I'm a mathematician/computer scientist, these days in San Francisco. My normal office lies in Waterloo, Ontario, but I'm here on sabbatical, working at the Simons institute in Berkeley for a few months during one of their long-term programs. Rent is rough here, and my wife is quite picky, so we decided to stay temporarily at a friend's place until we find an apartment in Berkeley. That friend works in AI.
My friend and I (and some random other AI people at a party) got to talk through the craziness of the past week as everything came out: the Navier-Stokes solution, the competing group of researchers, and all of the surrounding drama. It was an unforgettable experience, and one that will stay with me, especially as I stare down the abyss that is the future of my career in research mathematics. Getting an academic job was my dream for a long time, and the amount of work and effort it took to get here was immense. And now I'll have to decide whether to leave it for something else.
I also had some thought-provoking (read: anxiety-provoking) conversations about AI and the future of research mathematics. Perhaps it's not a surprise, but the general suggestion was that it might be time to leave the discipline. Understandable, I think. But these conversations have left me with a feeling of a sort of mismatch between the math community and the "AI math" community, which I plan to touch on here. More and more I am realizing that this mismatch is at the heart of the general debate about the future of research mathematics. And so I think it's worthwhile to try to express it, mainly for myself, but maybe for other mathematicians and other onlookers who aren't really sure what to think.
There is a style or philosophy of doing research math, attributed to Grothendieck, which is described via metaphors for how things should ideally go. One metaphor says that math progress should be like a "rising sea" - as we build up theories in math, the "sea" rises and rises until pieces of "land" (open problems) are covered. The sea does not make much effort to cover the land; it simply rises and eventually the land is covered. The other metaphor then compares two opposed styles: solving a mathematical problem should be like soaking a hard nut in water, patiently waiting for the shell to slough off without effort, rather than just using a hammer to crack it open. The really oversimplified idea here is that the "right" theory is more important than solving problems.
Of course this is easy to say, but as all research mathematicians know, solving problems has still generally been the main metric by which we evaluate the quality of a mathematician. Terry Tao has discussed this a lot, describing how research mathematicians do many things besides just solve problems, but that problem-solving is a proxy metric for all those other things. One big metaphor he gives for this fact in the new world of math AI is: when food (i.e. math proofs) is scarce, hunting (i.e. problem-solving) is the task that matters. When food is not scarce, hunting can become a sport, while other careers are born (e.g. chef, nutritionist, etc.). The tasks of those other careers have always been done; people have always decided what exactly to eat and how to cook things. But those things matter far less when actually putting food on the table to begin with is not guaranteed.
And this is where the first cracks of the math-AI mismatch begin. It seems to me that AI companies looking to demonstrate their value through math problems are taking the nut-cracking/hunter view. They feel that research mathematics is about solving problems, and reasonably so, because almost all prestige-bestowing mechanisms route through that one thing. Solving a Millennium problem then becomes the peak of progress in mathematics, and since they can do that, it becomes very hard to give a good answer to the question of What Next? for research mathematicians.
On the other hand, Tao and others looking for a way forward for the discipline are saying something else. They say that problem-solving is what some of us always sort of felt it was: a proxy. At the end of the day, what we really want is understanding and insight. The problems of course are important themselves, but the reason we fight to solve them is that we know from experience that solving such problems lead to new theories and ideas which allow us to understand more succinctly the structure of mathematics, and in turn, the universe. Having a proof is nice, but having a theory that makes the proof easy (like the rising tide so effortlessly submerging the land surrounding it) - that is the real goal. The sort of "short-circuit" AI solving of math problems skips this generalized understanding, this "tide submersion", and in some sense misses the point.
So then, who's right? I don't really know of course, and my point here is not to answer that question. I do think though one thing, that both sides here are missing a bit. What I feel to be the essential question moving forward in research mathematics (and in various other disciplines perhaps) is this: to what extent is "human compression" necessary or desirable? Let me try to explain what I mean.
I think one way to interpret what AI is doing (and I am sure people have said this before) is compressing human knowledge. Of course the overly simplistic model of what's going on in an LLM is the whole probabilistically-choose-the-next-token thing. Each output word answers the question, "given the internet (and more), and given all the words in this thread that have come before, what is the most likely next word?". That is, what are the conditional probability distributions which describe human language? Thus an LLM can be viewed as compression in the following simple way: instead of having the whole internet sitting around, we can just keep the parameters determined during the training of the model and use those to produce whatever information the user wants. And even better, we can even produce things that no one has ever come up with before! The data of the training parameters can be used to produce true novelty in research disciplines in a matter of hours. What better evidence for the models being very compressed versions of human knowledge is there than that?
The problem with this compression is that it is likely to become more and more inscrutable. At some point it'll just be faster to let the model agents interact in some more math-efficient language that we can't read, to produce solutions to open problems faster. Of course there are some security issues here, but not really for mathematics. Assuming the agents are only talking about math (yes I know about hugging face, but just go with it for now), and all we care about is solving open problems, then it doesn't make sense to burden them with English words, except perhaps at the end when we want to see the final proof.
By why stop there? If Lean-verification is so easy to do, why put it in English at all? And even Lean is probably inefficient. Let the AI make up its own verification language which is faster and more efficient, and then we'll really be cooking. The burden of human language can be lifted in order to maximize speed and efficiency.
Of course the problem with this comes when we do actually want to see the proof. When we want to understand why some new AI-produced theory or technology, say, works as intended. And the comments of the previous paragraph then really crystallize the key question: Why? Why understand the proofs at all? The idea that mathematicians understand what's going on is a nice one, and was useful in the past, but maybe it actually just hinders progress in the age when AI can solve any open problem we throw at it.
The answer to this, which I'll now describe, is what I mean by "human compression" and what I believe is a way forward for research mathematics that allows us to retain at least some of our identity. The point is that when the rubber hits the road, we want a human at the wheel. We are, so far, not yet ready to be completely governed (in some general sense) by AI. We still want to decide for ourselves. And as long as that is the case, then up and down the chain of human knowledge, we will want to have a simple (relatively speaking) conceptual understanding of what is going on. And to do this, we need our own sort of "compression" - a distillation into ideas and concepts that at least some collection of humans can understand quickly, at-a-glance. And it seems to me that this type of compression is essentially at odds with a more AI-driven compression because it is inherently about our in-mind conceptualization. Translation between the human and AI, at least in mathematics, will continue to become more interesting and complicated as the AI systems themselves do. And to do that translation, we will need more and more robust and encapsulating math concepts that can capture the AI outputs and allow them to be expressed in a way that is tight, compact, compressed. These concepts of course are what we always developed in math; they are what we built over time, guided by and eventually leading to the big open problems in math. We still want and need those concepts, just now the direction of causality might go the other way.
My theses here generally are that this "human compression" is inherently distinct from (though of course helped by) AI, and that it is something that we want to retain. I could be very wrong about either of these things of course, and in a way this writing is to try to convince myself that there is a way forward at all for research mathematics. And I don't have much in terms of future suggestions or ideas for what to do next, other than basically what Terry Tao (and probably others) has already said: we must reckon with our new crisis of values. "The mathematician is on a tightrope," between the old world and the new, and it is not impossible that we fall off. But I think with the right understanding of what it is we do, of the value that we have actually provided over millennia, we can successfully navigate this to become an integral part of captial-K Knowledge moving forward. We are still human, we still want to choose, and we still want to know, as humans. And as long as that is the case, I believe that research mathematics can provide. Perhaps not new theorems any more, but rather, the right way to think about things. And no matter what AI can do, we will always want humans to have that.