We met at the publication of your book, On the Other Side of the Blackboard, written with Emmanuel Sander. We were only just beginning to touch on artificial intelligence in mathematics. Since then, much has happened, and I know this period has been transformative for you.
Transformative is the right term, in the sense of a shift in practice, a kind of revolution. It was one of the very reasons we wrote this book with Emmanuel Sander. We watched artificial intelligence advance month after month. A few years ago, when you asked these systems how much is 5 + 3, they answered arbitrarily; but progress was visible, and we felt the need to explain what human mathematical practice looks like.
What we have seen in the last two months is the moment when the most advanced models, those perhaps not yet accessible to the general public but available to large companies, have become better than the best humans at solving conjectures—open problems, questions we did not know how to prove. This has been happening for two months now. On the eve of our interview on France Culture, for instance, we had a first alert in my field: the largest conjecture in the area had just been solved. Then there was Navier–Stokes, one of the Millennium Problems, labeled as one of the greatest questions in mathematics. And then, on October 6, that moment almost tipping, where we see an automation of proof arriving, with results published by OpenAI: more than 700 at once, precisely 722.
It was a upheaval, because you must understand that we have been working on these problems for a very long time. In my case, these are problems I have worked on since the beginning of my career, for fifteen or twenty years. We become emotionally attached. This attachment drives years of investigations, progress in small steps, months spent stuck on an idea: this attachment is what fuels us. And then, suddenly… I remember, it was one o’clock in the morning, I was scrolling through the list and all the problems, all the big questions I had been asking myself appeared. It was overwhelming, even though, of course, it also opened extraordinary prospects. It marks a sharp shift in the state of research.
Tell us about these problems, among the 722 papers published.
They span all branches of mathematics; OpenAI has demonstrated that its system is effective everywhere, not just in a narrow domain. In my case, it concerns sixteen or seventeen problems that I was really questioning on a day-to-day basis, very precise and very important topics for me. The others matter to others, but I am not the right expert to speak about them.
These sixteen problems are highly specialized, but essentially they ask what happens when matter, or a system, suddenly changes its behavior. You heat water; at 100 degrees it becomes vapor—a sharp change, a state transition. But it could also be a healthy population in which a virus arrives; we unfortunately lived this a few years ago: an epidemic spreads, the population’s behavior changes abruptly, many people are infected, many must be protected. This is the kind of thing we study in my field. We try to understand what phenomena are universal, that is, those that recur systematically in these behavioral changes, what is the right way to describe them, what are the right quantities.
I don’t know how many remember, but at the start of Covid there was the question of clusters, and a strong political will to reduce their size. The cluster effect falls under what is called metastable effects: once a cluster reaches a certain size, it cannot be stopped, it will spread and infect the entire population. You find this phenomenon in many phase transitions, abrupt changes in matter. Since there are many different phenomena, there are many models, each raising different mathematical difficulties, and the results of October 6 demonstrated a lot about these different phenomena.
When we say that AI has demonstrated, what is the nature of these demonstrations, compared to what you had developed yourself?
The first thing we understood in spring is that many people either laugh or worry, depending on their point of view (both sides make their case), about the hallucinations these large language models, like ChatGPT or Claude, can have. They say false things, they make things up. We therefore asked for a moment how these models could become good at mathematics, since in this discipline, saying something false is catastrophic: the entire demonstration collapses. But for months now, what these models produce is checked by the computer itself, which verifies logical coherence. These models no longer really make mistakes. They make them occasionally, but it’s very rare, no more than humans. One must abandon the notion that these proofs are false. There may be a few that collapse, because the machine also relies on a corpus of texts written by humans and can therefore err, but it is very rare: it is as accurate as humans.
Secondly, which we noticed quite early, in April (in my case, I realized this because lightning struck quite near us): AI is very good at making long-distance analogies. This is a topic we discuss a lot in the book. Of course, logic is the cement of mathematics; a proof must be logical in the end. But to find that proof, one must make analogies, be creative, and understand the links between things. Now, AI is by definition encyclopedic: it has no trouble knowing all mathematical articles. I am incapable of that; I have a very bad memory, a human memory. By April, we began to see articles in which there was a long-distance analogy that no human had noticed before. Yet, the proof was still very simple. It was like someone very good at finding the right book, opening it, and saying, “Look, this is what you must do,” but who did not yet have the ability to construct elaborate proofs.
