[This is a guest post by Francesco Fournier-Facio, Yara Kyrychenko, Jon Roozenbeek, and Julian Wykowski, cross-posted from the Proofs and Prompts blog. This blog post was initially written in a different file format and converted using AI. — T.]

Today we launch Futures of Maths, a new, living platform that asks active research mathematicians about the most pressing questions around maths and AI. A string of solutions to famous maths problems, found with the help of AI, has provoked a great deal of discussion among research mathematicians. We built Futures of Maths to understand how mathematicians are dealing with this change in their research and teaching, and where they see the field heading.

How it works. The platform shows you short statements about maths and AI. You rate each one from “strongly disagree” to “strongly agree”, or skip it, and you can mark the ones you find exceptionally important. Once you have answered, you see how everyone has rated the statement so far. You don’t have to finish in one go: you can leave and come back at any time. Most importantly, you can post your own statements. A moderator checks each one, and then everyone else can rate it, so the set of statements keeps evolving with the community’s input.

Results. Once we have collected enough responses, we will publish the results and make them available to the community. We will post again on Proofs and Prompts when this happens.

Who can participate. You can take part if you are 18 or older and have at least one article published in the last 5 years in a mathematics journal, meaning one indexed by zbMATH Open (recently accepted articles also count). The quickest way in is to sign in with ORCID. The article must be on your public ORCID record, added by Crossref. Articles you enter yourself don’t count. If it isn’t there, accept Crossref’s permission request in your ORCID inbox and make the work public. If that doesn’t work, for example because your article is accepted but not yet published, you can request a manual review from your institutional email address. We review these case by case.
We appreciate that this is only an approximation of what an active research mathematician is, but we chose it for its simplicity, objectivity and efficient verifiability. We welcome proposals of alternative criteria that cover a more representative sample, and have these same three qualities, they could be implemented in future iterations.

Privacy. The survey is anonymous. Verification happens on a separate website, and the survey itself never learns your name, email address or ORCID iD. When you first register, you receive a personal link that lets you return as often as you like as the consultation evolves. Please save this link. We can’t look it up or resend it, because that is what keeps you anonymous. Questions about yourself are optional, under “My session”, and you can withdraw your answers at any time.

Inviting wide participation. The results will only be meaningful if a large and diverse group of people takes part, both proposing statements and rating them. Please share this widely, and especially with people whose point of view you feel is underrepresented in the discussion on maths and AI.

If you have any suggestions or feature requests, please leave them in the comments!

I am continuing in this current era to open up my blog to a broader range of voices on math and AI, with the majority of posts now being guest posts by other authors. My own views on these topics (which often overlap with, but are not identical to, those of the guest authors) are summarized on this page. I have also given a number of recent interviews and talks on these topics; I am listing some of these here for reference.

• “How maths underpins almost every aspect of our lives“, Instant Genius podcast, BBC Science Focus, October 18, 2026.
• “Terence Tao & Yu Deng: Why We Build Open Math Model“, SAIR Foundation, October 19, 2026.
• James Tapper, “Terence Tao: AI race turns beautiful mathematics into ‘toxic waste’“, The Observer, October 23, 2026.
• “Can maths survive AI? Fields medallist Terence Tao calls for a ‘Maths 2.0’ world“, World View op-ed, Nature, October 22, 2026.
• “Terence Tao on how AI math solutions miss the point“, interview with Flora Lichtman, Science Friday, October 28, 2026. [Youtube version]
• “Inside AI’s Brutal Race to Solve a Famous Math Problem“, New Scientist, October 28, 2026.
• Rivka Galchen, “The Year Math Changed Forever“, The New Yorker, November 1, 2026.

Disclosure: while many of these articles were solicited due to current events, some are also related to a long-planned launch of a book I wrote, to be released Nov 10.

[This is a guest post by the AHM communications group. — T.]

We are happy to announce that the Teaching Working Group of the Association for Human Mathematics wrote a set of recommendations on doctoral advising aimed at departments, advisors, and students. The document contains five guidelines centered on prioritizing human-authored writing, avoiding overreliance on AI, confronting the ethical concerns of AI use, and improving the wellbeing of students, as well as concrete proposals on how to implement them. We encourage everyone to read and share the document, and we especially look forward to feedback and suggestions.

[This is a guest post by Karthik C. S., Amit Sahai, and Jie Xue. This blog post was initially written in a different file format and converted using AI. — T.]

This is a good time for mathematicians to think carefully about the purpose of their work. The development of artificial intelligence brings that question before us with unusual force. Much of the discussion concerns which tasks machines will perform and which abilities will remain distinctively human. But our response also depends on a prior question: what do we understand mathematical work to be, and what gives it its worth? If our answer has become inseparable from the production of results and the demonstration of individual ability, the present uncertainty may reveal a difficulty in our conception of the work itself.

There are broadly two perspectives on mathematics. According to one, we invent it as we construct it: we introduce concepts, choose axioms, and develop theories. According to the other, mathematics exists independently of these activities, and our task is to discover it. These perspectives need not disagree about every aspect of mathematical practice. We plainly develop formalism and arguments. The deeper question is whether the truths we come to understand owe their existence to these acts, or whether our constructions enable us to recognize something that does not depend on us.

While discovery of mathematical truth can sound passive, a paleontologist searching for fossils must decide where to excavate, develop techniques, and recognize the significance of incomplete remains. The fossils existed long before the investigation, but finding them may require extraordinary imagination. Their prior existence gives direction to the investigator’s activity. Similarly, a mathematical discovery may require the invention of an entire language and framework through which a previously unnoticed relationship becomes intelligible. The discovery may require great creativity, but that does not make the truth discovered a creation of the individual. We encounter this distinction whenever our constructions surprise us. A definition introduced for one purpose reveals an unexpected connection elsewhere. A conjecture that seemed to organize our examples admits a counterexample. Two theories developed through different questions turn out to express a common structure. In each case, our thought encounters something it did not freely determine. We can revise our definitions or abandon our conjecture, but we cannot make the original argument valid merely by wishing it so.

