> ## Content Index > Fetch the complete content index at: https://www.slightlyacidic.com/llms.txt > Use this file to discover other available public pages before exploring further. # Mulling AI and maths - URL: https://www.slightlyacidic.com/2026/10/09/mulling-ai-and-maths/ - Published: 2026-10-09T15:17:02.000Z - Updated: 2026-10-09T16:53:49.000Z - Author: VM - Tags: artificial intelligence, mathematics, mathematics research, Martin Hairer, AGMAI, Ai alignment, OpenAI, Navier-Stokes equations, Anthropic, Leiden Declaration, Erdos problems, Caltech Mathathon, frontier models, Association for Human Mathematics, Hugo Duminil-Copin, Joseph Tooby-Smith, meritocracy, Severe Misalignment, Fields Medal, Siddhartha Gadgil, Vamsi Pritham Pingali, natural science, Thore Graepel, AI safety, reasoning, David Robinson Some AI + mathematics things I read recently and am mulling below. I've been struggling to pull these and various threads on the topic into a cogent whole, even if only for myself, because the developments have been coming fast and thick, with no indications they will slow in the near future. There are also discussions starting in physics, for example see [here](https://4gravitons.com/2026/09/25/it-got-to-my-field/?ref=slightlyacidic.com), [here](https://proofsandprompts.com/2026/10/05/ai-and-the-mismeasure-of-science/?ref=slightlyacidic.com), and [here](https://www.nature.com/articles/s41586-026-10898-6?ref=slightlyacidic.com). I appreciated Rohini Subrahmanyam's deeply reported [feature for *The Hindu*](https://www.thehindu.com/sci-tech/science/as-ai-marches-on-math-research-scholars-ponder-the-future/article71463202.ece?ref=slightlyacidic.com) (paywalled), published on September 21, on how mathematicians are feeling and thinking about AI models' assault on open math problems because it offered something of a checkpoint. But a lot of things have happened since, including a tremendous amount in the last week alone. (What follows is a collection of posts, articles, etc. on my mind rather than something that will help you catch up.) --- ### Guiding AI companies > Martin Hairer, one of the other members of AGMAI (the advisory group on mathematics and AI), has written a post on the Proofs and Prompts blog that will I hope clarify what our purpose is. The rest of the blog is highly recommended too.[https://t.co/mhT7jaS82i](https://t.co/mhT7jaS82i?ref=slightlyacidic.com) > > — Timothy Gowers @wtgowers (@wtgowers) [September 22, 2026](https://x.com/wtgowers/status/2102510624219533693?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) From [‘Why I agreed to join AGMAI’](https://proofsandprompts.com/2026/09/22/why-i-agreed-to-join-agmai/?ref=slightlyacidic.com) by Martin Hairer, *Proofs and Prompts*, September 22, 2026: > In the ‘Severe Misalignment’ statement, which has now been endorsed by nearly 8000 mathematicians, we conclude by saying that “These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies, \[…\]”. It would be rather hypocritical of myself to then chicken out at the first opportunity of actually having a chance of setting up some form of communications channel between the mathematical community and the AI labs just because I could get some flack for it. > > Regarding the actual advice we intend to provide, it is a fact that AI companies have in recent months been producing some high profile mathematical results and that their publication and dissemination has been falling far short of acceptable mathematical practice, whichever way that is defined. However, the situation at hand is sufficiently unprecedented that we genuinely haven’t completely made up our minds yet on the best way forward and we fully intend to gather as much feedback as possible, be it through [our feedback form](https://agmai.org/input/?ref=slightlyacidic.com), discussions with colleagues, the discussions on this blog, etc. However, OpenAI doesn’t seem to have learnt anything from l’affair Navier-Stokes or what mathematicians at the front of their fields have been saying about how they would like AI companies to approach solving important mathematical problems. From [‘OpenAI Is Pissing Off a Bunch of Mathematicians—Again’](https://www.wired.com/story/openai-is-pissing-off-a-bunch-of-mathematicians-again/?ref=slightlyacidic.com) by Isabella Ward, *Wired*, October 6, 2026: > OpenAI assembled an [advisory group](https://openai.com/index/advisory-group-on-mathematics-and-ai/?ref=slightlyacidic.com) in mid September to help determine how the company assesses and communicates new results, upholds academic and professional standards, and builds tools that