it's in the literature
connections obvious and obscure
I’m writing this because I am going to need to refer back to it on Friday - I think that potentially this cute story from 2017 might be quite important for understanding what’s going on with the extremely impressive results that LLMs are getting in cutting edge mathematical research.
Basically, Thomas Royen was a retired statistician who had worked in the pharmaceutical industry. While brushing his teeth, he realised that a couple of formulas that were quite standard in that field could be put together to prove something about functions of correlated gamma distributions, and that this thing was equivalent to proving a general theorem that had as one of its special cases a famous unsolved problem called the Gaussian Correlation Inequality.
What’s interesting about this case is that everyone who had been working on the GCI had assumed that it would be solved by extremely tricky convex geometry or some such; as a result it was mainly worked on by different kinds of mathematicians who weren’t familiar with the same formulas as medical statisticians. The proof of the GCI was always there implicitly in the literature, but nobody who cared about one thing knew about the other and vice versa, until one person made the connection.
I think - let’s have a heated debate! - that this seems to be, from the press reports and comments from mathematicians close to the most recent LLM proofs, something like a model for how LLMs do it. If you’ve got an extrapolation engine, and you’ve interacted with lots of mathematical literature, then you are very well placed to industrialise the process of filling in those gaps.
On the one hand, this is really impressive, and absolutely gives the lie to any idea that “it’s just next token prediction” is an important thing to say about anything. If you’re pointing the vector in the right direction, next token prediction can be very useful!
But on the other hand, I am not yet convinced that this is the kind of thing which produced the real era-defining breakthroughs in mathematics, and more importantly, I don’t think it’s a trick which will transfer all that well to other sciences, even less well to the humanities and not at all to my personal area of interest, general management.
What do you think?
This really speaks to me as a former math olympiad kid who was for a time a management consultant
I think the post and a lot of the comments are drawing upon a distinction between exploration within a known problem space and the creation of new problem spaces - and whether AI can cross that line.
I really like this post from mathematician Daniel Litt (https://www.daniellitt.com/blog/2026/2/20/mathematics-in-the-library-of-babel), who engages with this. He sees little sign of models autonomously building theory, i.e., creating new frameworks rather than proving results within existing ones. The Royen story is a perfect case of the latter. The more recent Erdős unit distance result from May fits the same pattern
When thinking about what makes new-space creation hard to automate, I think its connected to James C. Scott's concept of metis; the messy contextual (or what I remember of it). Mathematicians are drawn to structures that feel productive in ways shaped by their moment, their aesthetics, the tools they happen to know, the generation of people they draw upon (connecting to Felix Salmon's comment). That's part of what creates areas that are exploitable by AI (forgotten paths).
But the same contextual pressures that cause mathematicians to overlook certain paths (creating the gaps AI can exploit), I hypothesize, are also what drive them to create new frameworks in the first place. You compress because you have to, and the shape of that compression is itself a creative act shaped by the specific moment, the specific metis. It defines the "taste" of what's worth pursuing and directing one's creativity towards.
That said, this interview (https://www.youtube.com/watch?v=DRcFvXAcxMg) with Tudor Achim (CEO of Harmonic, which tied OpenAI and DeepMind for IMO gold in 2024) is quite the provocation. He describes their process as using hallucination as the engine of creativity: models generate a wide tree of approaches, mostly wrong, and a formal verifier prunes them. Hallucination also injects entropy, novel combinations the model hasn't seen. It could be that throwing scale at the problem then allows some stumbling into genuinely novel territory.
Where I most agree with you is on management being a harder transfer case. The metis in business isn't just hard to codify, its itself dynamically being redefined, as it feeds on itself. People act against their conditions, which changes the conditions, which regenerates the metis. The frontier is always a moving post. You don't get that with math - unsolved problems don't change their nature just because someone is looking at them.
Dwarkesh Patel just did a podcast episode with Grant Sanderson (creator of 3blue1brown) about AI for math that spent a lot of time on this topic. I don't think they come to any conclusions that these comments haven't mentioned (it all really depends if the models of 2029 can do what Galois and Grothendiek did) but it's a really nice and illuminating discussion.