of AI isn’t going to change that.
August 14, 2026 at 5:40 am
hahahahah true
July 31, 2026 at 7:49 pm |
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August 1, 2026 at 4:40 pm |
[…] seltsam an und nicht besonders angenehm, so den Boden unter den Füßen weggezogen zu bekommen", schreibt er in seinem Blog. Andererseits sei er froh gewesen, die Probleme gelöst zu […]
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August 2, 2026 at 1:20 am |
[…] an und nicht besonders angenehm, so den Boden unter den Füßen weggezogen zu bekommen”, schreibt er in seinem Blog. Andererseits sei er froh gewesen, die Probleme gelöst zu […]
August 2, 2026 at 9:54 am |
[…] passé un temps considérable. Deux fois, le modèle l’a résolu à la première tentative. Le médaillé Fields écrit sur son blog avoir éprouvé une sensation étrange, pas franchement agréable, comme si on […]
August 3, 2026 at 2:06 am |
I prefer not to give details about my professional background but I am a Silicon Valley insider who has seen several hype bubbles come and go.
I also know well the Silicon Valley playbook of which the hype machine is an intimate part.
I could be 100% wrong, of course, but my take is that we are at peak AI hype, not the beginning of something interesting or new.
The kind of math AI agents seem to be good at is what I would describe as “problem solving/trickster” mathematics. Put it differently, IMO problems on steroids.
So sure, if your main contribution to the field was to excel at solving IMO style problems, very soon it will be harder for you to justify why you should get paid astronomical sums of money for doing something that a computer can do for much less.
Conceptually, and I insist “conceptually”, this is no different from the situation in the 1970s and 1980s when many engineers lost their jobs because they were good at the slide rule, something that became obsolete with scientific computation -first with large computers, then with scientific calculators. Engineering, as a discipline, didn’t end in the 1980s nor will math die because of AI math agents. Human math will refocus its efforts to solving big problems rather than doing low hanging fruit.
For me to be impressed, I would need to see one of the remaining 6 Millennium Prize Problems solved autonomously and with zero, I insist ZERO, human intervention or guidance.
I am not losing my sleep.
Hopefully the legacy of AI killing “trickster mathematics” will be the rise of more mathematicians like Andrew Wiles, Grigori Perelman and Yitang Zhang and the death of the celebrity mathematics style of which Terry Tao is the most prominent example.
Long live mathematics!
August 7, 2026 at 7:18 am
yeah no one like terry tao these days. he has become an openai shill
August 4, 2026 at 1:03 pm |
Long time lurker here. I like to comment on this sentence. “It is extremely satisfying to struggle with a mathematical problem for months or even years and eventually solve it.” Agreed however this sounds a bit like sports. We need to ask ourselves what do we want sports or knowledge. Preferably both, but what good would a cure to cancer be to many patients if came 20 years later because sportsmanship was valued over knowledge.
I do think that modern mathematics is one of humanity´s greatest achievements and I can understand the intellectual challenge and the sportsmanship like competition trying to solve problems, I think it makes for a great discussion when we view pro and contra AI usage arguments like you did which is getting rare in the mathematical community . Thank you for writing so many high quality blog articles.
PS:
LLMs assisted me with spelling, grammar and LaTeX , which enabled me to focus better on developing theorems … when I was writing about QSpire Algebra
I consider this an acceptable usage but I guess others would have a different opinion.
August 4, 2026 at 4:27 pm |
To Christian Oppel. I am the anonymous who is a Silicon Valley insider who posted before you did. I have a comment awaiting moderation because, I assume, it has links documenting what I said, so I will put another comment saying similar things but without the links.
AI won’t cure cancer.
The Nobel awards to Geoffrey Hinton and Demis Hassabis will be seen by future historians as a travesty, probably the product of Google’s hype machine.
Take AlphaFold for example. I encourage you, and others, to watch any of the talks by Jennifer Listgarten, a UC Berkeley professor who works on protein engineering, available on youtube abput her work on using machine learning to tackle the problem. Most of her talks have few views, in some cases a few hundred.
