AI can do math. What now?

Léonard Brice, 02/08/2026

The news was not widely discussed in media, but it produced a small earthquake in our research communities: in May 2026, the company OpenAI announced that one of its internal Large Language Models (LLM) had solved an 80-years-old open geometry problem, in a fully autonomous fashion.

This achievement comes after a few similar announcements. In the long list of math problems that were proposed by the Hungarian mathematician Paul Erdős, several of them were announced to be solved by AI (often even by the publicly available models) in the previous months. But the nature of the accomplishment was always questionable: the first ones appeared to be already solved somewhere else under a different wording, and the other ones were considered, after the fact, quite simple problems that would have been solved easily by a human if someone had spent enough time on it. The unit distance conjecture lies however undoubtedly in another category.

The wording is simple: given n points to be set anywhere on a plane, what is the maximal number of pairs of points that can be at distance exactly 1cm from each other? For example, four points can be arranged in a square, where each edge is 1cm long. By flattening the square into a diamond, one diagonal can be made 1cm long as well: that maximal number is then 5. With a large number of points, the Erdős conjectured in 1946 that it was impossible to do significantly better than arranging them in a simple grid; a conjecture that many expert mathematicians, including Fields medallists, tried to prove or disprove, until OpenAI’s LLM succeeded (and disproved it, by exhibiting a much more intricate construction)¹.

A few years ago, AI was barely able to solve secondary school math problems; but it has now become clear that logical and mathematical reasoning, one of the abilities that we have always considered monopolies of the human genius, will be shared with machines that even outperform us, not only by working faster, but by producing some reasonings that humans fail to have. Let us leave aside all fantasies about artificial superhumans, which seek to see in AI the moment where we became gods. AI is a machine, and as a machine, it automates tasks. What we urgently need to discuss is which tasks it can automate, and which tasks we want it to automate.

Humankind will not benefit from an AI that passes Turing’s test (that is, that cannot be distinguished from a human through exchange of written messages), and that ensures an always available but fictional presence, in a society where actual social interaction fades away. Neither will it from AI systems embedded in fridges, glasses, and light bulbs for the mere purpose of selling those items at higher prices, disregarding the tremendous material and energy cost of such a blind large-scale implementation – not to mention all security issues each such item would face. But it can benefit from AI systems performing tasks we would have collectively considered worth delegating to them: reducing painful workloads, curing diseases… and, why not, solving math problems. But this cannot happen without deeply transforming research (and it already does), nor without questioning the relation we have with our own knowledge and intelligence.

What kind of math does an LLM do?

It is not yet clear what the problems are that AI can solve, and above all how it will evolve. Is AI only able to mimic human reasoning, and therefore to solve problems humans can solve? The latest news suggests the opposite: OpenAI’s LLM solved the unit distance problem by using results from two different fields, discrete geometry and algebraic number theory, and techniques that could have been usable only by someone who is expert in both, which is not easy to find. As the mathematician Thomas Bloom says with some emphasis, “AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries”. AI compresses and mobilises knowledge that no human being can master entirely.

Moreover, one should bear in mind that AI training is made of several steps. LLMs are initially designed by being trained on huge amounts of already existing texts, mostly found on the Internet – including math manuals and research papers. From that, it learns how to generate sentences that are plausible compared to the corpus it has been exposed to – like an extremely powerful smartphone autocomplete tool. This phase is usually called pre-training, and is followed by fine-tuning, where it learns how to behave as desired, based on interactions with humans who give positive or negative feedback on the content it generates. But more and more of the AI power comes from an additional layer, automatic reinforcement learning: the model is tested on tasks for which its output can be objectively and automatically verified, and learns how to perform them in an autonomous way. Unlike fine-tuning, automatic reinforcement learning can be carried out at a speed that is unthinkable when there is a human in the loop; and unlike pre-training, the AI generates its understanding of how to carry out tasks based on its own experience, not on the human way of doing. AI does not only repeat a mashed version of what it has been trained on: more and more, it creates its own knowledge.

One can then ask the opposite of the above: will AI be able to solve all problems a human can solve, and beyond, making human intervention futile? This question is obviously much harder to answer, but what is certain is that we are not there yet. OpenAI’s breakthrough may also be explained by some peculiarities of the unit distance problem that made AI intervention particularly useful. A summary of the model’s chain of thought is available online, and suggests that it has quickly concentrated its efforts, after analysing where the attempts to prove the conjecture have failed, on trying to construct a counter-example. This is very similar to what a researcher should do in the same situation, although the widespread belief that Erdős’s conjecture was true misled most mathematicians into the wrong direction. But the strength of AI is speed, and patience: it can explore thousands of directions that humans would discard at first glance, and check in a few seconds an example for which a human would have required hours. In the model’s chain of thought, this ability to explore seems to have played a significant role – together with the sudden idea of using tools from a completely different field. This highlights two aspects of mathematical reasoning in which AI can easily outperform humans: example exploration, and recognising which tools can be useful in a given situation. Both were very useful for this specific problem, but may not always have the same weight.

