August 12, 2026
Welcome to the second episode of my Retreat in Lozère series. Today I want to talk about a question that is becoming uncomfortable for us mathematicians: what will remain when AI becomes better than us at doing our job?
This may have sounded strange a few years ago, when AI was sloppy and failed at high-school problems. But the line keeps moving—high-school problems, then undergraduate problems, then graduate problems, and now AI is starting to help prove conjectures that humans could not solve for decades. A recent example in my field of numerical analysis is the proof of Crouzeix’s conjecture by a neurosurgery resident a couple of days ago, with help from GPT-5.6 Sol.
So what will be left for mathematicians? Us mathematicians, we prove things. But what is a proof?
A proof is a sequence of logical steps that starts with assumptions and proceeds to a conclusion. We mathematicians like to think of mathematics as the noblest of sciences—once the assumptions have been established, everything follows from logic. Things are either right or wrong. What we do is pure science. Or is it?
A proof is also a social compact. I discussed this in a previous blog post about mathematical rigor. When we write a proof, we do not define everything. Instead, we give enough details so that another mathematician can follow the argument and fill in standard gaps. Without this compact, every proof would have to start from the foundations of mathematics. Nobody wants that.
This social compact also means that many proofs are not checked line by line. If the author and/or the institution is trusted, and if the paper looks correct at first sight, many people will believe it, including reviewers! Of course, this is different for a major theorem. A very important result will be checked carefully by experts. But most papers are incremental and reviewers will not check every line.
Let's be a little bit provocative. Sometimes I feel that academic mathematics works like an old gentlemen’s club. (And entering this club is hard; see my previous post on The French Research Mafia.) But once you have a seat at the table, people trust you almost blindly. Your papers look serious because your name is on them. You review your peers, your peers review you. In the end, everyone protects the same system. And I think AI is about to change this.
I would bet $100 that if you pick a math paper at random and run a strong model on it, it will find at least one mistake. Try me! Most mistakes will probably be small and fixable, but sometimes the problem is more serious.
The more technical the paper is, the easier it is to make mistakes. Moreover, if the paper is also hard to read, fewer people will check it carefully, then the author gets less help and the mistakes stay hidden.
I also think that, in 20 or 30 years, people may be surprised by how we used to do mathematics. They may ask how we could publish so many formal arguments without AI. They may look at us as artisans, much like we look at people who built the pyramids without cranes or any kind of machinery.
Recently, I took a very technical draft that I found almost impossible to read. To my surprise, some people around me had the opposite reaction: they said the draft was deep and impressive! However, when I asked more questions, it became clear that almost nobody had really read it carefully—this is the kind of hypocrisy that is everywhere in mathematics.
Complicated writing creates the impression of depth and a dense proof can make the writer look smart and serious. It can also make the reader feel that the result must be important. Sometimes the reader also wants to sound smart and serious by saying that they understood it! I'm not even joking, this is real.
Anyway, I ran a strong language model on the draft. Ouch. It found many mathematical problems: not only style issues, but also some very serious gaps. The main result may actually be wrong.
In the future, it may become much harder to hide weak mathematics behind complicated writing. And I think this is a good thing. My provocative posts are not always appreciated by the community, and this one may hurt a bit. Sorry!
Where does this leave us? If AI becomes better at producing and checking proofs, then one thing may become much cheaper: correctness. This feels and sounds strange because correctness has always been the heart of mathematics. But since you can give a proof to an AI and get a detailed report in a few minutes, writing a correct proof may no longer be enough to impress anyone. And I think this will make clear writing even more important!
Suppose an AI gives you a 40-page proof of a conjecture. Great. But somebody still has to read those 40 pages and understand what really happened. Where is the main idea? Is there a simple example that explains what is going on? Going back to Crouzeix’s conjecture, only a few days later, a couple of mathematicians produced a five-page proof instead of the original 24-page one. Thanks, humans!
I think part of our job may therefore move from producing arguments to digesting them. We may spend more time taking long AI-generated or AI-assisted proofs and turning them into mathematics that humans can understand. In other words, mathematicians may become a little more like editors. I think I actually like this future.
Let me say this one more time: clear writing will matter a lot. Mathematics can become more about ideas again, and less about putting complicated symbols everywhere. I have never been a big fan of the French Bourbaki style, or of the way mathematics is sometimes taught here. Bourbaki and I broke up—let's make math fun again!
Of course, there is another possibility: maybe humans will remain useful for coming up with strange questions and completely bold and unexpected ideas and connections. I hope so but I would not bet $100 on that one! By now we realize that every time we draw a line and say, “AI will never do this,” then the line seems to move a few years (if not months!) later.
So perhaps the better question is what we still want humans to do. I still want to learn mathematics from another human being. I still want somebody to tell me why a theorem is beautiful. I still want a teacher or a mentor who knows when I am confused. I still want a colleague who gets excited by an idea and spends time finding the clearest way to explain it to me. I still want to think about mathematics at my own pace without looking at a computer screen.
Mathematics is also a culture. It is about transmitting rigor, critical thinking, intuition, curiosity, and the joy of discovery.
Maybe this is what will remain.
“The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.”
— Henri Poincaré, Science and Method