Daniel Litt

I think of my goal as a mathematician as something like: to better understand the fundamental concepts “shape” and “number,” and to convey that understanding to others. I think this is a valuable goal for many reasons. The primary reason discussed on social media is the (admittedly slow, but I think very real) diffusion of mathematical ideas into technologies that better our lives. There are others, though: production of human capital (which plays some role in this diffusion), and diffusion of mathematical understanding into culture, for example. It is not obvious to me that even the bare technological need for mathematics can be filled without human experts; this seems to me to be a tricky question of institutional design.

For me the most compelling reason is simply that we feel a pressing need to understand. While deep understanding is largely limited to professional mathematicians, the existence of mathematicians and of the institutions they make up is a huge part of what makes mathematical understanding available to those who want it. I believe that the capacity to understand mathematics (and the universe as a whole) is part of a good life, and we should try to provide people with a good life.

We operationalize the goal of understanding in a number of ways, problem-solving and theorem-proving among them. My own research is oriented around certain open questions that I believe measure our lack of understanding (of numbers, shapes and so on), and I think resolving those is genuinely important. I would welcome a solution with any provenance. But I think proof is only a measure of progress—what actually matters (for technology, human capital, culture, a good life) is a bit harder to get at.

Crucially it is very hard to measure progress—plausibly theorem-proving is used in large part because it is easy to measure rather than because it is a perfect proxy for what we care about. I worry a bit that we may automate many easy-to-measure things without automating the correlates we care about for which they are proxies. This presents a clear problem for the design of institutions aimed at achieving progress.

In part because I think problem-solving is a proxy for something else, I don’t feel that my goal of understanding “shape” and “number” is particularly threatened by a machine that is good at problem solving. I expect that machine to be a substantial boon, instead. Ultimately I think we will operationalize mathematical progress in better ways. But how to do this is a difficult question and I think many on here are too quick to dismiss concerns about it.

"Undeniably, things are going to change: typing code into a computer with your fingers already feels anachronistic, yet engineering remains challenging. There is so much more to the job of a mathematician than solving problems. I am optimistic that the era of “Big M” Mathematics and “Big P” Physics has now well and truly arrived: we can finally take much more ambitious steps and reconsider the big questions that have driven our fields for centuries."

About 77% of the OAIs slopdrop manuscripts have no lean certificate. Of them, a vast majority (though not all) are extremely difficult to parse even for domain experts.

What then makes OAI so confident in their correctness?

Persumably it reflects their confidence in the capabilities of the prover or of some auxilliary verifier.

Either way, the traditional standards of mathematical proof dating back to hilbert deem any such model to be part of the requisite evidence bundle!

Presenting the proofs as if they stand on their own is dishonest and in practice offloads the work of proving the claims onto the mathematical community!

So, in response to the mathpocalypse, I’ve seen people comparing math to the arts. I’m very sympathetic to this as a philosophical proposition, and just in general to the suffering people are going through. I’m not sure it’s a good argument to convince anyone outside of math.

Well worth reading: proofsandprompts.com/2026/10/08/100…

One of the three withdrawn manuscripts was “Algebraicity of Kuga–Satake Correspondences for K3 Surfaces” which I had started trying to understand; it’s a question I quite like. The ideas seemed to be exciting, though of course this is no longer clear.

In cases like this with a human-authored manuscript, one can often still extract something interesting from the wrong argument (which is often still based on some perhaps-correct intuitions of the author). It’s less clear if that is possible here, and my experience is that AI is not as helpful for this kind of fuzzy task as one might hope.

We've updated our GitHub math repo with 6 new Lean formalizations, 19 modifications, and 3 withdrawals. The repo now has ~42% top-line results formalized. We will continue to update the repo with new formalizations and with any errata we notice.

github.com/openai/math/bl…

I do think these are clearly something like "research findings" or "raw data" at this point. Very unusual situation in math.

Based on a hot tip: I had Astra check the OpenAI paper "Algebraicity of Weil classes on split abelian eightfolds", and it seems to be flawed. This is one of the nonformalized results. Concerning.

A lot of people—me included—do not enjoy reading LLM outputs. There’s an alternative!

Personally I have always been very bad at reading mathematics. But you don’t have to read it by reading it, if that makes sense. Rather, you think in its general vicinity.

I think there’s a lot of truth to this. I welcome these results but if we want humanity to understand them it’s also important to support and reward human expertise in mathematics, and we (the mathematical community and society at large) need to find ways to do so.

as unbelievable as it may sound to you, some people actually enjoy thinking about math and are happy to get paid a small sum to do so

Keeping up with my effort to explain to the Twitter crowd why the situation is less black-and-white than our sadly tribal times permit:

The same person with 20+ years of math research and mentoring experience (me) can, at the same time:

1- Be excited by today’s theorem drop, be impatient to find the time to look more closely at some of the theorems in my area, and try to understand whether the LLM just reshuffled the cards — which would still be impressive, but the problems in my area had already been worked on quite extensively, so an ingenious enough reshuffling may have done the job — or whether it came up with some genuinely new ideas. The latter possibility would be amazing. Some of the more striking results in the release, including the quasi-Riemann-Hypothesis result, make me hopeful that there may be something of that kind going on.

2- Still see various reasons why, if we don’t figure out the right way to process these inputs, to manage and study LLMs, and to recognize, integrate, and properly value human contributions, this may end up being a Pyrrhic victory for mathematics.

I understand that some are here just for the snarky comments or for joining one of two opposing camps. But what I’m trying to say is that this is one of those situations where doing that fails to capture both the complexity and the potential of what is happening.

I also think that the above dual state of excitement and prudence may be shared by many other people with actual experience in math research.

waiting for Greg Brockman to quote tweet one of my papers on the arithmetic of differential equations with this line

I think it's easy for outsiders (maybe also insiders?) to fail to appreciate just how big math is (to us); how many "famous" questions there are.

As a community we manage to have huge number of problems whose solutions would be considered "holy grail" type mega-level math.

I don't believe the rumors it has been resolved!

may age poorly but I think the Hodge conjecture, BSD, and Riemann hypothesis are probably all true

They seem to use the idea of getting quadratic growth in the height of rational points of a rank one elliptic curve that we introduce in this problem with Natalia and Xavier (ref [22]). I really need to see the details of this!

Fun! Looks like mathematicians have a lot of exciting work to do. If I understand correctly one of the results is a (very special) case of a conjecture of mine, which is also a consequence of stronger work in progress by a student of mine.

Dude, where’s my slop?