The Platonic perspective gives a natural account of this independence. On this view, there were infinitely many primes before anyone proved that fact. Mathematical truth neither waits for our recognition nor receives its authority from it. We may choose how to describe a structure, but we do not choose its properties in the same way. This perspective also changes the place of the mathematician. Our intelligence becomes a means through which an order independent of us can be understood. The significance of the work lies in its relation to that order.

The distinction between a correct derivation and an understanding of its significance is equally important. An axiomatic system permits indefinitely many deductions, but the ability to produce them does not by itself supply a worthy direction of inquiry. Mathematical statements may be formally correct without being enlightening. We can introduce countless definitions and prove countless statements about them. What makes one definition illuminating? Why does one theorem disclose something fundamental while another merely adds to the available material? These questions concern the relation of a particular investigation to the principles it brings into view. Mathematical taste has a place here, provided we understand it as more than individual preference. We value a theory when it reveals a unity among things previously understood separately, or enables us to recognize why a phenomenon occurs. Although mathematical truths may exist independently of us, their discovery can transform our understanding and enrich our experience of the world. The ability to carry out longer chains of reasoning does not by itself explain such understanding. Reasoning serves understanding by articulating relationships and making their justification explicit. However, the success of reasoning does not mean that all knowledge can be reduced to it. Nor does our ability to reason more extensively determine what is worth knowing or why we seek knowledge.

Mathematics is only one field of knowledge. Not everything can be understood mathematically, nor is mathematical achievement the ultimate purpose of intellectual life. Yet we increasingly measure intellectual life through publications, citations, awards, and other indicators of performance. While useful, these measures become problematic when they define the worth of the work itself. We begin to see mathematicians as collections of productive abilities, and mathematics as merely their output. From this perspective, anxiety about being replaced by AI is natural: a machine that performs these tasks better appears to threaten our worth. But the problem began before AI arrived. We had already come to understand ourselves in terms suited to machines. AI merely exposes the consequences of this view.

The response cannot simply be to seek refuge in distinctively human abilities such as creativity, intuition, or taste. If we value these abilities primarily because they give us an advantage over machines, we remain trapped in the same conception of intellectual worth. Why should the value of our pursuit depend on being better at it than someone or something else?

Mathematics offers an opportunity to reconsider this whole orientation. Its truths do not become more true through the distinction of the person who expresses them. In understanding a proof, we bring our thought into conformity with a relationship whose validity does not belong to us. The particular path by which we arrive may bear the marks of our individuality. What is understood is available beyond that individuality. We can allow this experience to loosen the belief that knowledge has its highest value as an achievement attributable to ourselves.

Egoless mathematics begins with taking that possibility seriously. Humility then reaches beyond modesty about our accomplishments or generosity in sharing credit. It concerns the place from which we pursue knowledge. We may seek truth to serve our existing purposes, or allow what we discover to change those purposes themselves. If we approach truth on the condition that its discovery must preserve our importance, we have already made it subordinate to ourselves. Accomplishing this reordering can be demanding, as intellectual sophistication can furnish elaborate defenses of the very attachments that prevent us from understanding. What matters is whether our abilities remain answerable to truth, including when it requires us to reconsider what we have been using those abilities for.

From this perspective, the possibility of mathematical discoveries through AI need not be a source of sadness about our purpose. A truth has become accessible; its worth is not diminished by the absence of our authorship. We need not immediately turn the event into a judgment about our relative importance. The search can continue without requiring every advance to vindicate the seeker. Indeed, the demand for such vindication has always burdened the pursuit, whether the competing claim came from another mathematician or from a machine.

The opportunity before us is therefore more substantial than a rearrangement of mathematical labor. It invites us to reconsider what gives an intellectual life its worth, and to let the pursuit of truth shape not only our work but also how we approach life.

This may sound like asking us all to become saints. Insofar as it asks us to grow in virtue and relinquish our attachment to our own importance, that impression is not far from the truth. But virtue is not something to cultivate for the admiration of others. It is necessary for genuine joy, even in our work. The satisfaction of solving a difficult problem or winning a prize can be fleeting when it rests on our standing relative to others. The joy of understanding is different. Something previously obscure becomes clear, and we recognize a beauty that does not diminish when someone else discovers it first. We can share this joy without depriving anyone of it. Growing in virtue allows us to experience it more fully, free from the need to establish our own importance.

This perspective may also help reconcile two positions within the mathematical community regarding AI. Some are comfortable with or even enthusiastic about using AI, but wish to develop appropriate processes and principles for its use. Others prefer not to use AI at all. Both positions have their merits. Egoless mathematics offers a way of reconciling them by introducing a higher consideration: both approaches can reflect the same underlying commitment to mathematical truth, regardless of how we choose to pursue it. Our aim here is not to settle particular questions of policy, but to develop a higher-level perspective that can guide our deliberations and ultimately our actions.

Seen in this light, we need not be afraid of what AI may achieve in mathematics. Mathematics was beautiful before any of us were born, and it will remain beautiful long after we are gone. Its worth does not depend on our accomplishments, nor does our joy in understanding it have to depend on being the ones who make the discoveries. Our purpose rests on our willingness to let truth govern what we seek and what we become.

We, the authors of this essay, are still far from having attained such egolessness. Nevertheless, we recognize it as an ideal toward which we wish to strive.

We conclude with a thought experiment. Ramanujan is reported to have said, “An equation has no meaning for me unless it expresses a thought of God” (Robert Kanigel, The Man Who Knew Infinity, 1992). If he were alive today, witnessing the advances of AI in mathematics, would he be perturbed by what they might mean for his purpose as a mathematician?

Statement of AI Usage: While the ideas expressed here are entirely our own, we gratefully acknowledge the assistance of GPT Astra 6.0 and GPT 6.1 Sol in drafting this post.

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

Why we should not be sad

The release by OpenAI on October 6 of over 700 research papers was a watershed in the history of mathematics. AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics. So why are some mathematicians so unhappy?

