Mathematics is in turmoil following the increasingly striking results obtained using frontier LLMs. Understandably, mathematicians (me included) fear for their jobs and worry about changes to the status quo, and many recent statements and opinion pieces reflect this. However, these often conflate such concerns with the aim of mathematical research and with the value of mathematics in education.
Pure mathematics sits on the borderline of science and art. It grew out of attempts to describe the natural world, but, unlike most sciences, it is not experimental. Large swathes of mathematics were developed because people found the subject aesthetically pleasing and its questions intellectually challenging. Interestingly, many of these results turned out to have practical applications later on. This duality has often made it difficult for mathematicians to justify their work to the general public, including funding agencies.
Recent media statements, petitions, and personal opinions (I will intentionally not single out any particular author) argue that the goal of doing mathematics is for humans to develop personal intuition and understanding, and this must come through the slow, contemplative approach that has characterised mathematical research until now. Some even argue that certain problems should be ring-fenced from AI attempts, while others want a blanket ban on AI-assisted mathematics. To a non-mathematician, this might sound strange. Surely, the goal of mathematics should be to advance public knowledge and understanding, not to provide researchers with a rewarding journey or opportunities for personal growth. Questions about sustaining the mathematical community, education, and human agency are important, and I return to them below. Here, I want to consider another question: what justifies deliberately excluding a route to a discovery that would benefit the public?
Let’s first take a utilitarian view and replace mathematics with medical research, and the problem immediately becomes apparent. Imagine cancer researchers stating that, even though we could find a cure for cancer in a few weeks using AI, what really matters is the human process of running numerous experiments and the deep understanding the researchers gain from them. Consider the suggestions to exclude AI from mathematical problem solving simply to preserve the human process of discovery: either the problems mathematicians study are unimportant, in the sense that solving them has no value for the wider public, or these suggestions go against advancing science and the public good. If the former were true, that would raise the uncomfortable societal question of why we pursue these problems. Now assume that the problems are indeed important, which is my personal opinion. For example, dynamics, partial differential equations, and topological data analysis have all been applied to research on cancer detection and treatment. Even seemingly abstract branches of mathematics can have future applications or provide foundations for more immediately applicable areas. When an LLM produces a checkable, human-readable proof rather than a yes-or-no answer, that proof and the new techniques developed become available for study and generalisation. Those proposing to exclude the use of AI from tackling certain problems should explain what public benefit justifies their position.
Prohibiting the use of AI in mathematical research to preserve a familiar form of human work even when AI can produce substantial discoveries is, in my view, a form of Luddism. Imagine a coachman threatened by the first cars arguing that the real goal of travel was not getting from A to B, but rather experiencing the journey itself and taking in the beauty of nature and the cultural experiences along the way.
Even if we do not take such a utilitarian view but consider mathematics to be more art than science, it should enrich society as a whole culturally. It is not clear to me why a purely human solution is superior to an AI-assisted discovery when presented to the general public. Value can still be attached to a human proof or theory as a human achievement, as with handcrafted products. I think the two can coexist, and the human study of AI-assisted results can enhance our understanding and the aesthetic value of the subject.
Some argue that when big open problems are solved, the corresponding areas of mathematics will lose their guiding beacon. While I agree that certain problems have been instrumental in guiding research, saying that reaching these goals is damaging makes one question the importance of those goals in themselves, for the aforementioned reasons. Most solved problems open new doors, and our horizon broadens, leading to deeper questions we can ask. This is how mathematics has always functioned. It will just happen at a faster pace now.
Mathematical activity roughly falls into two categories: problem solving and theory building, with some overlap, of course. Until now, the majority of mathematicians seemed to value problem solving highly and many viewed theory building as a means to an end. The older the problem, the better. This is evidenced by publications in top journals and prizes awarded (with some notable exceptions, such as Grothendieck). Solving a famous 100-year-old problem has been the pinnacle of one’s career and has almost guaranteed publication in a top journal. Now that AI has surpassed the problem-solving ability of humans in some areas, views have started circulating that these problems are just proxies for developing theories, which is what really matters and something AI is not yet good at. However, even that might soon change.
