Mathematicians need to act

Do not collaborate with AI companies for the foreseeable future.

This morning, I woke up to a text from a postdoc (soon-to-be-professor) friend, saying that OpenAI’s new model, Astra, had proved that a nonsofic group existed. At first, I was shocked. Then I became angry.

Let me begin with a few technical notes. I first learned about the problem of nonsofic groups in undergrad; it is one of Gromov’s famous open problems, who in 1999 essentially asked if every group was sofic (though the reader may notice he did not use the word sofic in that paper, which was coined by Weiss a year later). For the last 15 years, the mathematical community widely believed that nonsofic groups existed, but it is quite hard to show that a given group is nonsofic. This was known to be a very important problem in group theory, with real implications; sofic groups are known to satisfy interesting properties, such as Gottschalk’s conjecture and Kaplansky’s direct finiteness conjecture. As recently as March of 2026, I went to a conference at UT Austin which was solely about sofic groups and the Connes embedding problem, although as a percolation theorist I have never worked directly on this problem or even adjacent problems myself.¹

Nine other results were announced by OpenAI in the same press release, including new bounds on sphere packing and a superexponential lower bound on multicolour triangular Ramsey numbers. Of all the facts accompanying this press release, the following stood out the most to me: the cost of these results was roughly $2,000 USD of tokens at Sol API rates.

In lieu of continuing with the technical details of this quite impressive result, let me now speak as a human being. This is a polemic, not a press release. This ruined my morning. I know that others feel the same way; I have spent today crashing out and learning that top mathematicians I know are crashing out. We in the mathematical community are all grappling with the implications of these highly effective large language models. What does this mean for our field? How should the practice of mathematics continue in light of increasing capable artificial intelligence systems? How should we interact with AI companies? Should we partner with them and work for them? Should we use these technologies to accelerate mathematical progress as quickly as possible and initiate scientific breakthroughs? What role do human mathematicians have to play in a world where artificial intelligence is increasingly capable of autonomously proving important theorems?

Many prominent mathematicians have articulated visions of what the future could hold, such as Terence Tao, Tim Gowers, Jacob Tsimerman, Daniel Litt, and others. Some, like Terence Tao, predict that we may have to adapt to an “abundance” environment where mathematical insights are plentiful rather than rare. Others, like Jacob Tsimerman, are broadly pessimistic about the implications of artificial intelligence for mathematics. Let me also draw attention to the Leiden Declaration on Artificial Intelligence, which offers a good number of intelligent and unobjectionable statements, and simultaneously assert that it does not go far enough.

I have talked to mathematicians who believe that in the future, we will simply be like artists; is this so different from what we usually do, which is already useless? One talented PhD student told me that while photography may have the obliterated the industry of portraiture when it made the production of images extremely cheap, at least photography still exists. Of course, artists and photographers are famously well-respected and paid in this society of ours.

I am sure that many obsolete careers of yesteryear were quite wonderful. Perhaps it was a great gig being the court astrologer. Pick up Silas Marner and you will realize that weaving used to be a well-paid and respected profession; while “Luddite” may be a pejorative today, the textile workers who destroyed mechanized looms were not behaving so irrationally from an economic standpoint.

I believe that today, mathematicians may be in the same position as these bygone professions. While I believe mathematical activity will continue and that some version of the profession of “mathematician” will continue, I think that LLMs pose an existential threat to the mathematical community and the values of that community. I believe our lives and livelihoods are important. I believe in chasing fantastic scientific and mathematical progress with the help of LLMs, we risk imperiling our institutions, our professional norms, and human mathematical understanding. Moreover, I believe that we are capitulating and surrendering in advance to AI companies that are broadly building a world that we do not want to live in. Mathematicians are not the only victims of these disruptive technologies, and we are far from the most important ones. There is a reason why AI companies have turned their attention to our field. Every new press release of this type enriches the shareholders of AI companies.

I am not a Luddite. I think that it is fine if human mathematicians decide that we want to use LLMs to accelerate scientific progress and introduce new ideas to mathematical fields which have been stuck on the same conjectures for a long time. However, it needs to come from us. We need to be making these decisions collectively. We cannot preemptively lie down and surrender control to AI companies. Their success hinges on our continued collaboration: our technical knowledge, our careful verification, our write-ups. Whether we offer that collaboration is a choice.