Some of it is the apparent disregard by AI companies for the practice of mathematics. The OpenAI papers — despite some tremendous ideas — are atrociously written and violate our community expectations of attribution. Some of them are even wrong. We have much more control over how we hire mathematicians, publish papers, and train students than over the behavior of trillion-dollar companies, and that is where our efforts would be better spent.

Much of the reaction seems to reflect a feeling that the subject that they love is under attack, and is being “hollowed out”. The recent OpenAI dump made barely a dent on the general problems of functoriality and reciprocity in the Langlands program. But even if it had solved every problem in the Langlands program, to imagine that would have “solved number theory” would be as naive as saying that the construction of the periodic table solved chemistry. Mathematicians should do a better job explaining that when we talk about the central problems in any area, we are merely discussing the most natural boundaries of our understanding — and that unlocking them is not the end but the first step to finding new playgrounds to explore. Of course, this takes time. I looked for areas of mathematics which were “harder hit”. It is hard not to be struck by Hugo Duminil-Copin’s lament:

Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.

I expected to see a few of them in the list. But not all of them. Not all at once. Not with such nonchalance.

How would I respond to that?

The classification of finite simple groups.

I can’t say much about percolation theory, but there is another historical precedent I think mathematicians should take note of. In 1972, Dan Gorenstein announced a program towards the classification of finite simple groups. After a flurry of activity over around a decade, the “classification” was announced by Gorenstein in 1983 (somewhat prematurely, with the case of quasithin groups not being completed for another two decades by Aschbacher and Smith).

The classification of finite simple groups was in many ways a huge success, but it was also a failure in that it — apologies to my group theorist friends and collaborators — cooled a lot of the interest around the subject. The “famous problem” had gone, and mathematicians on the whole turned their attention elsewhere. And yet, there remain very rich questions in finite group theory and even about the classification itself (to name just one: John Conway’s favorite question is why does the Monster group exist?). Perhaps for some fields, the OpenAI dump feels like how a group theorist may have felt in 1970 if the classification was dumped in their office, not in the form of the actual papers, but printed out on a spirit duplicator, smudged a little, then spread randomly like confetti all over the floor.

Right now, we need efforts like that of Gorenstein, Lyons, and Solomon and their work on the revised classification. If this was all we had to do, then it would be a fairly thankless task. But we also have the responsibility not to pronounce the subject “done” but rather — with these new tools available — ripe for further investigation. When mathematicians lament the winding paths missed by the construction of a galactic bypass, they need to realize that those old paths still exist and seek them out. And we need to reward that behavior when it comes to offering people jobs.

The promise of AI, and the (dis)-unity of mathematics:

Since I was 15, I’ve wanted to know what was. First, there was the childhood fantasy of finding a formula that Euler missed. But soon one learns more: how Grothendieck’s period conjecture, together with a sufficiently fine understanding of the geometry of mixed Tate motives (following work of Don Zagier, Sasha Goncharov, Francis Brown, and others) gives a completely satisfying structural, conjectural explanation of what’s going on. The irrationality of , or even its transcendence, is a mere shadow of the actual interesting underlying problem. Apéry’s proof of the irrationality of has inspired me for over years not because of its conclusion, but for how it reveals the tiniest hints of some still secret story relating algebraic -functions to arithmetic.

I hope AI can help mathematicians break out of the straitjacket of being constrained by the first natural conjecture they cannot prove, and work on unraveling the deeper mysteries woven throughout our subject. In the last thirty years, progress in mathematics by humans has been remarkable. But at the same time, I have felt a growing unease; the number of times I have seen mathematicians of all levels being unable to explain to someone even slightly outside their field why what they are doing is interesting has been distressing. The natural grooves of our profession have very much been bent toward more narrow technical expertise than a bigger vision. AI allows us an amazing opportunity to broaden ourselves as mathematicians.

I have tried (not always successfully) to be broad as a mathematician. I had some help by the fact that, while I knew since I was 17 that I wanted to work in algebraic number theory and the Langlands program, I grew up in the crucible of Melbourne University, which was a home to many Thurston students and low-dimensional topologists. This meant I was bathed in the sweet liqueur of hyperbolic 3-manifolds. That exposure has helped my career immeasurably. As with any two deep areas of mathematics, hyperbolic three-manifolds and arithmetic are much more closely connected than would first appear. But it’s still hard to be an expert on both, let alone pursue a crazy idea combining them. Let me get technical for a moment. For every imaginary quadratic field , there are many commensurability classes of compact arithmetic hyperbolic three-manifolds indexed by non-split quaternion algebras . Manifolds in each of these commensurability classes give rise to elements in via the decomposition of into hyperbolic tetrahedra, and their volumes are always rational multiples of . I have always had this idea that one could conceive of elements in these different commensurability classes as forming some type of geometric Kolyvagin system (“Heegner points”) associated to , which one could yoke together by some unknown integral arithmetic version of the Jacquet–Langlands correspondence. This is not a question I have pursued, for two reasons. First, there were so many underlying conjectures involved that any result I could prove was going to be conditional on many of them, something that’s not always appreciated by the mathematical community. Second, assembling all the technical background to formulate such an idea precisely would have taken a huge amount of background work. Now, with AI, the barriers to trying such a problem are much lower. It’s not trying to prove a theorem. It’s trying to explore some completely new direction that may or may not be nonsense. There are so many problems in mathematics that are still so fresh and vital but were previously untouchable that the opportunity to approach them is massively exciting. We need to be more ambitious, to create bold research plans that weren’t even conceivable a year ago.

This is not to say that everyone must (or must not) use AI. But we have to rethink and readjust what’s important. Proofs of major conjectures, using AI or otherwise, are going to remain part of the new mathematics. But working toward broader programs (new and old) and structural understanding is going to play a more fundamental role. Christian Szegedy raised the following point:

Mathematics prided itself on being meritocratic, with deep reverence for the explorers: the people who could climb what nobody else could. That specific kind of heroism will matter less, and other skills will matter more: choosing which terrain is worth exploring, steering fleets of airplanes, and explaining what they find to the rest of the world.