support mathematical research and learning. But the company doesn’t seem to have made much progress, mathematicians say. > > Many are frustrated that OpenAI and Anthropic continue to release results through blog posts, rather than scientific papers, which makes them harder to verify and often excludes prior work from other mathematicians. Piecemeal announcements aren’t limited to the labs: \[Levent\] Alpöge himself announced he had disproved an 87-year-old conjecture through a [tweet](https://x.com/%5F%5Falpoge%5F%5F/status/2079028340955197566?lang=en&ref=slightlyacidic.com) sent shortly after the World Cup final. > > Some have set up new tools this year in reaction to the increase in machine-assisted proofs; the idea is to help the community both use new results and sort through what needs to be further digested. … While OpenAI has been directly encouraged to use these tools, “they haven’t changed their behavior,” Kra notes. Nor does she feel the company’s actions are aligned with [the Leiden declaration](https://leidendeclaration.ai/?ref=slightlyacidic.com), a call by more than 4,000 mathematicians for AI companies to meet mathematicians’ standards that she helped write. > Would it not have been helpful for mathematicians and scientists if OpenAI's math drop came with better documentation about the process. What prompts were used, what checks done? I am a little surprised that the advisory council did not advise this. > > Thoughts? > > — Nirmalya Kajuri (@Kaju\_Nut) [October 8, 2026](https://x.com/Kaju%5FNut/status/2108234451646103723?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) --- ### Questionable flex Publishing results formatted one way is one thing; attempting them at all is something else. > The scale of this is staggering. > > But... Why? > > Why is OpenAI trying to solve hundreds of math problems to be released all at once? I think we got the point when they found a counterexample to Navier-Stokes -- internal model = great. > > I am not sure at this point what this… [https://t.co/eOzpu2z6kQ](https://t.co/eOzpu2z6kQ?ref=slightlyacidic.com) > > — Alvaro Lozano-Robledo (@mathandcobb) [October 7, 2026](https://x.com/mathandcobb/status/2107643298873569692?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) From ['Erdős problems in the OpenAI proofs'](https://www.erdosproblems.com/forum/thread/blog:10?ref=slightlyacidic.com) by Thomas Bloom, *Erd*ő*s Problems*, October 8, 2026: > One thing to stress is that none of these should be viewed as 'closing the problem', in that people do not need to work on it anymore. There is a huge amount to be done: to actually read these proofs, verify them, simplify them, understand them, explain them, work out what else those ideas can be used for, and so on. > > Furthermore, many may have other solutions available along lines completely different from what the AI explored, and to focus entirely on the AI proofs risks missing an alternative much more 'human' and comprehensible proof that may exist. > > Mathematics is a big tent, and there is room for many different types of mathematician. Everyone should find a way of doing mathematics that they personally think is meaningful and useful. But this may be downplaying a problem that "a coalition of current and former Caltech mathematicians" flagged in [that statement](https://proofsandprompts.com/2026/09/10/open-letter-about-the-mathathon/?ref=slightlyacidic.com) opposed to the 'Caltech Mathathon' competition: > Participants are not required by the rules of the competition to communicate their results on ArXiv after the event is over. It will again fall to the mathematical community to carefully verify hastily-produced AI-generated results. What OpenAI is trying to achieve here is not clear, beyond the value of ["flexing"](https://x.com/Ananyo/status/2107768856068370555?ref=slightlyacidic.com) their frontier models' abilities ahead of what are widely expected will be blockbuster IPOs. --- ### What is lost The cost may be too high for mathematics in particular. As mathematician Yiling Wang [wrote](https://proofsandprompts.com/2026/09/10/a-response-from-mathathon/comment-page-1/?ref=slightlyacidic.com#comment-335) in response to a [statement by the Caltech Mathathon organisers](https://proofsandprompts.com/2026/09/10/a-response-from-mathathon/?ref=slightlyacidic.com) (who were themselves responding to the coalition's statement), mathematics has certain precious resources and AI companies and their frontier models are blatantly draining them without doing anything in return. > Let me explain one point, that is implicit from the open letter