What about the future, then? AI has limitations, but progressively overcomes them. Hallucination is of course the main issue: since an LLM aims to generate text that looks plausible with regards to its training, it will often tend to write proofs that contain handwavy arguments, or even use theorems that never existed – a very frustrating experience that all researchers have now faced. But LLMs are also increasingly good at reading proofs, understanding them… and spotting mistakes and shortcuts. And if that is not enough, they are not alone there: a concurrent dynamic research field is that of proof assistants, that is, logic-based software that can verify proofs and guarantee their correctness; provided that proofs are written in a specific language that they can interpret. So far, proof assistants have not succeeded in imposing themselves as a routine tool for substantially long proofs: they require too much precision, and phrasings such as “it is obvious that”, “the same reasoning applies”, or “by a classical result”, often legitimate, are not an option. OpenAI’s proof is written in plain English, and had to be verified by flesh-and-blood mathematicians. But proof assistants are already used for reinforcement learning, and will probably play a more and more important role, now that AI has reached this stage.

If some human-solvable problems keep resisting AI, that would probably be due to the limited presence to the world that AI has: despite its outstanding performance, it exchanges with its environment only through chains of zeros and ones. Researchers, on the other hand, often find inspiration by drawing on a board, or by exemplifying their problems on instances that mean something to them. Intuition is a mysterious process, hard to explain after the fact; and it might be the case that it sometimes originates in areas of our brain and of our environment which have supposedly nothing to do with the problem we are studying, and to which AI may never have access. It is in that regard worth noting that the unit distance problem was solved by a generalist AI, and not by one specialised in writing proofs and trained only for that purpose. Should we deduce that all this other knowledge, from poetry to economics, also helps AI, in an indirect and diffuse way, to be smarter at math? If that is the case, then there is still hope that the knowledge that we have and that remains alien to AI, the one that we get from our senses, our emotions, and our immediate experience of life, sometimes helps us in the same way to beat it on its own turf. But that hope is quite uncertain.

What researchers and AI can do together

For now, and most likely for still some time, AI and human researchers will have to work as a team. In 2024, the famous mathematician Terence Tao compared an OpenAI model to a “mediocre, but not completely incompetent, graduate student”. With this comparison, which graduate students have probably all appreciated, he meant that he could use it to outsource some parts of his work, provided that he supervises its reasoning, suggests directions, and flags its mistakes through repeated interactions. I may have shared that impression at the time, but in 2026, it no longer accurately describes the situation. Personally, my use of LLMs in my research has significantly increased across the past few months, but never really to outsource segments of reasoning. In line with what I said earlier, I use it mostly to suggest techniques and results from the literature that could help me (which is closer to what a supervisor does) and for example exploration (for which it exhibits a patience neither a graduate student nor a professor has). Concretely, AI-assisted research takes the shape of a long dialogue between the researcher and the machine, where each compensates for the other’s weaknesses.

Comparing AI to a specific category of researchers is quite reductive for both sides, and hides the fact that AI is taking a genuinely new position which is not yet stabilised, and which is about to reconfigure the whole landscape of the research world. To some extent, this is just a new step of an evolution that is far more ancient. Calculation (including for the purpose of example exploration) used to be a painful task of researchers, whose barriers have been pushed back decade after decade. And as for recognising which tools and which theory are useful to address a problem, it was already much easier five years ago than forty years ago, when researchers had to search their institutions’ libraries every time they needed some literature. Each such step in this technical evolution made us more efficient, and was in many respects salutary, since the cumulative nature of science entails that we study more and more complex matters and build on a richer and richer ground. However, they did transform research, and even though it made us more efficient, it is worth wondering whether they lead us to the right direction.

A cognitive crisis?

The most obvious negative impact those tools may present is on researchers’ training. Working on a problem almost always serves a dual purpose: solving the problem itself, but also allowing the researcher to acquire new technical skills and a new understanding of the field. After all, before we can actually produce genuinely new results, we all had to solve hundreds of already closed problems during our schooling, from primary school to Master’s degree. And even after starting actual research, it usually takes years, if not decades, to acquire the deep understanding that makes a researcher able not only to solve problems, but to build theories. The arrival of AI seems to mean that we will need fewer problem-solvers, and more theory-builders: but how to create theory-builders, if not through problem-solving? Will we need to spend more time forcing ourselves to work on problems that AI could solve, only for training purposes? Will we have to extend Bachelor’s or Master’s studies, or to make study-only PhDs, asking students to write their theses on questions that AI has already answered?