I’m ready to board the plane, because I want to see what’s there. At the same time, I would add that people who previously have been walking around on earth with their eyes closed are not going to recognize anything however high they fly; I still believe that many of the skills that mathematicians have now will not only be transferable but vital to the new world order.

The difference in reaction between the optimists and the pessimists is that, for the pessimists, the game is over. For the optimists, the game has just changed. I acknowledge that there are aspects of our mathematical culture that we treasure that won’t survive, and it is only human to grieve that loss. But there now exists an amazing new tool to study mathematics. To turn away and say that the community of mathematics as it currently exists is more important than mathematics itself would be a terrible mistake.

The ugly truth.

For all that everyone has been saying on this blog and elsewhere about how mathematics is about understanding, it takes one look at the arXiv to see many people in our profession rushing out AI-assisted papers in a somewhat unsightly rush to be the “first” to prove things. As many others have noted, there is quickly becoming a gulf between the ability to produce novel proofs and the ability to either demonstrate understanding or push the subject in new and interesting directions. As people crank out more and more results, the Leiden declaration is being honored more in name than in spirit, and it seems like “business as usual” except with a much more compressed time scale and the constant threat of being scooped hanging over people’s heads.

To me, someone who chooses a worthwhile problem, obtains a proof completely using AI, understands it thoroughly, and uses it to develop interesting further questions or directions has made a substantial mathematical contribution. I would give that person substantial credit even if they supplied no new ingredient in the original proof. Choosing worthwhile problems may end up mattering more than producing proofs.

Our obsession with conjectures tends to greatly undervalue the exploratory work that makes answering (or even asking) these problems possible. We are in danger of making that mistake a hundredfold now by encouraging or rewarding the rush for everyone to press the button to get these new results out as soon as possible. I would encourage all senior mathematicians (with the privilege of tenure) to take up the work of creating future problems and exploring new ideas rather than simply using AI to answer existing questions. If you, as a senior mathematician, are stressed about being scooped by AI, then you are not being ambitious enough.

So what are we actually going to do?

Think pieces are ultimately pointless unless they propose some mechanism for actually affecting how people act. The tools that are going to be the most effective are going to be economic rather than rhetorical. Mathematicians want recognition, mathematicians want jobs, mathematicians want grants. The tools that we currently have to affect this are hiring and accepting papers at journals.

Let me offer some very basic concrete suggestions, followed by a few details.

1. Evaluate all work as if it is AI-assisted.
2. Make hiring much more about personal interactions rather than a paper file. Interview postdocs and graduate students in person.
3. Demand more from journals, and abandon those that do not adapt.
4. Put a greater emphasis on breadth in graduate education, but do not abandon technical skills.
5. Give better seminars.
6. Take more ownership of the results of AI.

Evaluations:

Nelson Cole famously spent three years of Sundays discovering the prime factorization

In , this was a contribution to mathematics. Factoring the same number in by hand would be an equally impressive feat of arithmetic, but its contribution to mathematics (even if it had not previously been known) would be far less, since this can be done on a computer algebra system in far less than a second. It’s obvious that AI is changing the economics of what is “easy” or “hard” — much has already been said about the coming collapse of the theorem economy. In time (perhaps quickly) there will emerge a new sense of what remains a substantial contribution to mathematics, whether it be asking interesting questions, developing new programs, or something completely different. In the future, I see no reason to reward a paper based purely on a result that could be produced by machine, whether it was or not. A new proof can still deserve substantial credit for the ideas it introduces, even when its conclusion is already obtainable by machine. So why not start now? Importantly, I do not think this in any way means that access to AI will be required to make progress in mathematics — as long as we continue to value human understanding. While I feel that this is the most controversial recommendation, I also feel it is the most inevitable. It requires relinquishing an aspect of our reward structure that has been around for a long time, but at the same time a structure that will make no sense in the future. Right now, the use of AI carries a stigma, and the phrasing of many AI disclosures reads more like an attempt to minimize the contribution of the machine. If we evaluate every paper as potentially AI-assisted, then we must insist that its authors explain why the mathematics is interesting, what questions it raises, and how it connects to the broader subject. Don’t we want people to be doing this anyway?

Hiring:

It’s not going to be possible to get people to change unless we change how we evaluate people at the level of hiring. Lazy committees in the past have sometimes pointed toward a JAMS paper, an Annals paper, or some other collection of publications to justify how a candidate is impressive. I am looking for a narrative of where you want mathematics to go, the technical ability to explain why this may be possible, being able to converse about many areas of mathematics, and a demonstrable record of having done all of this in the past. This is not in any way a radical new idea. The hiring process is going to have to be much more about personal interactions than a paper file.

It should become standard practice to interview not only for tenure-track jobs, but also for postdocs and graduate student positions. I’m sure we have all had the experience of admitting a graduate student who was “working on higher homotopy theory” to only find out after they are admitted they can’t compute the homology of the circle. At the same time, I have interviewed candidates at the tenure level whom I was initially ambivalent about, but after a 30 minute conversation in my office I was totally convinced that here was someone who not only had a deep vision of mathematics but who was also able to communicate this to their colleagues. Once we take this approach, we then have the responsibility to train our own graduate students to be better able to communicate mathematics, something that we should take much more seriously. Note there is also a difference between knowing mathematics deeply and explaining it fluently; they are related but different skills. Ideally I want both, but it is the first that I ultimately value more highly.

Journals:

The journal system was in dire need of reform even before AI. Our current incentives already are to produce more and more incremental mathematics which is only flooding the system. The radical overhaul required is not going to happen through consensus agreements with editorial boards. (Do you know how hard it is to get academics to agree on anything?) Being accepted by a journal has long since ceased to be a guarantee of correctness (if it ever was) and has simply acted as an imprimatur of quality. I explained above the philosophy of how I think we should evaluate manuscripts going forward. On that basis, I would say that a significant proportion of papers posted to the arXiv should not be published anywhere. If journals demand much more from a paper in terms of writing, ideas, and exposition, I think that would make life better for everyone — editors, reviewers, and even ultimately authors.

