It’s interesting to ask mathematicians about why they do mathematics. Some tell me that it’s a pursuit of truth. Others seek understanding. Occasionally there’s an admission of the pursuit of glory. My personal claim is that I’ve always wished to have a voice and to contribute to The Great Conversation, the sharing of thoughts, insights and perspectives with all those around me and who came before me. The question now is: “What’s this have to do with AI?”.
Well, recently I read an article by Henry Farrell, Alison Gopnik, Cosma Shalizi and James Evans that awoke me to an obvious fact — one I think deliberately obscured by AI companies — about what LLMs actually are. They are not “..intelligent agents but .. a new kind of cultural and social technology.” I think this is basically the correct place to start. Large language models are interactive games that let us play pretend with voices from The Great Conversation. They are built out of our books, papers, websites, computer programs and conversations. So, as Nestor Guillen puts it, there is a sense in which, when an AI spits something out at me, I am interacting with a large amount of accumulated mathematical culture through a new interface, albeit with hallucinatory garbage thrown in every now and then.
This interface is now capable of much more mathematics than I thought it was just a year ago. There is, of course, the whole Navier–Stokes situation, which Tao and many other people have been discussing. A colleague of mine also seems to have accidentally solved a seventy-year-old conjecture in topology and set theory by asking ChatGPT for help with a lemma: it came back saying, essentially, “yeah, I can prove your lemma, but I needed to solve this conjecture first.” I do not know exactly what to make of all this yet, but it has changed my view of what AI can and cannot do. I had a much smaller experience recently. I was working on a lemma that I was convinced should be true, and I had an idea of how to prove it, but the proof looked really fiddly. I asked ChatGPT to prove it, and it did so right away. But instead of just reading the proof and moving on, I immediately made my own life more complicated. I started asking what the canonical proof should look like. What argument would make the result absolutely obvious? This took me most of a day, basically the same amount of time I would previously have spent fighting through the ugly proof. But at the end of that day I understood the result much better.
I have started to think of this as mathematics above AI, or perhaps more personally as thinking hard above AI. In parameterized complexity we sometimes study a problem “above a guarantee”; this amounts to the situation in which some easy argument or relaxation gives us a baseline, and we study the additional difficulty that remains beyond it. It is a little like what is know as FPT above LP (fixed parameter tractability above linear programing), except that the baseline is now whatever the AI can spit out. In a sense, I increasingly see asking an AI for a proof as a second step in the literature review: first I check what people have already written down and made explicitly available, then I check what can be reconstructed or produced fairly easily by an LLM.
It’s important to note that these answers are the starting point, not the end result. We definitely don’t want to just volley automatically generated, poorly written and undigested math into the The Great Conversation. That’s a pretty antisocial thing to do. And it creates mathematical technical debt as Henry Cohn calls it, which is the equivalent of flooding The Great Conversation with incessant and half-assed tweets in the midsts of measured, well-thought arguments. All of this to say that when I say “Mathematics above AI”, I mean the mathematical activity that involves treating an AI-generated answer as a starting point rather than a conclusion.
So if we accept the existence of “Math Above AI”, it still might not be clear what the goals of such mathematics should be. Emily Riehl recently wrote that one of the things she likes in math is to search for the simplest explanation of a phenomenon, which often requires finding the “right” language and the “right” level of abstraction. My personal aesthetics agree with hers in this respect and it is in pretty close to what I was trying to do with the lemma I mentioned: I already had a proof, but I did not yet have the proof that could speak to God and tell me why the result was true. If proofs become cheaper, then great, questions about why a proof works, which language it belongs in and how it should be contextualised become more valuable. In my mind, this is a chance to engage in this Great Conversation as better interlocutors: we can ask more of ourselves and of our arguments and of their context.
Some might find my aspiration for a “Mathematics Above AI”, as wishful thinking. I’ve definitely had many conversations with students and colleagues who fear for the end of mathematics and most other intellectual disciplines. Although I too have some pessimistic days, for the most part I find this a little nonsensical.
First of all, it is always useful to ask oneself who is to gain from a particular narrative. In this case, if I were in the business of trying to sell a little “robot oracle” touted to have near mystical powers, then my biggest problem would be to try to prove that such powers do indeed exist. And what sounds more mystical and impressive (not to mention palatable to many, as Ian Bogost writes in The Atlantic) than the ability to end the world?
Second of all there is also a more abstract reason why I do not think that AI crossing some particular frontier means that mathematics is finished. If we assume that these systems are not capable of a fundamentally stronger form of computation than we are, then sure, perhaps they have more resources (e.g. electricity or memory or a bigger PR team), but that should not prohibit me from asking an even more meta-mathematical question.
Third, I haven’t even mentioned the obvious fact that any “theorem-proving robot-oracle” can itself become a new mathematical object: we can ask why its proof works, what features made the proof accessible to it and how its notion of an easy or natural argument differs from ours. This is very closely related to Lionel Levine’s recent essay Math for AI Safety: An Invitation for Mathematicians; I recommend the read. Either way, you might object that perhaps some of those questions will also be quickly answered by a later system. Fine. That system will then produce another object and another frontier. I don’t see any reason why this process should terminate.
All of this makes me want to learn more mathematics, and in particular more foundations: more logic, more category theory and even more set theory. This is my bet and you’re free to make your own. I obviously don’t know what math will look like and perhaps this is just me finding an elaborate justification for learning more about the subjects that I already enjoy. However, as I just said, if the act of finding a proof gets easier, then it seems reasonable to me that finding the best proof becomes more interesting and, to me, foundations supply the kind of language one needs to argue for mathematical beauty (obviously this is just my very biased opinion).
There is then the more immediate question of how we should interact with AI, and which interactions are productive for our cognition, our understanding and our society. I have been preaching to my students about this a lot. There is an interesting Atlantic article about the effects of AI on learning, and my takeaway is that AI seems capable of making us learn either much worse or much better. If I simply ask it for a solution and copy what it says, my brain turns off. But I can also use it to manufacture a mathematical opponent. When I am learning some new mathematics, or preparing to teach, I ask it to give me questions of increasing difficulty. I ask it, “by any means necessary, find something I do not understand in this chapter.” I ask it to behave like a clever or bored student and find the question that will derail my lecture. Recently I was preparing to teach logspace computation. I don’t work with this often and the last time I looked at it was when I taught the course last year. To be able freshen up on the details of Savitch’s theorem and the Immerman–Szelepcsényi theorem, I used AI to turn my turn my lecture preparation into a lemma-proving game. I had much more fun, and I actually understood the material much better than I would have by passively rereading the lecture notes I wrote in a rush while I was still learning Portuguese.
For now, I remain optimistic, perhaps to a fault. I am hopeful about the activity of doing mathematics, but less certain about the profession of being an academic mathematician as it is currently conceived: far too focused on publication numbers, scores, priority and speed, to the detriment of science and our mental health. This is probably too big a topic to engage with here and likely worth a whole other post. Either way, a time of great unrest is also a time in which we can think deeply about what we were doing in the first place. Perhaps some absurdities of metric-obsessed academic evaluation will become impossible to ignore. Perhaps we will be forced to be a little introspective and think more honestly what kind of understanding we value, what teaching should mean to us, and hey maybe even what kind of life is worth living. Ok, perhaps that’s a bit much for a post about math. But I guess a more modest ask is that we can become more demanding participants in The Great Conversation. Thinking (hard!) above AI seems like a good way of interacting with what some wish to sell as a “robot oracle”.