Or: Why searching for a solution to Navier-Stokes is fundamentally different than searching for a cancer cure
There are two kinds of mathematics: “simple” ones, with which you can create a complex video game, implement a working replacement for a worldwide financial ecosystem (Bitcoin), or even raw intelligence, with neural networks. This math is simple in the sense that it is basic linear algebra (for video games and AI) and even simpler (but extremely convoluted) math, for Bitcoin. What makes this type of math powerful is that when you combine it with a computing substrate, you get very impressive and useful results. You can simulate reality, get conversational intelligence, etc.
The second type of math is much more complicated, profound, philosophical and scary. It is the world of longstanding very difficult, possibly impossible to solve problems and conjectures. The simple task of understanding these problems is often a very difficult task, in itself. Are the “zeros” of a very exotic and weird function, defined in the complex plane, all lying on a particular strip? It seems that yes, but we’re not sure. Is there a fundamental wall separating a certain class of problems (P) from another class of problems (NP)? Nobody is certain, but there is a movie about it.
Before the advent of LLMs, the problem of searching for very difficult to find things (like the solution to a hard problem) was often performed with search algorithms, which implements the act of searching in a very straightforward way, akin to the way a chess engine might search for the best move, by moving in a tree of possible ones. This is a very powerful idea, but the problem is that it is not very general, and you have to adapt your search procedure in a very specific and non-obvious way, to every new problem you want to tackle.
What LLMs brought to the table is fluid linguistic and conceptual understanding and mastery. This extra, very powerful ingredient first began to impress us with its ability of writing poems, but very fast, we realized that it also allowed AI to reason. Reasoning is another kind of search, in the more abstract substrate of concepts and ideas. So the combination of fluid conceptual intelligence and reasoning allows AI to search for a solution in spaces that are much more subtle, which results, as we are witnessing these days, in solutions to math problems that proved, historically, extremely hard to solve.
Now what is the fundamental nature of the difficulty of a problem like cancer? I think it is probably more computational in nature, in the sense that it will probably require “simple” math, combined with huge computations. It is quite possible that the extra fluid intelligence ingredient will also provide tremendous help, in making sure that the full body of scientific knowledge has been properly read and integrated, but in the final resort, I suspect that the core engine of the eventual solution will be more in the raw computation domain, than in the highly abstract realm of extremely subtle ideas.