And the question applies not only to individual researchers’ training, but to the whole science community. We do not solve problems only to solve problems: most of the time, we do it to create a broader understanding, individually and collectively, of the objects we study. And this will be a crucial question, that I would expect to appear in front of us in the coming years, if not months: what does it really mean for science if machines generate and verify a proof for a theorem, but no human being really understands it?

One may argue that, if outsourcing tasks to AI teaches us much less than performing them alone, this impact is compensated by the acceleration made possible: we learn much less from an individual task, but we conduct many more of them, and may end up learning roughly at the same rate. Quantitatively speaking, this hypothesis is reasonable. But the critical point is qualitative: we do not learn the same thing, nor in the same way. And we do not even know what we need to learn anymore.

This loss of bearings is already palpable, especially among undergraduate students who realise that so many things they have learnt are now easily done by some LLM. As a teaching assistant, I have seen how this creates a new distance between students and the content they are taught, epitomised in the temptation to use AI to cheat in exams – we would be quite wrong if we interpret that as a mere new tool in a cat-and-mouse game. We are facing a crisis in our relationship with our own cognitive abilities; a crisis corresponding to what the French sociologist Émile Durkheim used to call a situation of anomie, where the sudden lapsing of social norms generates a deep feeling of insecurity.

It is not the first time that humankind has faced such a crisis. In an ancient Greek society that was progressively abandoning its ancient oral traditions, Socrates was known to reject writing: unable to create a dialogue with the reader, unable to adapt to them, and conducive to creating the illusion of knowledge in a person that reads, but does not really understand. His fable on the matter, as reported by Plato, sounds strangely familiar: Socrates tells us how, when the god Thot offered writing to the Egyptian king Thamus, the latter declined it with these words: “this discovery of yours create forgetfulness in the learners’ souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves.” Interestingly, Thot’s gift is presented as a pharmakon, a Greek word that means simultaneously cure and poison.

The introduction of printing in Europe, in the fifteenth century², did not only spread new ideas: by making the written records of knowledge much more accessible, it also entailed a long process of redefining the norms of cognition. Across Antiquity and the Middle Ages, the cultural conception of a powerful mind gave a strong importance to memory. Illiterate low-class workers needed to remember everything they had to know; monks and intellectuals would spend years learning the full text of entire books, and the ones performing the best at that task were also the ones more able to produce new intellectual content, and were admired and celebrated as such. Of what use, however, was that ability in a world where printing was suddenly making books widely reproducible and accessible? In two centuries, learning long texts by heart became a mere game for bored and old-school aristocrats. There starts a long process of reconstructing social norms around cognitive abilities: through the Enlightenment, the focus slid to reason, and it is only with the emergence of psychology as a discipline in the nineteenth century that the modern concept of intelligence, as an individual and measurable skill, was forged. Crucially, memory is still necessary in every reasoning process, and as such, it is still one aspect of intelligence among others: but it is no longer the bottleneck, and that made it possible to construct a broader understanding of what our brains are actually able to do.

This modern concept of intelligence has already been challenged, from the very beginning of computing. The Internet has, of course, extended the impact of printing to the role of memory, by making a large part of human knowledge almost immediately accessible. Mental calculation, which used to be considered an immediately visible sign of intelligence, does not have the same importance now that we all have a calculator in our pocket. Around the 1950s and 60s, computing has even brought such optimism in the scientific community that many researchers believed that there would soon no longer be hard or interesting problems in pure mathematics, and that future challenges would only lie in modelling the world as it is, that is, in physics, biology, and other applied disciplines.

In parallel with this movement, the concept of intelligence is continuously questioned, with approaches that, once again, tend to enlarge it. The notion of emotional intelligence, for example, was created only in 1990, and would have been considered an oxymoron a few decades earlier; it is now widely used. This is already a huge cultural evolution, and anticipates a new understanding of cognition of which intelligence, or what we nowadays mean by it, would only be a narrow aspect, analogous to what memory is to intelligence. But building that new understanding will most likely take decades; and in the meantime, we will have to navigate the unknown.

Alarms about AI intrinsically making us lazy, stupid, or unable to learn anything new, must therefore be taken with a pinch of salt. But perhaps the main danger lies elsewhere: like books, computers, and mobile phones, AI augments our cognition with an external tool, and thus creates a new dependence on something that is not completely ours.

The pope, Karl Marx, and Elon Musk

In his encyclical Magnifica Humanitas, Pope Leo XIV called to “disarm AI”. His point of view is summarised in the opposition laid out in the introduction: the development of AI may follow the path either of the Tower of Babel, or of the reconstruction of Jerusalem. On the one hand, a megalomaniac project competing with God himself, in the name of an artificial unity in which all differences are denied – one people, one language, one monument. On the other hand, a truly collective work, where each person brings their contribution, in their singularity. No need to be a devoted Christian (I am not, if the clarification is needed) to understand the metaphor.