The existence of a technology is not an obligation to use it. The rapid pace of AI progress is both impressive and alarming, but it alone does not dictate the uptake of that technology. There is no denying that AI companies have made spectacular scientific advancements: this means we are in the domain of politics now. Remember: As human beings, we have the power to write laws and enforce social rules. We instill professional norms in our mentees and colleagues; we ostracize those who behave in distasteful and predatory ways. We regulate dangerous and disruptive industries; we enforce legislation. We organize, form unions, and socially punish “scabbing” so that we maintain our collective bargaining power.

The use of a technology is always a choice.

The values of the current mathematical community

Let me first attempt to articulate some of the values of the current mathematical community and highlight how the use of artificial intelligence is in tension with those values.

1. Humility and correct attribution of work to others. As mathematicians, we generally spend the majority of our time understanding and digesting the work of others. As a result, honestly attributing mathematical ideas to other people is an important value in our community. In fact, sometimes a mathematician may write down a completely original proof and call it “folklore” or post it as an expository note, because they believe that the techniques were not essentially original and that the community at large owns this proof. Mathematical work is carefully-attributed collective property. Unlike many other scientific fields, in multi-authored works it is our norm to list names in alphabetical order rather than by who “contributed” the most or “proved” the most.

2. Accountability over mathematical work and admitting mathematical mistakes. Suppose that two mathematicians are having an argument, and one believes that (P) is true, while the other believes (Not P) is true. Both may have very convincing mathematical arguments at hand, but at the end of the day, only one of them can be correct. This results in a strong social norm in our community of admitting to mathematical errors. I have seen a person who has been arguing for 60 minutes instantaneously admit they were wrong as soon as they are given a valid argument. Similarly, when we write papers, we take full ownership of the mathematical content therein, and we expect a high level of rigour in mathematical arguments. If an error or gap is found in a proof, it is fixed or the paper is retracted.

3. A culture of open-access and knowledge-sharing. We upload our preprints to ArXiv, an open-access archive, to speed up knowledge sharing and avoid creating financial barriers (even if we ultimately publish in for-profit journals). We go to conferences to disseminate knowledge to each other. We teach other people what we know rather than hoard techniques which can be used to prove lots of results. We are far from the dueling Italian mathematicians of the 16th century; we learn techniques to solve problems so that we may educate other people how to solve the same problems.

Let me now say a word on how these values are threatened by the use of AI.

1. Large language models frequently misattribute theorems or altogether forgo citing the works of other authors. When synthesizing the vast domain of (human-authored) mathematical works they are trained upon, they often do not give adequate credit.

   1. This is generally quite convenient from a market-valuation perspective for AI companies, who stand to be enriched by an increase in “hype” and downplaying the achievements of the relevant mathematical field which helped lead to a result.

   2. For example, the proof of the existence of a nonsofic group heavily uses work of Andreas Thom and Gabor Kun, but details like these are generally de-emphasized in press write-ups.

2. Large language models frequently write nonsense proofs and double down on errors. In doing so, they waste human attention span, they confuse experts and non-experts, and they lead to human beings propagating these nonsense arguments further. This undermines our values of accountability and admitting to our mistakes.

3. Large language models threaten our culture of openness and knowledge-sharing, though perhaps at a second-order level. I believe that human mathematicians are increasingly incentivized to be more secretive about their mathematical ideas, since otherwise they risk being “scooped” by large language models. This is especially true when LLMs are regularly trained on large open-source databases.

   1. Let me give an illustrative example from my own life. I am currently working on the proof of a very interesting result with a tractable proof strategy: the sort that involves combining (x) technique from this paper with (y) observation from the classical theory, in novel and substantive ways. If I told a human about this proof idea, I could get an interesting mathematical idea across, and I would not be afraid of them writing up this result: it would take them about 3-6 months of work to do so, and about 70 pages of write-up. (More saliently, it would be viewed as in poor mathematical taste to do this, especially to an early-career researcher.) However, if a frontier model was told about this proof strategy, then perhaps after about $200 tokens it could write up the same result in a vastly shortened timescale.