If editorial boards are unable to adjust, then a more radical idea is to (at least temporarily) abandon journals altogether. (I still see a place for journals which emphasize exceptional utility and clarity (consider Essential Number Theory)). I am not advocating that people stop writing papers, but removing the fig leaf of journal publication forces the paper to stand on its own merits. Is the paper telling a compelling story? Is it Lean-verified? We should be producing fewer (and better) papers, not more. Autonomously AI-produced proofs, hopefully Lean-checked, also have their place, and should be recorded as part of our collective body of knowledge. (see hexagonmath.)

Graduate students, first-year courses, and seminars.

One of the thorniest issues we face is how to train graduate students. The typical thesis has always been to assign a problem in the overhang, an exercise that will teach the student the technical skills to solve a variation on some old problem, with the expectation that, as they continue, they can expand the techniques they’ve learned to new cases and then hopefully branch out to wider problems. In an age where such a thesis can be produced by the press of a button, we need to re-evaluate based on what we hope a good student will be doing in the future. Even with AI resources, no one is going to be able to explore the links between compact arithmetic hyperbolic three-manifolds and Kolyvagin systems if they don’t know what either of these terms means. (And even if somehow they could, why would they care?) It has become much more important to give students a broader overview of mathematics as a discipline. What I want a graduate student to achieve at the end of their PhD is a view of mathematics that extends well beyond their discipline, and yet with demonstrable technical skills on at least some class of problems. Often, students graduate with technical skills in a subarea, but their experience with other aspects of their area, let alone other areas of mathematics, consists of technical instruction which they have since forgotten, without any real vision. I advocated above that we need to evaluate work as if AI was involved. But this does not imply that we don’t need to teach students to do deep mathematics without AI just as the existence of computers doesn’t mean we should no longer teach children arithmetic.

This has consequences for what we teach, as well as how students learn to talk about it. A graduate course should leave students with some understanding of what the topic makes possible elsewhere in mathematics. We should expect them to retain a sense of the questions and connections, even when they have forgotten some of the technical details.

If we are to evaluate people more on how they can demonstrate understanding in person, we clearly need to be far more conscious of training our students to explain mathematics to other people, either in person, or in seminars, and to their peers. We need to work on giving our students opportunities to learn this skill. Remember that the most successful cohorts of students are the ones who learn together and who talk mathematics with each other.

We should also not forget that not everyone who completes a PhD in mathematics goes on to academia. We should also ask whether the length and structure of doctoral training serve students who will pursue careers outside academia.

Seminars:

Seminars are traditionally where someone presents the details of their latest result. Because math is difficult, this almost always consists of explaining some minor tweak to a previous method, often without anywhere near enough broader context to make it interesting. Instead, seminars should rarely be focused entirely on new work. They should be framed around the broader interesting questions, explaining the avenues on which we are stuck, on which we can make progress, on which we are hopeful, and then perhaps at the very end explaining some novelty, if there is any. This is how I have always encouraged my students to give talks! As senior faculty we all need to be vocal in our seminars demanding more from our speakers.

Ownership.

Mathematicians as a profession would do well to take more ownership of these results. What has come out of the machine has been built on the intellectual foundations of several millennia of human mathematical endeavor. Once you start to take credit on behalf of humanity (rather than attributing these results to either a Silicon Valley tycoon or a mathematical Twitter buffoon), you’ll immediately start to feel much better.

The world is looking at us.

My dean recently visited New York to meet with donors and other friends of the University of Chicago, and everyone she spoke to, regardless of whether they were a math major or not, wanted to talk to her about mathematics. The world is looking at us. There are many many people who care about mathematics without necessarily having a deep technical background (3Blue1Brown has over six million subscribers!). All of my friends in the humanities truly believe that having mathematicians in society both as custodians of one of humanity’s oldest intellectual activities and explorers of the frontiers of knowledge is important. I understand that some part of the culture we so deeply care about will disappear, and as humans, an emotional response to that is very natural. But this is a time to explain to everyone that there are so many great new opportunities, possibilities, and reasons to be optimistic. We are on the verge of a new era of mathematics, and I am excited to see what happens.

Notes:

For more context on Duminil-Copin’s remarks, see also this numberphile video. His full quote also says “We are a resilient community, and I have no doubt that we will adapt”.

Many of the themes mentioned here have been brought up in other posts. To give three examples:

• The idea that mathematicians should take more ownership of the results produced by AI was also explored in Nestor Guillen’s post.
• The idea that AI gives us the opportunity to be far more ambitious is raised in Ben Antieau’s post.
• The idea that we need to revisit our reward structure to create the right incentives was explored in Bryna Kra’s post.

The Academy for the Mathematical Sciences, which is based in the UK, has released a report “The mathematical sciences in an AI-enabled world“. To quote from the release:

The paper examines how recent advances in artificial intelligence are changing mathematical research and its applications, bringing new opportunities for discovery, verification and the use of mathematics across science, industry and society. It also considers the risks associated with these developments and identifies the support and infrastructure needed to realise the opportunities while addressing the challenges.

A central theme is that, as AI becomes increasingly capable of carrying out technical mathematical work, human expertise will be crucial to direct advances. Identifying promising questions, providing key ideas, assessing the reliability of AI-generated results, developing mathematical understanding and translating discoveries into wider use will remain essential.

The paper identifies priorities for the community, including education and skills, independent verification, access to AI capabilities, research assessment, publication and the infrastructure needed to support mathematical knowledge. It sets out areas for action in the near term and over the coming years.

[This is a guest post by the organizers of the Summit on Math Research in the Age of AI: Dan Freed (Harvard), Larry Guth (MIT), Bryna Kra (Northwestern), Emily Riehl (Johns Hopkins), and Mohammed Abouzaid (Stanford). This blog post was initially written in a different file format and converted using AI. — T.]

Last week we organized an intensive two-day summit Math Research in the Age of AI. It was held at the Harvard Center for Mathematical Sciences and Applications (CMSA). Thirty-two mathematicians from academia, government, and industry came together to help navigate this period of rapid change in mathematics and, most importantly, to offer ideas for building the future. There were five working groups organized around the following themes:

• Evaluation, production, and dissemination of research
• Coordinated actions for the AI era
• Open-source tools
• Formalization in mathematics
• Collaboration between academia and AI labs

Our report can be found on the summit website. There you’ll also find a link to provide feedback, and we also encourage open discussion in the comments section of this blog.