The encyclical’s focus on AI remains very general, and constitutes only a cautious call to think further, as is usually expected from a pope. But this framework can give us an angle to question the introduction of AI in research. From a model where each scientist had their area of expertise and their way of thinking, we slowly slide to one where a substantial part of our ideas come from the exactly same LLM that is used by thousands of other scientists around the world – and it is harder and harder, when I write a paper, to say which part could not have been written by someone else. By gathering all pieces of the “cathedral” of mathematics into one universal tool, AI presents an obvious risk of eroding its diversity. Science may face what agriculture faced: a colourful landscape of multiple practices and varieties being replaced by monocultures, better suited to massive production on the short term… but deprived of alternative options, and at the end of the day unable to adapt to changing conditions. A new Tower of Babel?

To follow the path of the reconstruction of Jerusalem, researchers need to keep, with their singularities, control of the knowledge production process. This is precisely what is challenged, when larger and larger parts of that process are made autonomously, outside of their minds. Here, a surprisingly relevant author is Karl Marx, in his Fragment on Machines in Grundrisse, and later on in Das Kapital. In Marx’s theory, machines capture the workers’ know-how into matter, dead labour, of which the workers, living labour, only become an appendage. The industrial revolution has therefore converted millions of artisans and farmers, who used to have full control and a deep understanding over their production process, into proletarians that only perform a fragment of that process somewhere in a factory. In his view, humans fulfil themselves through the conscious transformation of the world; by dispossessing workers of their knowledge and control, and by making labour alien to its purposes, mechanisation is therefore a source of alienation – a concept he develops in the 1844 Manuscripts. This evolution culminated long after Marx’s intuition, under Taylorism, when it was theorised by capitalists themselves. But what neither Marx nor Taylor anticipated was that in the twenty-first century, the very category of labour that used to be the most protected one, cognitive labour, has become the most threatened.

Marx predicted a disappearance of artisanal craft and traditional agriculture, which never fully happened: even though they compete with an industry that makes much cheaper products in larger quantities, they marginally survive because some part of the population is willing (and affluent enough) to support it, and keeps consuming the goods it produces. The same phenomenon is likely to happen at large scales with art, for example: even if LLMs become able to write excellent novels, reading is also a dialogue with an author and as such, AI-generated texts don’t hold the same interest. But as for science, I don’t believe any engineer will be willing to use only theorems that were proven without AI intervention.

What makes this framework relevant is that if technology is not neutral, it always serves some human purpose. The control workers used to have over their work in artisanry does not vanish: it is transferred to the company that owns the machine, or that produces it – and to those who own that company. AI is currently being developed by a handful of huge firms, whose names are known: OpenAI, Anthropic, Google, Meta, Mistral, DeepSeek, xAI. All of them seek profit, and are bound, with different approaches, to some geopolitical bloc and some political project. And the choices made in the types of AI they build reflect those allegiances. The control that some of those companies are progressively getting over the making of science should be considered the most worrying aspect of this evolution – all the more so when some of their owners seem to develop a strange passion for Roman salutes.

This may be the first question OpenAI’s achievement should have raised: what if the company decides, tomorrow, that it wants to keep its more powerful models for itself? Do we want to live in a world where a private firm, alone, owns tools that can solve any problem before any scientist had the time to consider it? Or, at least, where a few persons can choose which field of research can benefit from this enhancement, and which will lag behind? Yes, we all have the same fields in mind.

A very temporary conclusion

Never in human history have we seen a technology being developed so fast. Advances are now measured in months, and predictions are sometimes overturned within a few weeks. When you read these lines, many aspects of what I describe will probably be outdated. Even in my personal testimony, I am only talking about a tool I am currently learning to use – a tool that continuously transforms itself. Like so many of us, I anxiously feel the ground moving below my feet.

The closest to a certainty that I have is the following: AI is changing everything about science, and there is no way back. But there are multiple ways forward. Huge possibilities are now open, to produce new knowledge; but it will benefit us provided that it remains science, that is, production of a new understanding of the world we live in, enabling humanity to improve its dominion over its future. That is only possible through a Jerusalem AI, in which the choices made are the outcome of a truly democratic process. We therefore need to hold both ends of the rope at once, to propose a different direction: on the one hand, the governance, and the place of private companies, states, academia and democracy – here, proposals such as the CERN for AI³, or any idea to develop AI outside of the control of a private company or a specific state, are an interesting start. On the other hand, the technical and scientific challenge: is the AI we know, with its gigantic and fully centralised neural networks, with its training processes pumping up our private data and exploiting precarious click workers, the only way to make AI? Among the technical choices that have been made over the past years, which ones are scientific breakthroughs, and which ones are simply economically, politically, and ideologically driven? Answering this question is now one of the most important tasks academic computer scientists have in front of them.