   2. The way in which LLMs threaten open source movements is well-articulated in the following essay by Sean Boots:

      The process for creating and improving generative AI tools (scraping vast quantities of information without compensating the original creators) incentivizes restricting rather than sharing information. It also damages open source movements and collaborative human efforts (like Wikipedia) that society as a whole benefits from.

Why we do mathematics and the goals of doing mathematics

Why do we do mathematics and what are the goals of doing mathematics? We prove theorems and write papers for many reasons: appreciation of mathematical beauty; professional competition; a stable livelihood; career prestige; the joyful experience of problem-solving. As with all basic sciences, we are motivated by the pure desire to understand the world around us, including our mathematical world. The acquisition of this understanding necessitates proving and learning new theorems and lemmas.

In light of this, why are so many mathematicians reacting negatively to the use of AI in mathematics? Should we not want to understand as much of the mathematical world as possible? Studies show that humans enjoy artistic works less after being told they were generated by AI; clearly, humans have a pro-human bias. I have meditated a long time on my personal negative reactions to the use of AI in mathematics. After considering, I believe that it is rational that I have serious qualms about the use of AI in mathematics.

To me, the primary goal of mathematical progress is human understanding of mathematics. In this, I take essentially the same perspective as Thurston in his seminal essay, On Proof and Progress in Mathematics. I do not believe the sole aim of mathematics is to prove the most theorems or lemmas possible; it is to enrich human understanding of the mathematical world. While the use of AI has the potential to be a serious tool in the pursuit of that endeavour, I believe it poses a grave threat to the current practice of mathematics. In particular, what alarms me most about the use of AI in math is that I believe it has the potential to disrupt, mitigate, and harm human mathematical understanding. I believe it will push people away from mathematics and into other careers; I believe that access to an increasingly-powerful oracle will paradoxically disincentivize people from doing the work of mathematical understanding; and I believe that the median person (say, the person who takes a couple of courses in math in high school or university) will learn strictly less mathematics.

How the use of artificial intelligence risks disrupting mathematical understanding

Large language models are already disrupting mathematical understanding. Let me now give some examples, including some obvious ones.

1. Undergraduate students in scientific fields have a generally weaker mathematical understanding, because they do not solve problems autonomously but instead use LLMs. Those of who teach undergraduates know that free access to AI is generally to the detriment of undergraduate education (although of course, talented undergraduates can probably learn more with LLMs). When I was a master’s student in 2024-2025, students would often come to my office hours with nonsense proofs from AI. Now that AI capabilities have improved tremendously in the last year, many universities are observing an uptick in students performing extremely poorly on tests and in-person exams while obtaining perfect grades on homework, some of which are written with zero human input.

   1. This is likely very bad. Although research mathematicians are disincentivized to prioritize it, I believe undergraduate education is extremely important. Let us state the obvious: most people, including those with math degrees or substantial mathematical education, will never prove an original mathematical result in their lives. They work through exercises and learn or rederive proofs of known theorems; they discover the crown jewels of 19th-century and 20th-century mathematics. It is important that they do this, because learning mathematics is a valuable human activity. However, with access to an oracle which can basically answer any question you want, including frustrating homework questions, it is tempting and tractable to complete one’s mathematics course without actually learning mathematical content or how to prove lemmas and theorems.

2. LLMs have the potential to disrupt the pipeline through which we produce mathematical researchers. At the moment, they are most capable of doing mathematical problems that are pedagogically useful for PhD students to do.

   1. How do we produce research mathematicians? In fact, it is incredibly difficult. We often give graduate students quite tractable problems that involve reading a proof in the literature and extending it to a slightly more general case, or using a technique from one field in another slightly different context. It is through problem-solving and autonomously writing proofs like this that people internalize the techniques of their field and learn how to write original mathematical content. However, if artificial intelligence is used to write up proofs of this level, this raises the minimum level of a “publishable” result to something much higher, which is potentially out of reach for early-career researchers. The same concern is raised by Gowers in this blogpost, although he points out that if the bar instead becomes “proving something in collaboration with LLMs that LLMs cannot manage on their own”, this may actually be substantially easier.