Edgar Assing, Valentin Blomer, and Farrell Brumley are organizing an event at the Hausdorff Center in Bonn on Jan 11-15 to discuss recent developments regarding the Riemann zeta function and allied functions. From the announcement: “This includes in particular the recently established proof of the quasi Riemann Hypothesis over the rational numbers as well as over general number fields and its manifold consequences, ranging from Artin’s primitive root conjecture and torsion in class groups to equidistribution of periodic torus orbits of higher degree fields.

The target group are early career researchers with an excellent background in analytic number theory. The collaborative retreat will feature lectures by the organizers as well as exercise sessions and projects in small workgroups in the stimulating atmosphere of the Hausdorff Institute.”

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

The day after the Navier-Stokes result was announced, a student came to my office hours. They said, “Can we please talk about the Navier-Stokes result and what it means? I really need some therapy.” Two days ago another student mentioned, “I used to be really motivated, but now that all the models can do everything with just a prompt, I don’t feel like putting in any effort at all.”

Every time I have such a conversation with students, I appreciate how lucky my generation has been. We didn’t have to grapple with the temptations or threats of AI when we were students. Not being able to quickly look up answers forced us to challenge ourselves, and do it together with our friends. And years of challenging ourselves somehow made us love it [1], and led us to challenging jobs across academia and industry. These jobs are simultaneously wonderful and hard. But I wonder and worry for my students, for the next generation. What do we need to do to make sure these jobs [2] are (1) available for the next generation and (2) that we have enough students who are motivated to train for them?

I’ve been in conversations where people are concerned about whether mathematics will survive. I’m actually not concerned about this — I’m fairly sure it will, so let us set that aside for the moment. But what I want to think about is, what do we need to do to make sure all mathematicians survive? And not just the `great’ mathematicians, but also regular, ordinary mathematicians? Because even if we decide society only really needs the `greats [3],’ we need to train many tens of thousands of students to identify who those greats are. If we want one Steph Curry, we need to make basketball exciting for a whole generation of people, and we need to support college basketball, and after-school basketball, and neighbourhood basketball.

We’ve seen that by treating mathematical problems as a benchmark, humans have been able to train AI models to be very good at solving hard math problems and writing proofs. Soon we will create benchmarks about `asking good questions’ and `giving clear explanations’ and then AI models will be good at those tasks too. They will do the thinking and the meta-thinking. This is the reality we need to plan for.

For this post I will talk about “mathematics,” but I often mean mathematics broadly construed (the sciences, engineering, social sciences, and even many humanities subjects that require complex reasoning and critical thinking).

Background

People are attracted to mathematics and mathematical sciences for a variety of reasons. But importantly, whatever the motivation, mathematics has always been a `safe’ subject to study. Historically, being good at math has led to a solid job and a stable lifestyle. Why? Because along the way, by studying beautiful mathematics (which in itself might not be exactly what leads to employment), the student learns critical thinking and perseverance, and becomes stronger. The transferable skills are helpful for employment.

As a result, society has encouraged students to do math, even when they might not be `great’ at math. For example, as a teenager, I was part of a special program called Gold 2002, [4] which trained students for the International Mathematical Olympiad (IMO). I was not among the `greats’ in this program. I did not make it to the IMO, unlike some of my friends. And today, I have no hope of ever winning a Fields medal. But I still enjoy doing mathematics, and (thanks to some hard work as a teenager) I’m good enough at it to be paid to do it (again broadly construed, I’m a professor in an EECS department [5]). But one of the reasons that my parents let me spend months and years working on crazy math problems was that math was a `good’ thing. And so it was ok to do this, even if I didn’t win any gold medals.

But today people wonder — does mathematics still have the same status? Do we need critical thinking if a prompt can do it for us? My students ask — is it worth their time to get a PhD, in mathematics, or any theoretical field, without an immediate application? Or should they focus on learning to train better deep learning models? When previously I would have many undergraduate researchers from my lab applying for PhD programs, many now prefer to go straight to industry. What do we say to them?

I have heard people say — in this new era, math will become more like the humanities. We will continue to pursue mathematics because it is beautiful and because we love it. While this may be the case, will as many students pursue it if it does not lead to a good job and a steady life [6]?

Which leads to the thought — well, does society really need all these mathematicians? We will have only our strongest, most passionate students do the very hard math, or those who love mathematics enough to do it even as `starving artists’. With AI, if one mathematician can do the job of five or ten, should we shrink our departments and our graduate programs? What jobs do we expect our students to get when they graduate? And if there are no jobs for `thinkers,’ and `intellectual explorers’, then why should students pursue a path towards a PhD in math or a theoretical science?

This brings us to a public goods problem, which I elaborate on further below. I believe it is essential for us, the human society, to maintain our intellectual culture. And for this we need lots of students getting degrees in mathematics, sciences, and the humanities, but we need to find ways to actually motivate the students to want these degrees and the associated training.

What is our culture and why is it important?

As a child, I would see a big splash in the newspaper about the Nobel prizes every year. I didn’t understand the science or why the literature was meaningful. I just understood that this stuff was important, and that humans valued creativity, expression and intellectual and scientific achievement. And until recently, I had never consciously wondered about the cultural importance of Alfred Nobel establishing those prizes. I had always taken for granted that we should strive for excellence… but I see now that this was an impact of our culture, and that there are so many conscious decisions that have led to this culture.

But now, people have begun to worry about `The end of the age of heroes‘. In his blog post, Noah Smith makes multiple interesting points, but I will paraphrase one: even as mathematicians will likely retain their jobs as math-understanders [7] (since mathematicians are relatively cheap), they will lose their status as heroes.

My question is: while it may be ok for the mathematician to lose their hero status, what impact does this have on the next generation? In my younger days, and even now, I know I benefitted from being around, and hearing about, heroes — the brilliant, blazingly quick, hardworking minds, with the humblest of personalities to match. It gave me someone to look up to and goals to aspire to. With heroes gone, how would we orient our cultural compass?