but may not be obvious to younger people: A good open question is very precious. I am not speaking about big famous problem like Navier Stokes, but perhaps the ones you may find for your Mathathon. > > When you do research (at least in the traditional way) you will realize that, most of the questions that one comes up would either be trivial, or not that interesting, or way too hard. Finding an interesting problem that many people would also care about, not just incremental nor completely unreachable, needs a lot of vision and insight. It does not happen every day. Finding them is already a big step in the research. People who pose it, cherish it as well. > > We share it openly and label it as an open question, is a nice gift to the community. We also hope that people who try it, also learn about where it comes from, which then make them learn the subject and the field that we appreciate and dedicate so much of our energy to. Even if they don’t succeed, the knowledge is transmitted to them. Moreover, they may come up with other insights and many other people will still try it. The solution itself is nice, but the attempts, ideas people developed around it, the collaborations it generated in an open, respectful, and fair environment, are really something we treasure and which eventually make the field flourish. > > Sometimes you also see counterproductive behavior, such as people making some progress on an open problem, but would not jump immediately to solve it, but check with the person who posed it or other people in the field. If we notice competition with other group, we may try to collaborate instead of compete. It is not really needed, and does not always happen like that, but many of us do, just to keep the community healthy and open. > > Now seeing AI brute force a solution with immense compute, skipping the transmission chain, the discussions, the attempts, and spoiling the goodwill we dedicated to the community with the openness of sharing our questions, is hard to take. From ['AHM Statement on OpenAI’s October 6 Release of Mathematical Documents'](https://www.ahmath.org/statements?ref=slightlyacidic.com) on the 'Association for Human Mathematics' website: > Yesterday, on October 6th, 2026, OpenAI … 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. ([Caveat](https://x.com/Ananyo/status/2108529252878741671?ref=slightlyacidic.com)) For another, it is leaving accomplished mathematicians adrift vis-à-vis their professional programmes, with the line about "the community’s request not to destroy the open conjectures that orient research" being "absolutely ignored" reminiscent of what Wang had warned about: > SHOCK AND AWE: AI MATHEMATICS > > Fields Medalist: the release seemed intended to show the systems could “massacre us in the deepest sense of the term.” > > Here are other striking remarks from Hugo Duminil-Copin’s 8 October 2026 seminar talk “Un grave défaut d’alignement de l’IA en… [https://t.co/9Dtq2XVooX](https://t.co/9Dtq2XVooX?ref=slightlyacidic.com) > > — steve hsu (@hsu\_steve) [October 9, 2026](https://x.com/hsu%5Fsteve/status/2108364887605018884?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) --- ### Differing views But this 'march on mathematics', so to speak, does not mean there is no disagreement. For instance, University of Bath computer science lecturer Joseph Tooby-Smith [wrote in a comment](https://proofsandprompts.com/2026/09/22/why-i-agreed-to-join-agmai/comment-page-1/?ref=slightlyacidic.com#comment-576) under Hairer's post on *Proofs and Prompts*: > Mathematics is a team sport. It is a community, and any group that has assembled to speak on behalf of that community needs to pass a simple test. Can the average member of the community see themselves represented by that group? > > This group has been put together to contain the very best within mathematics, and I commend each of those nine people for taking up this baton and for doing something which will no doubt be very difficult. But, this group of nine people, as you yourself say, Martin, does not represent the average mathematician in the community. > > The group contains no PhD students, no postdocs currently struggling to find a job, no one from a teaching-heavy department, no one on the academic job market, no one actually at the stage of their career where the issues raised by these AI companies have substantial effects on livelihoods and putting food on the table. It doesn’t contain the voice of the ordinary mathematician, in an ordinary department and in an ordinary university. > > A distinguished career is a poor vantage point to