She explains that the problem that AlphaFold has solved -not 100% but with very high probability- is predicting, in a statistical sense, the protein folding structure from an amino acid sequence however that the hard, unsolved problem, in protein design is the opposite: given a desired folding structure, coming up with the amino acid sequence that would produce said structure.
The second problem is NP hard but having a tractable solution to it, even in a subset of cases, is what would revolutionize protein design. AlphaFold doesn’t take us anywhere close to solving it.
In addition, assuming that the second problem is solved for a meaningful subset of cases, you still have the issue of clinical trials on humans. Human biology being what it is, there is no workaround to the issue of doing clinical trials to understand the long term effects of any treatment.
We saw with the mRNA covid vaccines that the promising results seen in the early trials didn’t translate to the population at large. These vaccines didn’t prevent infection nor transmission for a number of reasons, including the mutations of the virus. Their real value was to lower the probability of death of the disease, particularly in the elderly or people with compromised immune systems. If you were a healthy individual below 60, you would have been better off getting covid and developing natural immunity. And if you were a young male (a teen or someone in his early 20s), you should have never taken the mRNA vaccine because the risk of developing myocarditis as a result of getting the shots would overweight any small benefit you would get from the vaccine.
To summarize.: don’t believe the Silicon Valley hype. Silicon Valley has a long history of using clever and sophisticated marketing to make money out of technology. You should google the ads by Bob Widlar promoting analog integrated circuits against digital circuits. You will see the same spirit OpenAI, Google, Anthropic and the others are using these days to promote their companies.
As I said in my original comment: long live mathematics!
August 4, 2026 at 5:38 pm |
Dear Timothy, thanks for taking the time to think and write about this.
It seems that you come to realize how paradigm-shifting AI can be to Mathematics, but I feel lik you are missing the bigger picture:
– what you describe regarding the loss of expertise will not only impact mathematics, but literally every other aspect of all of our societies;
– the idea (in essence) of steering academic work from practicing Mathematics to teaching them is moot. Indeed, AI itself may very well be better at teaching/writing books than all of us combined the by the time it is strong enough to take over the art of generating/investigating ideas better than us.
I cherish the fact that some (not all…) world-class mathematicians like yourself are collectively slowly realizing that unhinged AI development cannot come without huge repercussions, and I hope it will bring you to consider taking a stand against the global race for evermore powerful AI systems.
I strongly suggest to sign and share public calls for pause on AI development, until research shows it comes with ethical and security guarantees. For example this one from 👉 PauseAI 👈
It goes much beyond Mathematics… Our civilization is at play, not only the mathematicians’ ecosystem.
All the best!
August 5, 2026 at 4:58 pm
Of course civilization isn’t in any risk, if mathematicians use AI to solve problems. Because mathematics isn’t about, simply, obtaining the answer to some question; it’s about understanding the meaning of a question, its place in the scheme of things. The specific technical tools are of secondary importance, because there are can be more than one ways of understanding it. That’s why I don’t see the point of declarations about using or not using AI or anything else. A mathematical paper must explain how the problem is addressed and what its putative solution means, how it’s obtained is a means to an end. Whether AI is or isn’t used shouldn’t be of any particular interest, any more than a declaration about the size of numbers that were multiplied could be.
Stam Nicolis, https://www.idpoisson.fr/nicolis
August 5, 2026 at 3:48 pm |
For some reason it’s become fashionable to be overly impressed by AI developments and this has led to claims that these will render/have rendered humans redundant, in particular in mathematics. For mathematics I’d like to point out that calculators are routinely used to perform numerical calculations, computers are used to perform more complex calculations, without mathematics having broken down, on the contrary. For those that bemoan some mythical age when human intuition was the arbiter of mathematical proof, I’d like to recall the remarks of Vladimir Voevodsky that led him to develop his Univalent Foundations project. People can make many more mistakes than is commonly acknowledged, so understanding how to formalize proofs can help. It can’t replace human understanding, however.
The debate about Mochizuki’s work on the abc conjecture is nothing more or less than an illustration of the sociology of the field, as it expands. I doubt it’s the only example of an incomplete proof that has been published and it won’t be the last.
