Artificial Math

Every math department has the same conversation running in its corridors right now: what is AI actually going to do to research? I can't speak for every field. But I did spend a few years in academia as a mathematician, so I'll put that hat back on and give my two cents.

I've met people who want nothing to do with AI in math, and not only on the research side. The formalisation crowd is wary too, and Mathlib has to stay conservative about it or the library fills up with slop. At the other extreme you have mathematicians leaving academia to work on AI full time. Jacob Tsimerman announced he was pivoting to AI safety at OpenAI roughly an hour after collecting his Fields Medal in Philadelphia this week, which is about as loud as that signal gets. Underneath all of it sits the fear that AI will replace mathematicians outright, the same fear software engineers have been having about themselves. On this I'm a bit of a democristiano: for non-Italians, this means I hug the middle ground and decline to commit to either side. Dante would probably file me with the "ignavi" bunch, though I doubt my opinions rank anywhere near his list of things worth worrying about, so I feel safe.

Going sideways

I do think AI will reshape math. That doesn't have to be bad news. Plenty of real breakthroughs happen where two or more areas touch, and math goes deep and vertical very fast, so mastering one topic already costs you years and switching areas is painful. For a long time this meant the big cross-area discoveries came from a handful of people whose human CPU and GPU hardware was exceptional enough to hold more than one deep topic at once. That's probably one reason math research gets called an ivory tower. Very few people clear that bar, and a lot of good ideas are still sitting out there waiting to be found, or invented (I'm not opening that particular door today).

Models keep getting bigger, and with that they're starting to take on questions that aren't toy problems anymore. The clearest case so far is the Jacobian conjecture, open since Keller posed it in 1939. Levent Alpöge, working with Claude Fable 5, posted a counterexample on X that fits in 216 characters, a polynomial map from three-dimensional space to itself with constant Jacobian determinant -2 that sends three different points to the same image. That kills the conjecture in every dimension above two. The planar case is still open. The map was short enough that other mathematicians verified it by hand within a day.

Counterexample hunting is one obvious use. The other is learning, if you're disciplined about it and paranoid about hallucinations. The feature I care about most is democratisation. Hard topics used to sit behind serious moats, and now you can use a model as a study companion, or to interrogate your own understanding, or to get a working picture of some esoteric field you'd otherwise never touch. Math has always rewarded going vertical. Pick a topic, go as deep as you can, and if you're one of the rare ones, do it multiple times and find something new where a few areas meet. Models make the horizontal move cheaper. It used to be normal for someone strong in analysis to be shaky in algebra, and the reverse, exceptions aside, because there's a hard physical limit on how much RAM a brain has. Lowering the cost of cross-pollination is the part I'd bet on mattering most.

The bill

None of this comes for free. The bill right now is that the technology moves faster than people adapt to it, so teaching is where the damage shows first. It's getting genuinely hard to teach students who use AI not as a crutch but as legs. The pain the older generation went through was part of the mechanism. Solving something feels good because it hurt for a while first, and taking the hurt away flattens the whole range. A monochromatic world is flat. We need contrast, and we need to struggle a little, in a constructive way.

The second problem is the one almost nobody seems to be raising, which is that all this sudden AI-powered progress can steer where math research goes. Compute isn't cheap, or at least not pen-and-paper cheap. For decades a math department needed blackboards, chalk, pen and paper, and that was the equipment budget. Now there's inference to pay for, which means more funding, which means somebody has to provide it. Financial institutions are an obvious candidate. And a bank has considerably more interest in analysis, probability and stochastic calculus than in abstract category theory, logic or algebraic topology. That bias has always been there. What's changing is the size of it. The money advantage used to be a nudge, and it's becoming the thing that decides which departments can work at all, which widens the gap between the rich ones and the poor ones and gives funders a much heavier hand in what gets studied.

Why I'm still optimistic

But people are resourceful. Open models keep appearing, and they keep getting cheaper, and human curiosity hasn't diminished at all. So I think these factors add up to a new golden era for math, and I strongly hope I'm right. Long-standing problems falling. New theories being born. And with theorem provers now in the picture, the symbiosis between humans and machines is tighter than it's ever been. I spent years in areas where a good fraction of the useful results were folklore, and nobody was encouraged to prove them formally except in a few cases. Those can now (or will reasonably soon) be proved with LLM help, which means we get the foundations without having to take folklore on trust. The decisions and the responsibility stay in human hands, so I hope the shared wish to know more and understand better is what ends up guiding them. Exciting times are ahead, and I'm really curious to see what the future holds for us!