Third point, reached rather in autumn: these artificial intelligences managed to create proofs and become good mathematical architects. When a mathematician proves something, it is a full-fledged architectural work. You must then assemble the pieces, but finding the architecture is the most creative part. And the AIs achieved that this autumn, with very elaborate proofs. In my field there are thousand-page proofs. It is very rare in mathematics. To give a sense of scale, I have written 3,500 pages in my entire career, and a single paper is usually around fifty pages, sometimes less, sometimes more.
Last point: the October automation, which is a form of systematization. They published 722 papers, but they attempted to solve 4,000 problems. They try everywhere; sometimes it works, which is wonderful, and in other cases, it does not. It is a practice that differs somewhat from human operation, which is actually a good thing, because it means the machine will complement human capacities. From this point of view, the fact that the proof is somewhat different is not a problem.
What is the quality of these mathematical texts?
That’s an important point. I’m not among those who find these texts totally unreadable. They are not easy to read, and I think there is a tendency to avoid over-humanizing the writing to prevent mistakes. Indeed, when you ask these systems to write in a more pedagogical way, they still can’t do it without making mistakes. They oversimplify. There is a challenge. But it doesn’t bother me too much, because the key, ultimately, is understanding the proofs. Take into account the volume, even just for the sixteen papers that concern me out of 722: at the scale of the entire corpus, we’re talking tens of thousands of pages in October, and soon hundreds of thousands. If no one understands them, if no one reads them, it’s as if a tree, or rather a large thicket, had fallen in the forest without anyone noticing. Obviously, people must be able to understand all of this and draw ideas from it.
When these proofs are poorly written, it is almost like archaeological work: you must dig to find the gems in a text that isn’t optimal. That may not be so bad, by the way: the joy of discovering the gem, polishing it, reflecting on how it could be reused and the questions to pose to the machine, is thrilling. For my part, I feel a form of intellectual excitement when I comb through these texts to find passages that have genuine mathematical value.
What does “badly written” mean in mathematics? Initially, it was said that AI’s mathematical writing wouldn’t be elegant…
I disagree: I find there are extremely elegant ideas in the papers I’ve read. If these ideas had been the work of humans, I would have been astonished. So you cannot reduce this to a defect in elegance. The fundamental point is that there is a balance to strike between logic, which constitutes the formal part of mathematics (the so‑called formalism), and intelligibility for human beings. This balance is not easy to achieve. One can, for example, create purely formal proofs; these AIs are capable of that, they are formalizations in the Lean language. These formalizations are purely logical and show that everything follows logically. But they are hundreds of thousands of pages for statements that seem intuitively obvious to us. It is unreadable. Therefore one must find a better balance, and that is what the machine does when it tries to produce proofs in human language: it finds the right elegance. It is like having the idea for a superb thriller: there are thousands of ways to write it, some will fail to grab attention, others will reveal more of this or that. There are a multitude of ways to write a proof. Do not think a proof is something fixed that would have only one form.
Today, yes, these are texts that are difficult to read, and perhaps they will always remain somewhat so, because by definition the machine operates and creates proofs differently from us. What it writes is different from our understanding. Moreover, what would be good for me might not be for my PhD student or for my fellow researcher. Everyone has their sensitivities. It is up to us to engage with these texts to create transmissible versions and communicate these ideas more broadly.
Among these 722 problems there was a weakened version of the Riemann hypothesis dating back to 1859. The machine’s text produced so far was not understood. Is that still the case?
I am not a specialist in this field, so I won’t take a position on that. What I can say is that reading a mathematical text is extremely slow. Do not imagine you can harvest two thousand pages and read them in one sitting: even ten pages can take weeks. Even for problems very close to my field, I have so far read only one article that I understand. In fact, I have read two, because I had written one in parallel: I had already solved the problem before the AI, and while reading it I recognized the same patterns, the same ideas, to such an extent that I told myself I didn’t even need to read it. But it will be an extremely slow process. When people say that mathematicians are no longer needed, I point out that only a few dozen people have acquired the expertise to understand these problems, and it will take us months and months, even with that expertise. For some of these problems, I have no idea yet how the proof works; for others, I have read them in passing and understand a little.
