Care for a little more AI?

Hugo Duminil-Copin, Professor at Université de Genève and IHES

Disclaimer. The effects of AI are vast, and there is much to be said about its broader impact on society. One should also remind that mathematicians have a role to play in advancing research on AI safety. I do not feel sufficiently qualified to address these wider questions here, so I will instead focus on the direct effects that the current use of AI is having on the mathematical community and provide my experience as a practicing mathematician.

The progress of AI in mathematics is spectacular. I will not venture here to predict what AI may accomplish in the future, but I can already say that, without question, frontier models are far better than I am in many mathematical tasks. To mention at least one, they possess encyclopedic knowledge of my field and neighboring domains, while I sometimes struggle to remember some of my own proofs.

Some problems with direct and urgent applications will obviously profit from advances in AI. Their solutions will enable further progress in related sciences and contribute (one hopes) to the greater good. It would be absurd to deny this.

But what benefits some areas of our discipline may be ill-suited to others. It is entirely legitimate to ask whether we truly need and should really welcome the rapid proliferation of AI generated proofs in mathematics.

The $\theta(p_c) = 0$ conjecture. So far, I have been fortunate not to be directly affected by an AI solving a problem on which I was actively working. But lightning has struck close by. In April, some colleagues went through this painful experience: ChatGPT Pro proved a difficult problem in percolation that we used to call, among ourselves, $p_c < 1$. It now seems only a matter of time before the most famous conjecture in our field, $\theta(p_c) = 0$, also falls to the bulldozers. Indeed, it became clear that AI will probably prove $\theta(p_c) = 0$ before humans do. It is certainly an excellent candidate for frontier models as there may be a simple proof, something hidden that everybody missed. Will the proof come from an employee at OpenAI or Anthropic? From another company eager to “help” mathematicians in distress? More prosaically, from a mathematician armed with their favorite AI model? Or even from a casual user with virtually no mathematical expertise? Only time will tell.

A very simple statement, not much background, and a high difficulty (for human) is precisely what made the $\theta(p_c) = 0$ conjecture so beautiful. This conjecture stands out among percolation problems because it is inspiring and intriguing, not because solving it would obviously unlock a vast new area of mathematics.

My own story with $\theta(p_c) = 0$ is instructive in this respect. I no longer work actively on this question, but there was a time when this conjecture inhabited my mathematical dreams. During the early years of my career, I returned to it regularly. As you can guess, I did not find a proof. And yet I consider my relationship with this conjecture a success. A success measured neither in theorems, nor talks, nor prestigious prizes. A success far more precious to me.

Let me provide a small sample of the encounters made possible by the conjecture. I met my main coauthor in percolation theory, Vincent Tassion, because we were both trying to solve the same simplified version of it. It was while trying an analogue of $\theta(p_c) = 0$ for the Ising model that I started working with Michael Aizenman. Together with Michael and Vladas Sidoravicius, we did prove the conjecture for Ising. More importantly, we were able to breathe new life into a powerful tool that Michael helped develop in the eighties: the random-current representation. This tool later enabled us to solve a multitude of other problems, in particular the triviality of the model in four dimensions, and it is now used by many. There are many other examples I could cite off the top of my head. My failed attempts to prove the $\theta(p_c) = 0$ conjecture generated dozens of ideas that I later repurposed in other contexts, leading to discoveries I would never have imagined making.

A mathematical question is much more than a theorem waiting to be proved; it is a source of momentum. First and foremost, it is a lighthouse in the night: it illuminates and guides mathematicians in their wanderings, both esthetic and scientific. 

Seeing the beautiful story of $\theta(p_c) = 0$ coming to its final words would obviously not be the end of the world. But I had hoped that one day, a young mathematician, through the originality of their thinking, would come and show us what all of us had missed. I could then have marveled at the creativity of the human mind. But depriving me of this pleasure is obviously not the main issue here.

What is my problem with the way generative AI is being used in mathematics today? Certainly not that AI-generated proofs are systematically unreadable. This is not true. The proof of $p_c < 1$, for instance, is short and particularly elegant. 

