it's in the literature

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?