There is one thing that has been somewhat shocking for me: we were progressing very well in our field and we were at the interface of several others. We were doubly affected, because we were close to many major questions. Often, what we lacked lay on the other side. I am trained as a probabilist; algebra I can learn, but it’s not my specialty, and becoming sufficiently expert in these areas was very difficult for me. The encyclopedic AI managed to fetch these tools. It took up our ideas, built on what we had written, and that does not bother me. But we were very affected as well because we were not far off the mark.
In the case of the quasi‑Riemann hypothesis, if I understand correctly, people were extremely surprised and some of the ideas produced may be more surprising than those the AI used in my field. For us, I am fairly confident: we will very well understand these problems, because it is, in essence, a natural extension, but extremely impressive, I must say, of what we were doing. In other domains, there may be ideas so astonishing that a lot of work will be needed to understand them. As for a possible error, I do not rule it out, but I am not a specialist in that field. I believe the proof has been formalized in Lean, so there should be no logical fault. But you never know: even without a logical fault, it could rely on a false result or misinterpret a human text. Experts will have to judge.
Among these experts, Maksym Radziwiłł, from New York University, explained that the article on the quasi-Riemann hypothesis was so badly written that no journal would have accepted it for publication.
That, however, is true. I think even the texts in our field, in their current state, would not have been accepted by any journal. I see this more as if someone were telling us: “Here you have data; you’re sent a torrent of data, raw proofs, and it’s up to you to dissect them, to transform them into real arguments intelligible to humans.” It’s a task demanding creativity, emotion, collaboration, exactly as the search for new proofs required before. The change is that we no longer find new proofs ourselves, or at least we will likely have to do it with AI. There will be few completely new arguments produced purely by humans; there will be some, but probably marginal. To claim otherwise makes no sense.
But this work remains to be done. This morning, before we spoke, I was finishing the rewriting of a proof that, in its initial version, was completely unreadable. It had already been revised by a first colleague who had done an excellent rewrite. Then a Chinese team used it to improve the result with AI, but the proof still wasn’t much simpler or more readable. I had to do the work, and now I have a very short proof that I like, that I understand, and that I will share with others. This process does not end when some say: “Okay, you’re done; we have the proof.” That makes no sense. I’ll circle back to the same image: it’s not because a tree has fallen somewhere that you can use it.
In On the Other Side of the Blackboard, you’ve never hidden what resists you, much like rock climbing: you immediately think of sheer cliffs. What goes on in your mind when you face a problem that is hard to grasp?
Several things. First, a mathematician, like a young child, comes with his or her own intuitions. When I begin reading an article, I have intuitions about the problem, mental representations that arise from my experience. Some are mathematical, others not. I also bring my limitations. I am not someone who abstracts particularly well; I have colleagues far more adept at manipulating abstract objects. I come with this limitation. I also have a very poor memory: retaining notation is a real burden for me. So I approach reading the proof with these limits, just like a young child learning subtraction might think of “taking away”: when they reach school, they already possess many intuitions from the outside world. You will need to navigate with these intuitions, sometimes misleading, sometimes, on the contrary, powerful levers to understand new things. From there, you must develop new intuitions, advance in concepts, acquire a new understanding of certain concepts, certain intuitions, certain facts.
It’s a very slow process. I’m not a neurologist, but I almost feel it’s akin to reshaping our neural networks, with new connections forming. This shift in our worldview is extremely slow and can only be slow. AI does not offer a miracle shortcut to skip this learning process. That’s the real danger: beyond AI threatening certain activities, which is already worrisome, it can make us forget why we practice them. When I try to understand an article, I can of course ask AI to help on certain points. In the beginning, I did, and I didn’t get anywhere. But more recently, especially since April, I told myself: “No, I’ll return to what I did when I was young.” As a student, I rewrote everything myself. If I hadn’t rewritten a proof, it was because I didn’t understand it.