judge how the shift that we are going through right now is going to affect the average member of the community. While the nine-member group that includes Hairer is not responsible for managing the effects of AI-in-mathematics on students, his comment raises a good point vis-à-vis any other group grappling with that problem, and which the Caltech Mathathon itself stood for, together with the statements not opposed to what it was attempting. Quotable passages from two such statements: From ['Two responses to “A Severe Misalignment of AI in Mathematics”'](https://proofsandprompts.com/2026/09/18/two-responses-to-a-severe-misalignment-of-ai-in-mathematics/?ref=slightlyacidic.com): > The third claim of the \[['Severe Misalignment' statement](https://mathandai.org/?ref=slightlyacidic.com) signed by multiple Fields Medallists\] declaration is that the mathematical community urgently needs to address these problems. > > Yes. But who is “the mathematical community”? > > This sentence bothered me more than it probably should. The original signatories are twenty-five Fields Medalists. They include some of the most respected and institutionally influential mathematicians. These are not powerless observers, but the people who are in position to influence the norms and values of the community. and from ['Could AI Help Mathematics Escape the Meritocracy Trap?'](https://proofsandprompts.com/2026/10/09/could-ai-help-mathematics-escape-the-meritocracy-trap/?ref=slightlyacidic.com): > AI will not supply research funding, job security, or the judgment of an excellent mentor. But it may make some forms of intellectual support less dependent on admission to a privileged academic environment. That is the possibility I want us to take seriously. The wider global mathematicians' community is also not a monolith: > "Those who find ugly meanings in beautiful things are corrupt without being charming. This is a fault" - Oscar Wilde [https://t.co/4RpjegJseG](https://t.co/4RpjegJseG?ref=slightlyacidic.com) > > — Siddhartha Gadgil (@SidGadgil) [October 8, 2026](https://x.com/SidGadgil/status/2108208239582359885?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) In a [March 1, 2026, blog post](https://siddhartha-gadgil.github.io/automating-mathematics/posts/ai-and-the-pointy-haired-boss/?ref=slightlyacidic.com) Gadgil, who is a professor of mathematics at the Indian Institute of Science IISc), Bengaluru, had also articulated, at least by implication, the importance of distinguishing between 'most mathematicians' and 'top-level mathematicians': > Those who will ever make major breakthroughs number at most in the hundreds or thousands, and these people should obviously acquire the knowledge and skills they need for this. This post is to address the question for the rest of us, and will take as a premise that my estimate is roughly correct. > > … > > This post has been written more in the spirit of what can be and maybe what should be. From the point of view of a young person today, I hope this is an argument why they should take the trouble to develop their knowledge. But practically there is a more important consideration - while cognitive skills of AI have reached human expert levels in many cases already, our hands are far better at dealing with the physical world than those of robots. So learning something to do with the physical world - biology, chemistry, mechanical engineering, nursing, plumbing, carpentry - is much safer than a purely white collar skill. On May 9, 2026, Vamsi Pritham Pingali, another mathematics professor at IISc, had spelled out a more germane but also somewhat defeatist viewpoint in a blog post titled ['Thoughts on AI and its impact on maths'](https://math.iisc.ac.in/~vamsipingali/ThoughtsonAI.html?ref=slightlyacidic.com) (this excerpt omits all the parenthesised bits): > … many seem to believe that AGI \[artificial general intelligence\] is on the horizon. So let's assume AGI arrives. If any old noob with a computer can solve deep conjectures in maths, then the national pride argument \[for the state to fund mathematics research\] goes away. If computers can win fields medals, the signaling argument goes away. Students can learn from AI just as well as they can from a generic teacher. > > Now the argument that remains is unpredictable applications. Honestly, mathematicians must be more honest and try to apply advanced maths to the real world. Hitherto, the bottleneck was lack of knowledge of the real world. However, with the help of AI, even esoteric concepts like say Hermitian-Einstein metrics and Kac-Moody algebras can find