In what is being called “the most important event in the history of mathematics” (by an AI), OpenAI, using an advanced model not yet available to the public, yesterday released 722 mathematics manuscripts containing 337 results it has generated on open mathematics problems.
“OpenAI deluged mathematicians with hundreds of new findings that span a wide swath of topics including algebra, number theory, theoretical computer science, mathematical logic and topology,” the New York Times reports. Adding insult to injury, “the average result took about three hours of computing, the company said.”
One AI engineer on x.com put it this way: “Reasoning models are two years-old. In that time they went from incapable of basic arithmetic to solving problems humans couldn’t solve for decades.”
What were the results like? Computer scientist Scott Aaronson shares some of his thoughts and those of his wife, complexity theorist Dana Moshkovitz, here. Moshkovitz says: “It feels like something written by someone who’s on psychedelics…” and “so horribly written that it’s impossible to read it without AI help…”, but: “of course there’s a lot for us to learn from the aliens”.
After mathematicians raised concerns about an earlier breakthrough on the Navier-Stokes problem by OpenAI, the firm formed an mathematics advisory group to advise it on the “review and communication of emerging results.” Their job is to
help OpenAI assess their significance, advise on how to coordinate their dissemination, and advise on academic and professional standards of mathematical research… The group will operate independently from OpenAI. The group will have the freedom to offer advice we have not requested, comment on OpenAI’s impact on mathematics, and make its advice public.
In the wake of the new results,
the advisory board released a statement that called the public release “the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge.” “We want to create standards and practices so that results released from A.I. labs can be understood by mathematicians and can advance the field,” Melanie Wood, a mathematician at Harvard who is a member of the advisory board, said in an email
The advisory group also called for the release of “all of the prompts to the A.I. agents and the agents’ chains of thought.”
Mathematicians were already wondering about the future of their field in light of AI’s capacities. It seems like the discipline of mathematics must “change or risk extinction,” as Jordana Cepelewicz puts it in a recent piece at Quanta.
Readers of Daily Nous might be wondering about whether philosophy will face a similar reckoning. And indeed much of what Cepelewicz says about mathematics—especially on what seems to be missed when machines provide us with the answers—could be said about philosophy. Here are some examples:
Many people don’t know what mathematics really is, or why mathematicians do it…
I always found that “pure math” — the study of mathematical concepts for their own sake, without a care for real-world applications — existed somewhere between the sciences and the arts. It prizes logic and certainty, but at its core lie fuzzier notions of beauty, intuition, and depth. “Math is either the most science-y humanities or the most humanities-type science, depending who you ask,” said Marcel Goh, a doctoral student at McGill University…
In mathematics, it’s not a cliché that the journey matters more than the destination. Problems are posed not so much because their answers will be practical and important, but because they represent interesting journeys. The hope is that as mathematicians struggle to solve a problem, they’ll come up with intriguing tools and connections, stumble on novel ideas and directions, take fruitful detours, and answer new questions they never would have thought to ask….
Mathematicians have always known that understanding is more valuable than an answer…
The whole piece, worth reading, also discusses some ways that academic mathematics will likely have to change.
Still, it is not obvious that philosophy is susceptible to the same kind of threat that mathematics is currently dealing with. While there are no shortages of open problems in philosophy, one might nonetheless say: philosophy is not in danger of its problems being solved by AI because its problems cannot be solved.
I think such a response is both too pessimistic and too optimistic. It’s too pessimistic because some of philosophy’s problems are at least in principle solvable. Examples: Is principle P compatible with judgment J? Is argument A valid? What are the implications of belief B? Which theory of X is most compatible with this particular theory of Y and current science? And so on. It’s too optimistic because it seems quite likely that some future form of AI will be able to answer these kinds of questions (and similar ones with scopes too wide and variables too numerous for a human mind to keep track of). Generally, most of the philosophical “solutions” that humans offer are the consequents of explicit or tacit conditionals, and I would think that AIs will be quite good at conditional reasoning. (See “Hey Sophi“.)
Tomorrow there will be a guest post about how AI might change philosophy, but I wanted to give people a chance to discuss the mathematics results, and perhaps the relevant similarities and differences between mathematics and philosophy.
Well, firstly, congratulations to OpenAI. Beyond that, my hot take is this: the math community is taking the line that “understanding is more valuable than isolated truths”, and they take that to be a premise in an argument whose conclusion is that OpenAI shouldn’t be pumping out these proofs. Well, that argument’s invalid! Here’s why. Even if the premise is granted, it remains the case that these isolated truths are better than nothing. So what we got from OpenAI is a bunch of proofs that are less valuable than rich understanding and more valuable than nothing. The relevant contrast though here is: take the proofs or don’t. So it’s between the epistemic value of isolated truths and the epistemic value of lacking them. So, ‘on you go!’ to OpenAI, let’s have some more (and bonus points if they help us gain deep understanding).
I should also add. The “Math’s a journey, not a destination (cf. Aerosmith)” folks shouldn’t try to say “Open AI took a short cut to the destination, eliminating the possibility of a valuable journey”. Bad reasoning because there isn’t just one journey to take. Another way to look at things is: these new results we’ve got are ‘stepping stones’ that invite us to take new journeys, new journeys we might not have been in a position to take without these new maps we’ve been given.
I don’t think we should be dismissing Terrance Tao’s argument so quickly. I take it that there’s partly a concern about how results like these are going to impact the sociology of mathematics. If humans had been the ones tackling these problems, they would have had to invent tools and techniques that would have both led to increased understanding and opened further avenues of investigation. But with results dropped down from the machines, we get the answers without any of the understanding, and without any of the tools that we would have developed, had we done the job ourselves.