Did you rewrite it with your own vocabulary?
With my own vocabulary, first by using the text as support. I paraphrased almost, but with added details in some places and removed others, because some calculations posed no problem for me or I had already encountered that idea elsewhere. In a second step, I tried to write it again without aid. As long as I couldn’t do it, it meant I hadn’t understood.
It’s funny, I think I had almost forgotten this practice. AI had made me forget it: I tended to ask it to add details here and there, but in fact I learned nothing. I retained nothing. It added details, but since I hadn’t found them myself, I couldn’t learn. I therefore stopped using AI to read articles produced by AI, because it was counterproductive. And what’s true for me is true for anyone learning mathematics; I want to emphasize this, especially for the young. This illusion of instantaneity will have a harmful effect, because the human brain doesn’t learn instantly. Whatever the discipline, and that is what we describe in the book, one must go through this process of facing reality. In mathematics, this means facing our thoughts, our psychology, with the formal world.
Do you fear that with AI, steps are skipped, and that this is what makes things delicate?
For me, there are two risks, the second being major. The major risk is falling into the illusion that one must produce as many results as possible with AI, without those who produce them making the effort to ensure they understand them themselves and can convey them. Indeed, if we do that, a decoupling will arise: a vast library of mathematical results that no one will understand. If we reach that, we will have a problem. We cannot be sure there are no errors or dangerous things there. We will have no control over what happens. And we will no longer be able to ask the right questions, because to ask a question, you first need to understand.
That is what I always tell young people, increasingly numerous, who ask me what learning is for in the age of AI, if it becomes more capable than humans in almost any intellectual task. I tell them: just to ask that question, you had to have learned up to now; you could not have asked it without a lifetime of learning. Learning helps you grow, develop, become the person you are. It’s true at the individual level, and it’s true for humanity. If we stop learning mathematics, we won’t be able to ask the right questions; we’ll end up in The Hitchhiker’s Guide to the Galaxy, with an answer even the question cannot comprehend, the famous 42. That is the first risk: loss of control. It’s true for mathematics, but it will be true for all sciences, and even more problematic when there are public health or potentially military questions. There are real dangers. For mathematics there is already a real danger in security, cryptography: losing control is a possibility.
The second danger is more subtle and explains why, personally, I do not use AI in my own creative practice and why I will continue not to: we do not know what AI-driven creativity will become if it feeds only on itself. Questions arose when AI began to feed on its own texts: language became impoverished, stereotypes strengthened, and textual standardization grew. Do we want that for science in general? It’s a real question. A fringe of the mathematical community argues there remains a place for human problem-solving and creativity. Even if we believe the vast majority of output will come from AI, that human portion will, in a sense, provide the essence of novelty and originality that will feed these machines. This is also why I read AI-generated articles without using them: I want my understanding to be human, and perhaps different from what an AI could generate.
Thus, two dangers: loss of control and, perhaps, drying up of creativity. We are still in a transition phase precisely because the machine feeds on everything we have produced as human beings. It’s impressive: in my case, I am cited dozens and dozens of times, and I can clearly see that these are the ideas we have been defending within the community for years. Today there’s no problem of creativity, because all of this feeds on what we have done. But beware of the danger of standardization and drying up. We must preserve a human element.
In these texts you consulted, is there also a form of mathematical “AI slop”? An writing that could be improved, let’s say?
I understand why it is hard to read. But I don’t think it’s yet due to circularity. I think it stems from the fact that how these neural networks operate is quite different from ours, which is good news. Typically, and as I’ve said, they love to introduce notations. It’s probably because, in computer language, it’s not an issue to have what one might call a macro, a symbol that replaces a command, a short text, a formula, or a certain notation. For the computer, it’s almost typical for a program to have all these replacements. But for the human brain, that’s terrible. I much prefer, even if it adds three extra letters, to have the thing written out explicitly rather than a notation I will surely forget ten pages later and have to track down.