August 6, 2026 at 12:16 am |
[…] mathematician Timothy Gowers wrote in a recent blog post, “we might arrive at a situation where the mathematical literature has, in some form, been vastly […]
August 6, 2026 at 6:12 pm |
[…] matemáticas de Babel. Uno de los matemáticos entrevistados, Timothy Gowers, indicaba que esta trayectoria puede hacer que en el futuro dispongamos de una literatura matemática enorme, […]
August 6, 2026 at 10:49 pm |
[…] presented at the International Congress of Mathematicians in July 2026. In a July 26, 2026 post on his weblog, Gowers explained that he did not sign, not because he disagreed in any concrete way, but because […]
August 10, 2026 at 5:50 pm |
[…] starve.Timothy Gower, an accomplished (Fields Medal) mathematician, recently stated a version of this argument for mathematics that I found compelling and quote below:“But I worry about that argument, because it seems to […]
August 11, 2026 at 5:33 pm |
[…] remains to be break up on how mathematicians ought to method the brand new analysis methods. In a weblog put up responding to the declaration, Fields Medal winner Timothy Gowers questioned whether or not the affect of AI may change […]
August 11, 2026 at 8:07 pm |
[…] “Se chegarmos a um mundo em que os teoremas matemáticos não estejam mais associados aos matemáticos, talvez isso não seja mais problemático do que o fato de as estrelas não receberem nomes de astrônomos e de a maioria nem sequer ter nome”, escreveu Gowers em seu blog. […]
August 11, 2026 at 11:45 pm |
[…] nevertheless, persists. In response on his blog, Fields Medalist Timothy Gowers offered a more optimistic view, suggesting AI’s accelerating […]
August 12, 2026 at 8:17 am |
This may work for senior people, but young humans do not know their mathematical taste in a way they can express. We very likely find this taste by being exposed to the “well-written textbooks” in the mathematical cultures of today, and when doing so, we start to take part in those cultures as a result. If we don’t have that, then the companies largely get to decide on what taste means, and if this is motivated by profit, then one can look at the recent history of any media made into personalised recommendations to see if that’s desirable. For young people privileged enough to be taught traditional mathematical culture from caretakers (and I must stress, this is a lot of privilege), this is likely not a problem, but as we see from today’s media, unprivileged youth becomes even more vulnerable to being set astray from a healthy mathematical community that would develop their talents and help them.
August 12, 2026 at 8:58 am
And of course, one could ask what “desirable” and “valuable” means. The point is that those are communitary constructs meant to ensure the existence and happiness of a community, and therefore the mathematical community may want to preserve itself by imposing them in some way. But that entails the existance of such shared values and that democratic actions can be taken, which I think is not quite the reality in this case.
August 14, 2026 at 4:03 pm |
[…] article de Timothy Gowers est l’un des plus éclairants que j’ai lu sur ce sujet (et ça date d’avant la […]
August 15, 2026 at 4:31 pm |
[…] and that AI-assisted papers make reviewing more demanding. Timothy Gowers, writing about the declaration, pushes the thought further: he imagines mathematicians selecting from a vast body of AI-generated […]
August 15, 2026 at 9:50 pm |
[…] and that AI-assisted papers make reviewing extra demanding. Timothy Gowers, writing about the declaration, pushes the thought additional: he imagines mathematicians deciding on from an unlimited physique […]
August 15, 2026 at 10:28 pm |
[…] shows, and that AI-assisted papers make reviewing extra demanding. Timothy Gowers, writing about the declaration, pushes the thought additional: he imagines mathematicians deciding on from an enormous physique of […]
August 16, 2026 at 6:06 am |
[…] shows, and that AI-assisted papers make reviewing extra demanding. Timothy Gowers, writing about the declaration, pushes the thought additional: he imagines mathematicians choosing from an enormous physique of […]
August 22, 2026 at 3:25 pm |
I find the Leiden declaration complete nonsense, so I am very happy somebody takes a critical stance. My reasoning is the following: