A lot of the space outside of the convex hull is just untried things. You can brute-force trying random things and checking the result and eventually learn something new. With a better heuristic, you can make better guesses and learn new things much more efficiently. There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs, or that most of our new discoveries are not found the same way.

I think of most things you can get to by guess and checking as definitionally inside of the hull; most forms of guess and checking are you take some existing thing, randomize a bunch of its parameters, and see what you get. Whereas with something like relativity, there's not even a starting point that you can randomize and guess/check from the pre-existing knowledge space that will lead you to relativity. That's more like, adding a new dimension to the space entirely.

It's possible LLMs can handle this after all! But at least so far we only have existence proofs of humans doing this, not LLMs yet, and I don't think it's easy to be certain how far away LLMs are from doing this. I should distinguish between LLMS and AI more generally here; I'm skeptical LLMs can do this, I think some other kind of more complete AI almost certainly can.

I supposed you could just, I dunno, randomly combine words into every conceivable sentence possible and treat each new sentence as a theory to somehow test and brute force your way through the infinite possible theories you could come up with. But at that point you're closer to the whole infinite random monkeys producing Shakespeare thing than you are to any useful conclusion about intelligence.

I come back to something like this idea when I consider the distinction being made that LLMs can only combine and interpolate between points in their training material. I could write a brute-force program that just used an English dictionary to produce every possible one-billion-gazillion word permutation of the words within, with no respect for rules of language, and chances are there would be some provable, testable, novel insight somewhere in the results if you had the time to sift through and validate all of it. LLMs seem like a tool that can search that space more effectively than any we've had before.

It's also worth noting in that in very high dimension, the convex hull will contain massive volume. It could well be the case that humans established that convex hull millions of years ago, and all of our inventions and innovations sense have fallen inside it.

> There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs

This doesn't make any sense, by their nature they can't "guess-and-check" things outside their training set.

> You can brute-force trying random things and checking the result and eventually learn something new.

And most of the mathematicians seem to welcome this "brute forcing" by the LLMs. It connects pieces that people didn't realize could be connected. That opens up a lot of avenues for further exploration.

Now, if the LLMs could just do something like ingesting the Mochizuki stuff and give us a decent confirmation or disproof ...