This hard-to-read writing mainly comes from that. It also arises because the terms used aren’t the standard terms: you encounter a text that uses familiar concepts but not under the name you know them by. Imagine this in another context: it creates cognitive dissonances. You’re told about a tie, but they call it “a piece of fabric hanging from the neck,” or something less explicit still, “a fabric quadrilateral.” And you wonder: is it really a tie? Is it something I would have missed? These sorts of things accumulate and make reading difficult. But I don’t think the underlying concepts or ideas are themselves complex or poorly introduced. I think the problem is the form, which remains deeply improvable. That will be our work: AI will improve and be able to explain more and more complex things; the limiting factor will always be the human.
The abstraction is a prominent notion in your book, with the idea that some mathematicians, like Grothendieck, are more abstract than others. What happens to abstraction with AI? For us, there is something real; the computer, I’m not sure what it considers real, or how it develops abstract problems.
In my view, the question of abstraction is not fundamentally changed by artificial intelligence. Just as we, flesh-and-blood beings, produce abstraction, there is no obstacle to an AI, made of silicon or other materials, crafting abstract concepts. Today, I must say that in the proofs we have seen, the ability to create unified theories is not yet present. Will it be next month? Perhaps, but I’m not making predictions. We have not yet truly seen unifying concepts emerge. That is probably the next step.
But Grothendieck’s merit lies both in reaching an unprecedented level of abstraction and in having a great unifying capacity, with deep alternatives to what was done before him. Could AI do that?
I don’t see any obvious obstacle to it doing so in the future. For now, it really succeeds at linking what already exists. In a sense, that’s also what we did before; from all these links and analogies, unifying concepts emerged. Mathematics was the art of formalizing these analogies, these links we found, these resurgences of things that looked alike. We tried to introduce the unifying concept that allowed speaking about them rigorously. Indeed, the danger of abstraction in day-to-day life is that the more you ascend in abstraction, the more your concepts become vague. The mathematicians’ aim was to avoid this by remaining extremely precise at every step.
In any domain, there has been this period of nearly horizontal progression where, with the ideas at hand, we develop more and more material and theorems. Clearer connections appear, until a person or group invents the unifying concept that brings everything together. Grothendieck is one of the most striking examples, but there have been many others. Emmy Noether did it with symmetries and invariants, which had a huge impact in physics. Kolmogorov developed a theory of probabilities that unified a great deal, while probability had been practiced for three hundred years. This process of discovering unifying concepts does not currently manifest in the corpus produced by AI. But I see no fundamental reason why a machine, if guided by humans, couldn’t become a highly effective tool for finding them. I believe AI and humans together will be highly effective in this field. Will AI become so powerful that it will hardly need humans to find these concepts? Perhaps. But humans will still need to engage with these new concepts.
Indeed, one of the most fundamental impacts of mathematics isn’t the latest small theorem proven. It’s these unifying concepts that take tens, sometimes hundreds of years to become so effective and understood that they become shared knowledge. This is the case of zero, negative numbers, and the notion of a straight line: everyone knows what that is. One could think that a line is only used in geometry. But no: we rely on it constantly in architecture and urban planning; perspective is a story of lines. Many mathematical notions, like algorithms and, more recently, fractals, are becoming cognitive tools for the general population, but that happens through a very slow process. Even if the machine could discover unifying concepts, we would still need to work to make them accessible to the population.
Back to October 6: 722 problems solved, and let’s say 3,300 that are not. Do you have an idea of the nature of these 3,300 problems? Does that mean the machine isn’t so powerful?
Those problems may have turned out to be more difficult than expected. One tends to attribute a given difficulty to a problem a priori, which sometimes turns out to be lower, sometimes higher. The machine navigated that space, and some problems proved tougher than others. In my field, we were surprised by how close we were to the solution for many problems: some appeared far harder than they actually were.
But the same was true among humans. One of the most beautiful moments of the last month for me was last Wednesday, when some Chinese colleagues posted a proof they had long found concerning one of the problems demonstrated by AI. A new proof they had discovered a year earlier, but it takes humans time to write. It astonished me because it reminded me that, through the successes of our peers—I use that term deliberately—it’s a little bit of ourselves that grows too. There is a certain pride when a climber reaches the summit of a major Yosemite cliff. I’m not a climber, but I watch what they do and think, “Wow, there are humans who can do that.” When a singer produces something like that, there is something in us that grows. I found it very impressive to see that they had solved that problem and to tell myself that human beings remain incredibly creative.