applications. For fun, I asked ChatGPT 5.4 to see if AI safety can be helped by Hilbert-Mumford stability. It came up with some interesting AI safety paradigm. It is time for us pure mathematicians to slowly own AGI and use advanced maths in the real world. > > That being said, if one receives a terminal cancer diagnosis, one cannot be expected to go to the gym or file taxes instead of say watching Lord of the Rings and eating pizza. So if we are going to be replaced by AI soon, then for now why not do something we enjoy like pure maths? This is also valid. The golden age of humans using AI as a great tool is here. Not that anyone cares but I can't say I'm in line with these views, at least insofar as determining how the mathematics community as a whole should countenance AI companies' increasingly nihilistic approach to mathematics research. That said, I find Gadgil's and Pingali's views very interesting because they seem to be informed by a particularly Indian experience of doing mathematics. In an older post on January 12, 2026, titled ['Mathematics as Natural Science in the Era of AI'](https://siddhartha-gadgil.github.io/automating-mathematics/posts/mathematics-as-natural-science-in-ai-era/?ref=slightlyacidic.com), Gadgil had also written: > To me mathematics has always been part of natural science, both during the many years when I was primarily a mathematician, and in the last few years when I have focussed more on automating mathematics. By this I don’t mean that the goal of mathematics is only to answer questions in say physics or biology. Rather, natural science is defined by theories that are (ultimately) empirically validated, and mathematics can be viewed in the same way. Over the years, my understanding of how to view mathematics within natural science has evolved (including, somewhat surprisingly, from my years immersed in software development). > > This is by no means the unique view of mathematics and may not even be the most common one among mathematicians. An alternative view is one that can be roughly considered the cultural view of mathematics. Perhaps the most common view in practice \[of what mathematics is\] is a combination of the natural science and cultural views. While one’s view has not mattered much yet, it does so when adapting to a world with powerful AI. For those who take a natural science view the coming era of AI coupled with formal verification with the Lean Prover (or similar systems) is a great opportunity, though a lot of work will be needed to make the most of this. Others may view AI as a peril. > > … > > \[A\]s we enter the era where proofs are generated by AI (with Lean), it has become common to claim that the role of mathematics is human understanding, rather than advancing scientific knowledge. Indeed this is often stated as if it is an axiom or law of nature, or at least that everyone agrees on this – the former is absurd and, as for the latter, I at least disagree. As I said this is a subjective opinion and not something to be debated. I will instead continue discussing mathematics as a natural science… As Pingali said, 'flexing' on the ability to tackle real-world problems would also be better ahead of an IPO: > Sure. Now deploy your resources to applying say complex differential geometry to make progress on cancer or something. That’s better for IPO and for mathematicians (who will see a new direction given proof of concept)., [https://t.co/pLiiCOgBxX](https://t.co/pLiiCOgBxX?ref=slightlyacidic.com) > > — Vamsi Pingali (@vamsiprithamp) [September 22, 2026](https://x.com/vamsiprithamp/status/2102252517338067095?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) That said, my feeds and reading habits are exposing me predominantly to comments on this issue arising from the Global North. I haven't been seeing or reading anything much penned by mathematicians in Latin America, Africa, and South Asia. I intend to remedy that. --- ### AI and safety I also read two articles recently that went beyond the sensational, and sensationalised, statements by Jacob Coxon *et al*. about how AI companies are playing fast and loose with the demands of safety and alignment vis-à-vis frontier models — a concern not directly related to AI companies' "hostile takeover" of mathematics but which speak to the companies' attitudes towards existing guardrails and their inability, or unwillingness, to differentiate between guardrails that ought to be broken and those that shouldn't. First, from ['Don’t be fooled—LLMs don’t reason'](https://www.technologyreview.com/2026/10/02/1145639/dont-be-fooled-llms-dont-reason/?ref=slightlyacidic.com) by