Aspects of our culture are built on respect for facts about the universe. Nvidia CEO Jensen Huang said in an interview with Ezra Klein a few weeks ago that it doesn’t matter if the multiplication table is starting to be forgotten, and that “we’re going to lose some finer intellectual dexterity … I don’t know how valuable it is for most people to learn how to do surface integrals or partial differential equations. I don’t really know how important that is, but it’s important to some people.”

I was bothered reading this, because I was worried about the implications for our culture.

Multiplication tables matter not just for their own sake but for what they represent [8]. Yes, knowing builds fine intellectual dexterity. But they also show young children that there are fundamental, unchanging facts about the world. Invariants. And starting from this, we can build up confidence that allows kids to grow into students who are grounded and questioning because they can both see and understand why there are invariants [9].

Our understanding of partial differential equations allowed us to develop numerical computer programs that can solve them for us. And when the computer does solve them, it is amazing that we (as a human society, not every individual) are able to understand exactly what the computer did. But this is not true when AI does all the work for us; we lose the essential human understanding. I will come back to the value of this soon.

It is important to maintain a culture where we value and respect intellectual dexterity and expertise. And admire the young children who know their multiplication tables, and the older people who win Fields medals and Nobel prizes. And respect those people who care deeply about partial differential equations, and create space for them to develop their expertise, even if not everyone understands them. The goal of expanding and developing human expertise in subjects can and should be independent of the state of AI [10]. So that we can ensure generations of students will continue to want to study and experience the “unreasonable effectiveness” of mathematics.

Education: Canary in the coal mine

Every professor and teacher I know has grappled with AI-based academic dishonesty by students. Students cheat when `doing it the right way’ doesn’t seem worth the time and effort. And because our culture for students (especially in the United States) has focused on achievements/badges (or, as I tell my students, `sticker collection’), instead of building true understanding and intellectual depth, students want to do more and do it faster, even if they are not doing it deeper or better.

We need to realize that this is just the canary in the coal mine. Students are turning in proofs/essays/code they don’t fully understand. But this behavior is not limited to academic settings. It is likely that engineers are checking in code they didn’t fully read. It is only a matter of time before a researcher publishes a result they don’t fully understand, or an administrator signs off on a decision they don’t fully comprehend. And soon enough, AI systems will be making ever more complex decisions that we just sign off on, because we don’t understand the foundation of the system and we gave up questioning.

If we go down the slippery slope, we risk becoming AI-sock puppets [11]. Faced with AI-output, we will take quick looks and “sanity checks” to make sure AI output is reasonable and move on, but it won’t have the same gravitas and deliberation that comes from a human-authored proof/document. We will have no choice but to use AI, because that will be the only way to keep up. Sanity checks themselves may become illusions — because we will be out of practice at true understanding. It is likely that we will have verified Lean proofs that assure us of correctness, but now we have to evaluate the faithfulness of the theorem statements. (And Lean itself has a very steep learning curve. There is a lot of work to be done to make sure such verification tools are universally accessible to all [12].)

The only thing that can protect us against this is culture. A culture where the point of pride is in the understanding of the work and in the exposition of it. A culture where it is ok to say we don’t understand, ask for clarification, or (respectfully) push back. And where such requests are valued (instead of saying, “let us move on, we only have a few minutes”).

A culture that appreciates being slow and steady and careful. A culture where a student can pursue a more theoretical research area for their PhD, but also be assured that this does not destroy their chance to get a stable, enjoyable job.

So how do we cultivate this culture?

Such culture building must start in the classroom. We must teach harder, better, and stronger [13]. We need to train students so that they know what they know and don’t know — students who can identify proof by intimidation and by authority, and have the confidence to push back. We should provide homework problems that let students explore concepts in depth, and force them to confront ideas they do not understand. Students thrive when they are challenged (with support) and then succeed. This needs resources.

We must walk away from classes that teach students the “high-level ideas” but do not expect students to fully grasp the rigorous details “because AI can do this stuff nowadays, [14]” or because “it is too hard for the students [15].” Because these can create a generation of students who don’t know the difference between understanding things deeply versus superficially. And then this can propagate.

This is now a bigger deal than it was a few years ago. It used to be that knowing a solution strategy or the right algorithm to apply was sufficient for students to become productive members of society. But now that this work can be done by AI systems, we have to be able to understand what the AI has generated, process it and question it, which requires not just knowing the names of the algorithms but also the proofs of why they work, what the key assumptions are, and where and how they break down.

A theorem may be correct given its assumptions. But the gap between assumptions and what we actually care about might admit counterexamples. Identifying such counterexamples requires knowing the modeling and the math at multiple levels. Interpretation matters and ambiguity can creep in when identifying if a formal Lean statement is faithful to the natural language. For such tasks mathematical mental muscles are essential.

Amit Sahai, Po-Shen Loh, and others have beautifully argued why we are going to need more (highly trained) mathematicians. I won’t repeat their argument here. But I will add to their point — it is important that we have a large, diverse set of people, with different life paths and trajectories, becoming mathematicians. People who do not all think exactly the same way, who can interpret, debate, disagree and convince one another.

A public goods problem

This brings us to a public goods problem. We need to maintain our culture. For this, we need to engage and excite lots of different students. We know that if we don’t incentivize students to learn, many of them won’t, and our (humankind’s) mental muscles will atrophy. We know that when humans moved from working in the fields to having indoor desk jobs, our bodies suffered. And to compensate, we invented things like the gym to make sure our bodies stayed fit.

But there is a catch. We pay to go to the gym, not the other way around. Entry-level jobs in fact used to train people while also getting useful contributions from them. Entry-level jobs, or PhD programs, are like being paid to go to the gym. If these are taken away by AI, how do we train people? And how do we convince people that they should be trained?