In my opinion, the problem lies elsewhere: the current use of AI does not empower us, it petrifies us. These artificial discoveries risk decapitating entire fields before they have had time to develop to their full potential. Even worse, they nuke the mathematical landscape, making it increasingly difficult to inhabit after each blast.

Let the lighthouses built for us shine a little longer. We owe it to our predecessors. Let us not profane our most beautiful conjectures by using them as mere “benchmarks” for the next frontier model. 

What are we supposed to tell young PhD students entering a field? Which problems should they work on? How can they build mathematical abilities when the most accessible problems may fall at any moment under the battering ram of compulsive users of AI models? What Holy Grail should they pursue for the next several years if the proofs of major conjectures might suddenly appear, from one day to the next, on some social network? 

What is at stake is not only how we choose problems, but the culture and ethic that shape our profession. The social consequences for our community, especially for the younger generation, are devastating. Some will argue that mathematicians are simply an endangered species. This may be true, but I do not believe it.

A profound misalignment between AI and mathematicians’ interests. When mathematicians say that the process matters more than the solution, this is not an empty statement. The richness of what emerges from repeated attempts, failures, detours, and encounters is extraordinary.  I obviously cannot speak for everyone, but I am deeply convinced that what I have described from my own experience with $\theta(p_c) = 0$ resonates with the paths many of my colleagues have followed. 

Understanding the mathematical world requires both personal and collective exploration. The same formal concept takes on a different shape in the mind of each mathematician, through a long and gradual process of appropriation. The ideas that live and circulate within our community are the true jewels of mathematics. Like Peter Scholze says in the context of the Leiden declaration:

In my experience, mathematical ideas, like children, must be nurtured and grow over the years. Just like I do not want my children to be educated by AI, I am pondering my mathematical ideas without use of AI [$\ldots$]

Let us recall that our mission as mathematicians is not limited to producing theorems, we must before anything produce understanding. We remain the first link of the chain turning very abstract notions mastered by few experts into concepts that become part of the culture of next generations. 

Ideas need time to emerge, to be refined, tested against one another, understood by multiple mathematicians. That is the price we must pay for ideas to mature in our minds and ultimately become genuine intuitions. Once properly mastered, these intuitions can be passed to other mathematicians, then teachers, and kids. This is exactly what happened with 0, the = sign, negative numbers, infinity, fractals, etc. And this process is ongoing with more recent concepts.

Unfortunately, it is precisely this slow process of intellectual development that is threatened by the way generative AI is currently being used. The consequences for mathematics, education, and culture will be massive if nothing is done.

What should we do? Mathematics is far from being an isolated case. One of the fundamental questions raised by generative AI is how far we are willing to delegate our cognitive abilities to a machine. I have always regarded mathematics as one of the highest forms of human intellectual exploration: a way of pushing the limits of what we are able, individually and collectively, to understand, imagine, and abstract. 

If we mathematicians lay down our arms and accept delegating our creativity, even partially, to machines, what other form of intellectual adventure will not follow? Instead, we must persevere in our human exploration, accepting that it takes time, that it sometimes leads us astray, and that it often remains fruitless. 

Let us keep on the traditions of a discipline several thousand years old, and teach how to search, doubt, experiment, fail, and restart. Training future teachers, engineers, and researchers is impossible without personal experience of what it truly means to do mathematics ourselves. In my case, it is this experience that I like to share with students, school kids, and the public, more than the precise statements of my results. As a community, we have a responsibility to continue playing this role. In full honesty, we have even a great deal of room for improvement here.

While some areas of mathematical research may very well embrace the possibility of speeding up the resolution process leading to spectacular applications in other fields, we should not succumb to a purely utilitarian vision of our discipline. 

I believe that many areas of mathematics simply do not stand to benefit much from the assistance of AI. I certainly do not have a straightforward rule of thumb for determining which ones, and even if I did, I am in no position to provide a general recommendation. In fact, it has never been the intention of this text to offer a solution (one that I do not have). Still, I sense that many mathematicians share a similar view of recent developments, and I wanted to give voice to it, in the hope that some of them might recognize in these words something of what they themselves feel.

I will conclude on a more personal note: in my own research, I have chosen not to rely on artificial intelligence to replace my creative process.

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Received 30 August 2026.