3. In response to AI capabilities improving, talented, young people will leave mathematics because they believe they do not add value, and this will impoverish the mathematical community. This will be particularly true in problem-solving fields. AI companies will take advantage of this situation: they will hire these people and lure them to their safety departments to solve the problems that they are actively creating.

   1. I believe this may pose the gravest long-term threat to the mathematical community. Many mathematicians, including early-career researchers, react to new announcements of AI-led results with despair. This does not seem wholly irrational to me: we have been culturally conditioned to believe that proving theorems is how we add value to the world and to the mathematical community, and it increasingly seems that artificial intelligence technologies are autonomously capable of doing the same. If people believe that they do not add value in this career, they will choose different careers. Our profession, which already loses talented people every year because it demands so much personal sacrifice, will hemorrhage more people. Our human mathematical understanding, as measured by the number of people who actively understand and do research-level mathematics, will shrink further.

   2. Imagine the following scenario, where human beings had access to an oracle whom it could ask arbitrarily hard mathematical questions and receive a correct answer. As humans, would we understand more mathematics, or less? Would we talk to each other more about mathematics, or less? I believe the answer is less. This is clearly a defect of human psychology—nevertheless, we must grapple with the implications. To quote Tsimerman:

      1. “There are many people whose primary enjoyment of math comes through problem solving in one of its incarnations. If that disappears, that is not a trivial issue and many of them might not want to do it anymore (even if there were some way to proceed).”

Are we ethically obligated to use artificial intelligence?

It seems obvious to me that if mathematicians do not use artificial intelligence to the fullest extent possible, then mathematical progress will proceed at a strictly slower rate than if we did. Does this have any ethical ramifications? Are we obligated to use artificial intelligence to pursue mathematical progress at the fastest rate possible?

Fortunately, I believe the answer is no. Of course, some of us are working on mathematics that are directly relevant to other scientific fields or building new technologies. (For instance, a theorist working on quantum error-correcting codes may write an algorithm which makes building quantum computers more tractable.) If these technologies dramatically improved human life, perhaps we are obligated to accelerate progress in those areas as much as possible. However, the work of the vast majority of pure mathematicians does not directly impact other human beings (aside from other mathematicians).

I am inspired in this regard by the following philosophical analysis by my friend Marcel Goh, a PhD student at McGill University. Consider the following scenarios:

1. You are a doctor that [sees] many patients on a daily basis, and there is an AI tool that helps you diagnose twice as many patients in a day, and the diagnoses are accurate 50% more often.

2. You are a low-level employee at an office job, paid for eight hours of work a day, five days a week. Your work is not a matter of life and death, but produces value to some interested parties. Implicit in this hourly wage structure is that you’re expected to do as much work as you reasonably can every day. You have access to an AI tool which allows you to produce twice as much value at the same effort cost.

3. You are a digital artist commissioned to preparing some visual art that will go on some organisation’s website. You have the opportunity to use generative AI tools to produce in a day what would have taken a week in the past.

In Scenario 1, Marcel argues that it is unethical not to use AI. (I concur with him.) In the other scenarios, it is much less clear. Like Marcel, I believe mathematicians land in the latter two scenarios. I do not believe that, in the majority of cases, we are ethically obligated to use artificial intelligence to speed up mathematical progress. Of course, if we want to, we may do so anyway.

If the impact of AI on mathematics is dictated by AI capabilities, we will not succeed

I believe that in a world where AI-led mathematics directly competes with human-led mathematics, AI-led mathematics will ultimately surpass us. If the competition is “Who is better at proving theorems: humans or artificial intelligence systems?” I believe that humans will lose, possibly in the very short term (within the next 2-5 years). This seems fairly obvious to me, though I know some experts will disagree. Just think: every person who wants to say something interesting about symmetric spaces has to learn semisimple Lie theory, and every person who exists is born knowing nothing about it. Moreover, each person lives a finite life, perhaps only eighty to a hundred years, and when we die we cannot contribute anything to the mathematical literature. But a large language model can be trained on Lie theory and only has to learn it “once”.

Thus, if we want human-led mathematical activity to continue, we must avoid a scenario where large language models directly compete with humans. We must convince governments and nonprofits to continue employing mathematicians and those who do mathematical work, even if LLMs are more capable of autonomously proving and verifying theorems. We must find ways to incentivize humans to continue doing mathematics.