There is a tremendous investment needed in thinking about what future sustainable solutions to this will look like. This is just the beginning of that conversation. But in the short term, it is important culturally to celebrate and reward the work of understanding and communicating mathematical results — and not just by mathematicians but by society at large. We also need to advocate for educational institutions at all levels and for what they represent for our culture. Teachers need more resources, not less, to be able to teach in the AI-age, because we need to be teaching more, not less. There is no silver bullet. AI tools alone cannot solve problems in the education sector — these come with their own set of challenges. We should be able to offer individualized, 1-1 human tutoring to help students at different levels. I hope we can collectively build a positive future, where we use AI-tools effectively to help us in ways that raise everyone up together.

As a concrete first step in this direction, I invite university leaders to declare their commitment to hiring across departments in the sciences, engineering, social sciences and humanities, and announce their unwavering support for human intellectual growth and exploration. Similarly I invite philanthropists and funding agencies to declare their support for open-ended research, theoretical pursuits and rigorous, grounded education in the fundamentals. Students need to know that we have got their back, and that we are working for their future — because they are our future.

I would like to thank Terry Tao and Nestor Guillen for the invitation to write this post, and also Ted Theodosopoulos, whose 1st Relatorium dialogue on Mathematics and AI started part of this conversation. Thanks also to Anant Sahai for many conversations that have helped refine these thoughts, as well as Vinayak Ranade, Sukhada Fadnavis, Shubhangi Saraf and Amit Sahai for their helpful feedback. Thank you also to the organizers and participants of the Renaissance Philanthropy AI for Math Workshop 2026 — conversations there also influenced my thoughts.

Notes

[1] I understand that challenge and struggle does not work for all students, but it does help a good number. In this post, I will focus on that set. There is a longer discussion about education to be had more broadly at another time.

[2] Jobs that are typically held by PhDs that apply mathematical/other critical thinking skills. I understand that there are far more complex questions regarding jobs in the age of AI, and I’m not trying to tackle them. For now I focus on the narrow question of jobs for highly-trained scientists/mathematicians/humanists.

[3] I argue that we need far more than just the `greats,’ but just hypothetically.

[4] I am forever indebted to M. Prakash sir, out of the Bhaskaracharya Pratishthan. He intensively trained 200+ highschoolers across Pune, India, and several from the group went on to win IMO medals (another example of the Steph Curry argument). And the rest of us benefitted on the side. It was a privilege to participate.

[5] I teach courses on the mathematical side of the department, such as optimization, probability, and deep learning.

[6] I also love to dance, it is beautiful. But I am nowhere good enough to earn a living doing this. Luckily the math worked out ;-).

[7] Human understanding can never be developed by an AI, no matter how good AIs get. AI-augmentation of humans cannot generate human understanding.

[8] Perhaps we will find skills that are a suitable substitute for multiplication tables, that do everything that tables do and are `better’ in other ways. I’m ok with teaching those skills instead.

[9] As Picard said “there are four lights.”

[10] And every PhD granted is a new human expert in a subject.

[11] Not quite a meat proxy, but still bad.

[12] Some of our work to make lean accessible is available at leantutor.org

[13] not faster. https://www.youtube.com/watch?v=gAjR4_CbPpQ

[14] There is a space in this system for courses that teach students how to “use” tools (e.g. prompt engineering courses that have recently come up at many community colleges in California), or general AI literacy courses. But we must not confuse the role of these courses. Those may be helpful (in the current climate) for maintaining employment. But they do not replace the role of courses that fundamentally teach critical thinking, and should not take resources away from such courses.

[15] To cater to the fact that students come in at different levels, we need to have scaffolding classes at the lower-division level. A student may take longer to graduate, but should be able to take even the hardest class given enough time and hard work.

Daniel Carter, Kevin O’Bryant, and I have uploaded to the arXiv our paper On the diameter and distribution of Sidon sets. This is a paper stemming from an interaction on my blog last year, regarding the use of AlphaEvolve to improve the then state of the art on the size of Sidon sets. We obtained a series of further refinements to the bound until progress stalled; after several months of no further progress, we decided to release what we have in this preprint.

Recall that a Sidon set is a set of integers with the property that all the sums with and are distinct. Equivalently, all the differences , are distinct, except at where of course there is a large multiplicity. A basic question (Erdős problem #30) asks for the largest size of a Sidon set in . Several algebraic constructions of Sidon sets (by Singer, Bose, and Ruzsa) are known with and special choices of ; see this survey of O’Bryant for details. Erdős and Turán have conjectured that cannot exceed , with the bound also proposed by Erdős as “perhaps too optimistic”.

For some time, the best upper bound was of the shape

of which several proofs are now known. As tracked in the optimization constants repository, the constant factor in the second term has been gradually declining in the last few years; in our preprint we obtain the bound

but we appear to be reaching the limit of our current method. Very recently, a slightly sharper bound of 0.942809522 was obtained by a different method, but our focus here is on the method rather than the specific numerical constant obtained. (We also did not try to use 2026-level AI tools to optimize our own bound.)

To illustrate our approach, let us sketch a proof of the bound (1). The fact that all the non-zero differences of are disjoint gives rise to a pointwise convolution bound

where denotes the reflection of . Taking inner products with a non-negative autocorrelation function to be chosen later, we conclude that

Taking the inner product of with a further weight , and rearranging, we conclude the bound

On the other hand, if one has the pointwise lower bound

on , then the left-hand side is at least . This gives an inequality

that can be used to bound by choosing appropriately. For instance if one selects

and

then one basically has , and the above bound becomes (after ignoring some lower order terms)

which after some algebra gives (1).

One can improve this bound by replacing the sharply truncated cutoff functions with more complicated cutoffs, which obey some biconvex constraints which one can explore using various optimization algorithms. It turns out that one can do even better by working not with deterministic choices of , but with random choices of , in which inequalities such as are only true after performing an expectation. The calculations get quite messy (and highly computer-assisted), but eventually give (2). The method also extends to the more general class of -Golomb rulers, in which a given non-zero difference is permitted to appear up to times.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I live to serve the knowledge frontier and nothing else.

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

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

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

And honestly? That’s rare.

By Rikab Gambhir (The @finitephysicist)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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