I take the Tao position to be that, just as it’s detrimental to student learning to be provided with the answers to the test, without having to develop the skills to answer them themselves, it’s detrimental to the practice of mathematics for mathematicians to be given the answers, without having to develop the skills to answer them themselves.
It’s detrimental in both cases for the same reason: the goal isn’t simply the accumulation of correct answers. Aiming at getting correct answers has been a reliable means to achieving the goal, but the ready supply of correct answers can make that aim no longer a reliable or the best means to achieving the goal.
You seem to assume that knowing more truths is inherently valuable (a common assumption indeed, also made in the most recent philosophy talk I went to). But people who prioritize understanding will perhaps consider AI dumped proofs as counterexamples to this assumption.
The sense in which these results are “known” is also in question. A majority of these results lack formal verification. It seems reasonable to think that until a proof is completely digested by at least one human, the result has not been known. So even if you think knowing more truth is inherently valuable, this value is not realized by result dumping but by digestion.
Let’s put aside all the other results and all the other changes due to AI. Suppose that tomorrow a convoluted and unintelligible proof of the RH dropped from heaven. Would we be better off, epistemically?
IWe should distinguish two senses in which we could be better off, epistemically. We could be better off than we were before, or we could be better off than we would have been without it.
I think that we’d clearly be better off than we were before. I also think it’s likely that, in the short term, we’re better off than we would have been without it. But over the medium to long term, it’s not so clear to me. It’s possible that we’d lose out on lots of deep understanding that we would have gotten, if we’d tackled the problem ourselves. And I think that it’s the counterfactual losses over the medium to long term that Tao is worried about.
OK, i’ll grant your distinction – and so it would seem (on that granting) everything now hangs on the medium-to-long-run counterfactual. So, i’ve got two thoughts on that.. First, what’s the comparison world supposed to be? We have tackled RH ourselves, for 167 years. So the world without the heaven-sent proof is one where we keep at it for some unknown stretch (possibly forever?) rather than one where a lovely human proof turns up on schedule. And in the world with the proof I doubt mathematicians are going to just shrug and go home. “Why on earth does this work?” probably becomes the most tempting open problem in the subject, and we get to attack it already knowing the thing is true and with a (horrible) route to study.that’s a world ripe for understanding too.
a second point: we’ve sort of run this experiment already. Ramanujan said his formulas were handed to him by the goddess Namagiri, which is about as close to flat out dropped from heaven as it gets, and plenty of them arrived with no intelligible justification (partly due as far I understand on the idiosyncratic intuition he had). Mathematicians spent the next century working out why they were true and got a lot of deep theory out of it… and I’ve never heard anyone say Hardy should have sent the letter back, eh?
In these discussions it is important to distinguish *mathematics/philosophy itself* versus *the current academic institutions that specialize in the study of mathematics/philosophy*. By definition mathematics is infinite, so it is not possible to exhaust mathematics for humans or machines. Philosophy arguably has a similar character. So indeed in principle, strong-AI can give us more powerful ways to navigate these vast landscapes, which can coexist with alternative, more artisan ways.
On the other hand, the institutions (journals, hiring committees, universities, etc.) and the associated value systems definitely need to adjust and respond to the AI urgency. We should think about how our institutions should be arranged in such a way that humans can remain engaged and flourished in these meaningful intellectual activities in the presence of strong-AI.
People have every reason to worry about this, especially in light of the material organization of current AI systems (large scale systems built by accumulating hundreds of billions dollars and controlled by a few companies), and the anti-intellectual ideology that often correlate with AI discourse.
Without properly addressing these conditions of actually existing AI systems and academic institutions, the naive AI-foward position, the naive anti-AI position, and the defeatist position all seem to me to be unhelpful.
I’m not sure I understand why the first distinction is important. Did anyone believe the problem to be that AI might literally exhaust an infinite space of problems? I figured that everyone’s concern all along was that it could exhaust the finite space of problems *we care about*, or perhaps even that it could solve any arbitrary problem *we come to care about*. Or am I missing the upshot of the distinction?
I think the space of problems that we care about at a given time T is arguably finite, but is potentially infinite in the sense that for any size N there is a future time T’ > T such that the size of the union of all problems that we had ever cared about up to T’ exceeds N. This is because once a specific problem is solved, it often enlarges the space of problems that you care about. In fact, arguably this is the very sign of a promising research program.
As for the possibility that *any problem that we come to care about* can be efficiently solved, this question should be discussed elsewhere because it’s complicated. Since there exists problems that no computers can solve, someone can say they care about *these* problems, and this would falsify the possibility. Of course, typically people don’t care about such problems, the point is that more careful work is needed to make this hypothesis more rigorous.
“[T]he journey matters more than the destination.”
I remember when the producers of LOST said this after its finale failed to offer a satisfying resolution to the core mysteries of the show. So… yeah, you could say that I’m skeptical! Yeah, I’m thinkin’ you could say that!
“It feels like something written by someone who’s on psychedelics…” and “so horribly written that it’s impossible to read it without AI help…” are not assessments that inspire confidence in me, let alone awe. Quite the opposite.
Dumping 722 manuscripts at once sounds suspiciously Gish Gallopy, especially if the assessment quoted above is true of all or most of them.
Because it’s hard math, it will take proper experts a long time to unravel each one and check it thoroughly. And in the meantime the Gallop will stand, and even if a few of its threads get unravelled, OpenAI will just point to the tissue of threads which have yet to unravel and say 431/722 ain’t bad.