Some mathematicians may react to this state of affairs as follows: we should give up on protecting or preserving human-led mathematical activity. Perhaps the future career of the mathematician is that of a prompt engineer, and not an autonomous researcher. Perhaps we will simply read and digest AI-created output rather than autonomously prove theorems. Perhaps mathematics will evolve into a recreational hobby; well, chess engines are better than ever, but simultaneously more laypeople are interested in chess than ever.

I don’t find this scenario particularly appealing, although that may reveal a lack of imagination on my part. I also fear it is unlikely that many people will continue learning mathematics to the level at which they could digest (AI-written) research mathematics in this ecosystem.

On artificial intelligence companies

Cui bono? Of course, someone is enriching themselves from the latest AI advances in mathematical progress and the cultural power which comes attached. How should we feel about the companies which are building these scientifically impressive technologies? What about the immediate impacts of these technologies? To state the obvious, LLMs consume vast quantities of electricity; lead to the creation of widely unpopular data centres; introduce algorithmic opportunities for bias and discrimination; and are increasingly harnessed for warfare. The issue of aligning them with human incentives is a serious one that remains unsolved; these technologies come laden with catastrophic risks. As I write this, a populist anti-AI backlash is already brewing.

Like many of us, I myself have used LLMs in the last year. Much like the internet, I find these LLMs extremely fun to use, and I think they are helpful for deciphering mathematical content and learning mathematics. I am sure they will only get better and make fewer mistakes. I am sure that using LLMs will make human mathematics more productive. Should I continue using them or desist from doing so? What career consequences will I face if I don’t use them?

The main objection I keep coming back to is that they are creating a society and a future that I broadly do not want to live in. They threaten the community of people that I personally care about. I feel I owe a debt to my mentors and friends to preserve our community. If that involves me desisting from using LLMs, I would be more than happy to never use them again. The problem is that our mathematical community will be wrecked anyway if we do not cooperate.

A call to arms and some prescriptions

As time goes on, the use and uptake of these technologies by all industries will only become more normalized. But as mathematicians, we are in a special position compared to many other industries. We do not have heavy profit incentives which force us to use these technologies. We have an extraordinary amount of professional freedom and latitude compared to many other careers.² We by and large do not have ethical obligations to use AI.

Moreover, these AI companies need us. They need us to validate their proofs; to certify their write-ups; to use their free subscriptions; to work for their companies. They need us to imbue their products with soft power and cultural prestige. We are just one of many industries that they are prepared to disrupt. I am sure that many users are out there prompting an LLM: “Do a breakthrough! Do a breakthrough!” This strategy is unlikely to lead to success without serious mathematical training; indeed, even if a breakthrough resulted, the reader is unlikely to be able to understand the proof. As of this moment, mathematicians are still required for the production of AI-assisted mathematics.

In light of that, here are some prescriptions for the near-term:

1. Do not work for AI companies. This includes unpaid labour. Do not intern there; do not seek employment there; do not verify proofs for them. Take an explicitly hostile stance towards this industry.

   1. Work in AI safety, but judiciously. It is of critical importance that more people work in AI safety. However, ensure that you do not help legitimize and hasten the production of disruptive models; take an adversarial stance, and do work that reflects your values. I am skeptical that the net impact of the AI safety teams of companies like OpenAI and Anthropic is a good one.

2. Refuse all gifts and partnerships from AI companies. Do not take free or subsidized subscriptions; do not engage in “scientific partnerships” in which they pay you some heavy chunk of money in order to extract prestige from your scientific project.

3. Use LLMs for mathematics judiciously, or desist from using them entirely. Include chat logs with LLMs in mathematical papers.

   1. My personal stance at the moment is that, in the current climate, it is unethical to ask a large language model to prove a new result or theorem. Until we establish firmer professional norms about what counts a contribution, we should avoid engaging in "arbitrage” by using LLMs to quickly prove results.

   2. In particular, asking LLMs to prove new theorems can lead to unintentional scooping of human mathematicians who are concurrently proving these results. I am already aware of cases like this. Consider the following two scenarios:

      • Scenario 1: Mathematician A and Mathematician B are both trying to prove on Statement X. Mathematician B finishes proving Statement X before Mathematician A and publishes the result.

