Comment by paulceton

I’ve been a math professor for about ten years, and I think people outside the field misread what makes mathematicians angry, sad, and worried. Interviews that make it into the media tend to go with “AI is killing mathematics,” maybe because that gets clicks. The reality is more complicated.

Some mathematicians work on well-posed, famous problems, but most mathematical research is about questions that aren't well-posed yet. A typical example is finding where the boundary of existence lies for some equation: which meaningful conditions guarantee a solution, and how far can you weaken them? You can’t even start on that without understanding the problem. For the famous open problems, the meaningful conditions have largely been settled already, and what remains is to prove existence under those conditions or build a counterexample. That is exactly the kind of task AI is bound to be good at, and plenty of people expected computers to get there eventually, going back to AlphaGo. It just arrived much faster than expected, which is causing a lot of problems. Assuming it made no errors, AI would be a great research partner.

Which conditions are meaningful depends on context: where the field is going, what an individual researcher finds interesting, connections to actual phenomena. AI can’t decide what I find interesting. The OpenAI Navier-Stokes result shows this failure of context. OpenAI announced finite-time singularity formation with a smooth body force. Constantin, Ignatova and Vicol (https://arxiv.org/abs/2609.20803) showed that for solutions with the key features of OpenAI’s construction, the force can neither vanish identically near the singular point nor be real analytic in space. Fefferman’s Clay problem statement does allow a forcing term in the breakdown versions, so the announced result may fit that format, but this method cannot reach the unforced case, which is what mathematicians actually want to know. It is a step forward. Still, I think a mathematician who had spent years on this problem would have pushed toward better conditions, and the paper would have been better for it.

What makes me angry about the ~770 papers is that the author, OpenAI, takes no responsibility for their manuscripts. Many are badly written, to the point where you can’t tell whether they are right or wrong, and verification runs on unpaid labour from human mathematicians. People say, “but it’s checked in Lean.” As Bastounis, Circelli and Hansen showed for the Navier-Stokes case (https://arxiv.org/abs/2610.08144), the Lean proof can pass while not corresponding to the natural-language proof in the paper. A human author would be expected to dig into the Lean code and fix the paper. I doubt OpenAI will. If someone dumps more Lean code than any person can read and walks away, who is supposed to check it? The person who was working on that problem gets scooped if they finish their own result, and gets no academic credit for verifying OpenAI’s paper either.

I’m also angry at mathematicians, not just OpenAI, who use AI to mass-produce results they don’t understand themselves. Publishing a result you don’t understand is the same as not taking responsibility for what you wrote. It shifts the burden of verification onto referees, readers, and whoever builds on that result next.

My biggest worry is that mass-produced, un-understood papers will slow mathematics down. You need to know what we currently understand to see where to go next. If everything around you is papers nobody understands, you can’t find where you could make a real contribution. It’s not that your view is narrow; it’s that the papers block the view. It's like we want to go to space, and low Earth orbit is filling up with space trash.

I also worry about my students. A PhD student is something like an academic child, and I don’t know how to advise them anymore. Do I tell them to put out 20 papers a year whether they understand them or not, like some people do? Or stick with the traditional goal of one paper a year that they fully understand? My conscience as a scholar says the latter, but I’m afraid they’ll lose out on the job market.

And there is sadness. Solving math problems is a joy. It’s an experience of thinking through something with other people, being delighted and frustrated together and making progress, and that is being taken away. A friend I used to discuss problems with now feeds our ideas to an AI and writes the paper right away. I’m left with the experience already spoiled, doing the tedious cleanup and verification without the fun.

As for “mathematicians are just protecting their jobs”: mathematicians don’t make money from research in the first place. There are almost no experiments, so research grants mostly cover conference travel. Most of a math department’s budget comes from teaching, especially courses like calculus and intro statistics with thousands of students. If anything, now that AI does the homework, assessment is moving to oral exams and in-person tests, which takes more staff, so I don’t expect math faculty positions to shrink unless we give up on math education. Outside academia, better access to mathematics may even increase demand for mathematicians.