In what follows, I am not excusing the recent conduct of OpenAI over alleged failures of attribution and confidentiality, but would like to draw attention to the fact that such conduct is unfortunately more common than the community would like to admit. Several interesting statements have appeared in the press after OpenAI’s announcement of a proposed solution to the Navier–Stokes problem. These statements portrayed an open mathematical community where scooping is rare and punished. The history of mathematics is full of famous cases of scooping and controversy. I always think twice before discussing any of my ideas and ongoing projects with others. If they are honest, it can lead to receiving helpful advice or even to a fruitful collaboration. In less fortunate cases, they might just run with the idea. Sometimes, this is intentional. Such conduct is unacceptable, but nothing new, unfortunately. It can also happen subconsciously, as follows: the individual forgets the conversation, then “independently” rediscovers the idea. This is a bit like an LLM having the conversation in its training data (I am not claiming this was the case for Navier–Stokes, and OpenAI has officially denied this happened).
We need to separate the discourse around competition from the need for proper attribution and how ideas communicated in confidence are handled. Some were upset that OpenAI jumped in after it heard rumours about Alpöge and Buckmaster’s progress. They claimed that this is not how mathematicians operate. Again, I would dispute that claim. Many mathematicians grow up taking part in various competitions and go through competitive university admissions and exams. Getting jobs is extremely competitive, with relatively few reaching tenured positions. Competition is ingrained in the culture. The first person to solve a problem gets most of the accolades (publication in top journals, prestigious jobs, and prizes). There have been numerous competitive situations in history. The main difference in this case is that a big company entered the competition. I do not know exactly what OpenAI’s main motivations were. In my collaboration with DeepMind, however, I saw several motivations at work: improving and benchmarking AI systems, furthering science, and generating publicity through major breakthroughs.
Once large corporations solve a few more famous problems, they will likely lose interest in pure mathematics, as the publicity and benchmarking value of these achievements will diminish. My expectation is that scientific interest alone will not sustain comparable investment once the publicity and benchmarking returns decline. It is not yet clear whether mathematicians will have the computational resources required to continue making such advances. For now, maybe we should be grateful that big corporations are willing to invest so heavily in solving some of our most challenging mathematical problems. For universities to remain competitive and independent of corporate interests, it seems desirable for groups of universities to host open-weight LLMs in their own data centres. Such efforts are best coordinated at the national level and through funding agency policies.
Some have stated that OpenAI’s work damages mathematics as it did not arise from the usual process of doing mathematics in the community. In this respect, there are parallels between the proposed solution to the Navier–Stokes problem and Wiles’s proof of Fermat’s Last Theorem, though they were obtained on different timescales. Wiles worked in complete secrecy for seven years until he announced his result. OpenAI obtained the proposed solution to the Navier–Stokes problem internally over a few days. People are free to inspect OpenAI’s manuscript and try to understand it in depth. They can even use AI to help analyse the proof.
A separate issue is how AI contributions should be acknowledged in research. According to recent publishing policies, AI cannot be a co-author and one must include a detailed AI statement about how exactly AI was used. This seems to go against the spirit of how mathematics has been done until recently. Firstly, mathematics is very egalitarian. In mathematics, unlike in many other subjects, authors are listed in alphabetical order, and their individual contributions are rarely specified. It is possible that one of the authors simply asked a question and outlined a few ideas, which the other authors worked out. That does not diminish that author’s role as far as the outside world is concerned. The Hungarian mathematician Paul Turán said that asking about such details of a collaboration is like asking a couple about their bedroom secrets.