      • Scenario 2: Mathematician A and Mathematician B are both trying to prove statement X. Mathematician B asks an LLM to prove Statement X, verifies the proof, and publishes the result before Mathematician A finishes proving statement X.

   3. I am not sure why if I can articulate a coherent moral theory that makes Scenario 1 morally permissible and Scenario 2 morally impermissible, but I nevertheless think we should attempt to avoid Scenario 2 as much as possible in the near future.

   4. I personally find using LLMs to learn classical mathematics a morally grey area, although I am willing to revise my opinion on this. Others, like my friend Marcel, prefer to abstain from using LLMs entirely.

What could an ethical model of using AI in mathematics look like?

Imagine the following scenario. Field A has spent the last 80 years trying to prove Famous Open Conjecture. Mathematicians in Field A would like to make some amount of progress on this conjecture and inject new and diverse ideas into Field A. 60% percent of mathematicians in Field A would prefer to use LLMs to attempt to attack Famous Open Conjecture. In light of this majority opinion, they proceed to use LLMs to collaboratively solve Famous Open Conjecture, which introduces many interesting new definitions and conjectures into Field A.

A priori, I see nothing wrong with the model of doing mathematics outlined above. In fact, it is not so dissimilar to the funding model in fields like astronomy, where people put large amounts of money together to build and use high-technology telescopes. For some Famous Open Conjectures, I might be in the 40% who would prefer not to use LLMs, but I am not the only mathematician.

What is important to me is that these decisions are collectively made by mathematicians. I would like us to maintain a choice in how we do mathematics. I am angry about the exploitation of our field by AI companies and the work that other mathematicians have done to collaborate with those companies.

A dose of reality

I am broadly cynical about the future (for many, many reasons). I think it is likely that no matter what we do, the way in which mathematics is conducted will change forever. A technology, once introduced, cannot be put back in Pandora’s Box. However, I would encourage others in the mathematical community to act now to safeguard our values and preserve the mathematical community. To build a future we would like to live in, we cannot be passive and preemptively surrender.

Right now, we are living in the Wild West; these technologies have been freely introduced to the public. As a result, individual actors are using them in a variety of ways. Morally confusing scenarios are already playing out across mathematics every day. Each month heralds a new result, and we should expect many big conjectures to fall in the next year.

I urge that we quickly act to instill cultural and professional norms that safeguard the practice of mathematics. To me, this involves desisting from using LLMs to prove new results, and I wish that as a community we could agree on doing this for the time being. Of course, OpenAI and similar companies will use their models to prove new results anyway. In light of that, we must accrue political power and we must campaign for regulations against these companies which directly threaten our livelihoods and those of many other people.

About the author:

I am a rising 2nd-year PhD student at the California Institute of Technology. All the views espoused in the essay above are my own. I am 24 years old and have spent the last seven years of my life studying mathematics.

Further reading:

• Mathematics without Mathematicians, by Fernando Borretti.

• Existential Risk from AI: An Exposition for Mathematicians, by Xiaoyu He.

• The Dark Night of Mathematics, by Kirwin Hampshire.

1. I should mention that there had been some spectacular recent progress on this problem in the last 6 years; in particular, just two years ago, a closely-related conjecture called the Aldous-Lyons conjecture was resolved in the negative, by work of Lewis Bowen, Michael Chapman, Alexander Lubotzky, Thomas Vidick. They proved (non-constructively) that there exists a unimodular random graph which is not sofic; in particular, Cayley graphs of finitely generated groups give rise to unimodular random graphs.

I should also mention that the disproof of the Aldous-Lyons conjecture in turn built upon ideas from the MIP*=RE paper of Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen. This paper is also one of these papers with techniques that felt (at least to many operator algebraists) like they came from outer space: the authors used computational complexity theory and quantum interactive proof systems to disprove the Connes embedding conjecture, a problem originally formulated in the field of operator algebras. I write these historical details to emphasize that any new result in mathematics is built upon previous work, and is really the result of a community’s worth of effort.

2. Meanwhile, a friend of mine in software engineering told me that it is impossible for him to abstain from using ChatGPT and remain employed at his current company. Much of his job now involves reading subpar LLM-generated code and maintaining it.