[Davidzheng]

Actually i also think it's possible. Start with natural numbers axiom system. Form all valid sentences of increasing length. RL on a model to search for counter example or proofs. This on sufficient computer should produce superhuman math performance (efficiency) even at compute parity

[MrLeap]

I wonder how much discovery in math happens as a result in lateral thinking epiphanies. IE: A mathematician is trying to solve a problem, their mind is open to inspiration, and something in nature, or their childhood or a book synthesizes with their mental model and gives them the next node in their mental graph that leads to a solution and advancement.

In an axiomatic system, those solutions are checkable, but how discoverable are they when your search space starts from infinity? How much do you lose by disregarding the gritty reality and foam of human experience? It provides inspirational texture that helps mathematicians in the search at least.

Reality is a massive corpus of cause and effect that can be modeled mathematically. I think you're throwing the baby out with the bathwater if you even want to be able to math in a vacuum. Maybe there is a self optimization spider that can crawl up the axioms and solve all of math. I think you'll find that you can generate new math infinitely, and reality grounds it and provides the gravity to direct efforts towards things that are useful, meaningful and interesting to us.

[soulofmischief]

As I mentioned in a sister comment, Gödel's incompleteness theorems also throw a wrench into things, because you will be able to construct logically consistent "truths" that may not actually exist in reality. At which point, your model of reality becomes decreasingly useful.

At the end of the day, all theory must be empirically verified, and contextually useful reasoning simply cannot develop in a vacuum.

[staunton]

Those theorems are only relevant if "reasoning" is taken to its logical extreme (no pun intended). If reasoning is developed/trained/evolved purely in order to be useful and not pushed beyond practical applications, the question of "what might happen with arbitrarily long proofs" doesn't even come up.

On the contrary, when reasoning about the real world, one must reason starting from assumptions that are uncertain (at best) or even "clearly wrong but still probably useful for this particular question" (at worst). Any long and logic-heavy proof would make the results highly dubious.

[danenania]

A question is: what algorithms does the brain use to make these creative lateral leaps? Are they replicable?

Unless the brain is using physics that we don’t understand or can’t replicate, it seems that, at least theoretically, there should be a way to model what it’s doing with silicon and code.

States like inspiration and creativity seem to correlate in an interesting way with ‘temperature’, ‘top p’, and other LLM inputs. By turning up the randomness and accepting a wider range of output, you get more nonsense, but you also potentially get more novel insights and connections. Human creativity seems to work in a somewhat similar way.

[kmeisthax]

https://en.wikipedia.org/wiki/Monstrous_moonshine#Origin_of_...

[iczero]

I believe https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... (Gödel's incompleteness theorems) applies here