Secondly, if it is deemed that discussions with XYZ do not warrant including that person as an author, they are typically mentioned in the acknowledgements as follows: “We thank XYZ for helpful discussions.” I have seen very few papers where details of what those discussions entailed were given. However, now that one can have helpful discussions with an LLM, suddenly some are calling for mathematicians to give a detailed account of what the LLM’s contributions were, and maybe even provide a full transcript of prompts and LLM responses. That may be straightforward in some cases, but imagine a scenario where the authors conducted hundreds of conversations with an LLM over many months. How is it possible to decide who contributed what? Also, some of the questions the authors ask an LLM may seem silly in retrospect, so having to publish all prompts could lead to self-censorship. Disclosure should identify the system and its broad contribution. It should not become a demand to reconstruct the origin of every idea across a lengthy collaboration.
I think the cleanest solution would be to simply list the LLM as an author if the contributions were substantial, and mention it in the acknowledgements for minor contributions. The main objection to this proposal concerns accountability. Under current publishing ethics standards, authorship is not only about credit for intellectual contribution, but also about accountability, and hence publishers exclude AI systems from being authors. The Committee on Publication Ethics, a nonprofit organisation concerned with publication ethics, also points out that AI tools lack legal capacity to make conflict-of-interest declarations or manage copyright and licensing agreements. I would distinguish credit from responsibility: an LLM could receive authorship credit for a substantial contribution, while at least one human author remains accountable for the paper.
In the summer of 2025, I was of the view that if AI became so capable that it started generating meaningful papers by the thousands, then humans would be unable to process all the new information, and the whole endeavour would lose its purpose. However, recently I realised that this would still be of considerable value: we would have an extensive database of mathematical results that one could turn to when trying to answer a question arising from a particular real-world application. This mirrors the way some pure mathematics results found applications in the past. The main difference is that AI would produce and retrieve the results. This could take the form of a version of arXiv for AI. AI agents would submit papers written in natural language, each accompanied by a Lean certificate. AI would also moderate the submissions and retrieve relevant results.
So is this the end of human mathematics? AI may profoundly change how research mathematics is done. Here are two reasons why humans should remain involved. First, we cannot completely give up control to AI. A loose analogy is that a commercial airliner can pretty much fly itself automatically, but we still have pilots in the cockpit to make certain key decisions and in case things go wrong. For humans to have agency, it is important to have enough people who can understand and direct state-of-the-art mathematics.
Second, mathematics must remain part of education and human culture. And this is where many of the opinions about the value of deep thinking, gaining personal intuition, and restricting AI use fit in. I strongly believe the main purpose of mathematical education is to teach people how to think logically. Few people will ever need to be able to solve quadratic equations or use the Pythagorean theorem in their professional careers (though there are plenty of jobs where they are useful), but logical thinking is a skill that is transferable to any walk of life that requires problem solving. The perseverance and deep thinking that solving mathematical problems develops are qualities that we must strive not to lose. As over-reliance on AI can prevent students from developing these qualities, it seems desirable to limit AI use in the learning process and preserve human-led education. Restricting AI during learning is different from restricting discovery: the aim is to develop the learner’s abilities, not to keep a problem unsolved.
To ensure that grades reflect students’ own understanding, I would restrict graded assessment to invigilated written and oral examinations. It is difficult to establish which parts of a project a student produced independently as a declaration of AI use does not itself verify the student’s contribution. A subsequent invigilated assessment can gauge understanding, but understanding a solution is not the same as having developed it. A project submission alone therefore cannot be relied upon as evidence of the student’s independent problem-solving and research ability.
Teaching graduate-level courses requires people who can engage deeply with research. Hopefully, intellectual curiosity and the mission to educate the next generation will continue to draw people into the profession and sustain a human mathematical community.
We should preserve the human ability to understand and explore mathematics, not preserve unsolved problems merely to protect a familiar way of working. The goal should be to enlarge public mathematical knowledge and understanding, whether the discoveries are made by humans, AI, or a collaboration between them.
Crossposted from my homepage.
Received 22 September 2026.
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