Thore Graepel (who recently left Google DeepMind), *MIT Tech Review*, October 2, 2026: > Three shortcomings prevent what chatbots do from qualifying as reasoning (in a way that a scientist might recognize). First, these models typically maintain no explicit, persistent, and inspectable epistemic state. There is no open ledger that lays out the hypotheses a model is considering, the confidence it has in various explanations, the evidence it’s weighing, and the unresolved questions it’s holding onto—all things that should be systematically revised as new information arrives. > > Second, they lack a clean separation between what the system knows and how it manipulates that knowledge. Knowledge and reasoning are inextricably interwoven in the weights of the neural network—there is no independent, explicitly represented set of beliefs. Third, while the chains of thought chatbots produce look like deliberation, research has demonstrated that the bots often [concoct](https://arxiv.org/abs/2608.09867?ref=slightlyacidic.com) [them](https://arxiv.org/abs/2510.27338?ref=slightlyacidic.com) after the fact, reaching an answer by one route but reporting another. > > This is a problem because in the high-stakes applications we all care about, such as medicine, engineering, and scientific research, it matters not only what a system concludes but also how it arrives at its conclusion. When mistakes happen—for example, in medical diagnosis and treatment—we need to be able to pinpoint what went wrong: Was the system’s reasoning at fault, did it draw on invalid evidence, or did it make incorrect assumptions? > > … > > I do not think we reach trustworthy machine intelligence by making system 1 bigger. Second, from [‘I Quit OpenAI Because Its Culture Is Broken’](https://www.theatlantic.com/technology/2026/10/openai-safety-team-resignation/688881/?ref=slightlyacidic.com) by David Robinson (who recently left OpenAI), *The Atlantic*, October 3, 2026: > Given today’s risks, frontier labs need to run like nuclear-power plants or busy airports, with layers of redundancy and careful, time-consuming planning, so that the occasional and inevitable human error does not open a door to disaster. Right now, AI companies don’t know how—but other people do. In a nuclear-power plant, the technical systems and rules people follow are set up so that, if equipment breaks or someone pushes the wrong button, we still won’t risk a meltdown. OpenAI and other labs are growing and deploying frontier AI with far less redundancy and rigor than this, even though the harm from an irreversible loss of control would be much greater than the harm from any single meltdown. Even short of a full loss of control, we could see autonomous swarms of AI agents that act without human permission. Imagine “rogue” agents that work like teams of hackers (for example, holding hospital computer systems for ransom) but never need to sleep. > > After three and a half years at OpenAI, I was among the longest-tenured employees at the company. I led the drafting of our current Preparedness Framework, and oversaw the writing of safety reports on 12 frontier launches. But as far as I know, I never encountered a colleague who had experience making airplanes fly safely or nuclear reactors run without melting down, or helping the financial system grow without collapsing. To be clear, this is a new need: Today and tomorrow’s AI systems are far more capable and dangerous than the systems we were building even six months ago. Perhaps I should have stayed and fought for fundamental shifts in our staffing and culture, but in practice, my colleagues and I were so busy sprinting that we seldom had the chance to consider big changes, much less to actually make them. That’s why I concluded that stronger incentives for safety—coming from outside the company—are a big part of getting this right. Speaking of OpenAI and its culture: > Last week I was called into a meeting with OpenAI’s head of safety and told they no longer trust me. A security guard took my badge and walked me out of the building. Then I learned my colleagues [@balesni](https://x.com/balesni?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) and [@j\_asminewang](https://x.com/j%5Fasminewang?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com) had been fired too. Why did OpenAI suddenly stop trusting… [https://t.co/DttJjXKGcp](https://t.co/DttJjXKGcp?ref=slightlyacidic.com) > > — Tomek Korbak (@tomekkorbak) [October 8, 2026](https://x.com/tomekkorbak/status/2108266859397283953?ref%5Fsrc=twsrc%5Etfw&ref=slightlyacidic.com)