Why do I say this? Because it was a beautiful moment for me, but also because I felt this problem was far from solved. I had no intuition about it; to me, it was almost science fiction. And on the other side of the world, humans were figuring out why it worked and writing the proof. So we sometimes misjudge the difficulty of a problem, and sometimes the opposite happens: we dive into a problem, and it turns out to be much more complicated. Among the problems we, humans, judged important and probably solvable, the machine tried them and found them to be harder than expected.
Then, it will progress in power. And humans will manage to use it more effectively, because today, all of this is very automatic. It’s important to understand that at OpenAI, no one understands these problems, or perhaps only a few mathematicians who understand a few of them. For most problems, the company itself would be totally unable to explain its results to mathematicians. There is no collaboration. It is more like: here, you have this, you bear the burden, and you carry it to the mountaintop. I sincerely hope the future will be one of collaboration and that this tool will become effective, and not a replacement, as was hinted in October: “Here is your proof, you no longer need mathematicians, see how strong we are.” I hope they themselves have understood their mistake at this level and that we will now imagine something much more interactive. I also hope this tool will serve the community rather than be a danger, as it has been presented in recent months.
When we spoke in September, you told me they were language models working in natural language, and that there was no need to formalize problems in a particular way. Is that still the case? Were these problems given to them unprepared?
Yes, it’s the model itself that automatically searched the literature for problems presented as major in order to try to solve them. There’s no longer anyone to prompt, as they say. No one asks, “Could you answer this question posed by someone, please?” No, it’s all completely automatic. I think the next step will be for experts to take these tools into their own hands. That should multiply possibilities, because human expertise complements machine expertise and allows us to advance much faster. Still, we need access to these models. And that costs money; there is an ecological expense, an ecological debt. The question of using these tools is therefore very relevant. Some will use them, and others like me will continue to work without AI, in order to supplement, in a sense, our capacities and our way of thinking, which we do not want to delegate entirely to the machine.
If, by any chance, the machine resolved all these problems, if there were indeed a replacement, which is not the case, have you wondered what your work as a mathematician would become?
Our work has always been to be the first links in this human chain of understanding. Perhaps the way we talk about mathematics, through prizes and grand conjectures, has made us forget this deeply human, even scientific, dimension of mathematical practice. For my part, I’ve always viewed it that way, and that’s why I wrote this book with Emmanuel Sander: our aim was to remind this extremely human dimension of practice. From that point of view, I can’t imagine any scenario in which this aspect of the job would be deeply transformed or replaceable.
The only scenario I could imagine would be one in which this fundamental axiom is removed: we want there to be humans who can transmit these ideas because we think they are useful to people in everyday life. I sincerely think that if we lived in a society that denies this axiom, that denies the importance of control and human understanding, and accepts that all our knowledge comes from a machine, then we would be faced with a much more serious difficulty than the question of what happens to mathematicians. Indeed, from the moment we stop learning, we cannot even pose these questions. Our cognitive abilities will gradually decline; we will experience a regression of our capacity to think, and the problem will be far larger than mathematics itself.
Let us remind ourselves again: mathematicians are the first to understand certain mathematical ideas. They then pass them on to other scientists and teachers, who in turn pass them on to others. At each step there is a filter: the most useful, the most striking ideas are selected. It is a sedimentation process. If we cut off the top, if we remove the first step, those below will have to begin to understand these ideas with fewer specialized skills. They will understand less, and the machine will thus be less exploitable. And if they stop using it, they will lose skills themselves; perhaps one day they will be told they are no longer needed. We descend gradually and end up with a regression rather than a progression of our thinking abilities. These are questions we have asked in many other domains, and it will be the same in the arts, music, and humanities. People at the frontier of what is understandable and what humans can do indirectly enable humanity to advance. If you remove these people, it’s as if you decided, in a sense, that the movement should reverse and that understanding should decline. Hugo Duminil‑Copin, on the other side of the machine