alt Hacker News> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibniz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science.
For example, many belive that to sovle the Riemann Hypothesis, we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.
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"new kind of math"
Well I think the point is there is no "new kind of math". There's just types of math we've discovered and what we haven't. No new math is created, just found.
It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances!
This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are discontinuous functions! We just need to replace a few axioms (chuck out law of the excluded middle, and throw in some continuity axioms).
I think “new math” is ‘just’ humans creating new terminology that helps keep proofs short (similar to how programmers write functions to keep the logic of the main program understandable), and I agree that is something LLMs are bad at.
However, if that idea about new math is correct, we, in theory, don’t need new math to (dis)prove the Riemann hypotheses (assuming it is provable or disprovable in the current system).
In practice we may still need new math because a proof of the Riemann hypotheses using our current arsenal of mathematical ‘objects’ may be enormously large, making it hard to find.
what basis do you have for assuming an LLM is fundamentally incapable of doing this?
math more like an art than a science.
we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.
In concrete terms: could a thousand LLMs-driven agents running on supercomputers—500 of which are dedicated to building software for the other 500-come up with new math?
Creation is done by humans who have been trained on the data of their life experiences. Nothing new is being created, just changing forms.
A scientist has to extract the "Creation" from an abstract dimension using the tools of "human knowledge". The creativity is often selecting the best set of tools or recombining tools to access the platonic space. For instance a "telescope" is not a new creation, it is recombination of something which already existed: lenses.
How can we truly create something ? Everything is built upon something.
You could argue that even "numbers" are a creation, but are they ? Aren't they just a tool to access an abstract concept of counting ? ... Symbols.. abstractions.
Another angle to look at it, even in dreams do we really create something new ? or we dream about "things" (i.e. data) we have ingested in our waking life. Someone could argue that dream truly create something as the exact set of events never happened anywhere in the real world... but we all know that dreams are derived.. derived from brain chemistry, experiences and so on. We may not have the reduction of how each and every thing works.
Just like energy is conserved, IMO everything we call as "created" is just a changed form of "something". I fully believe LLMs (and humans) both can create tools to change the forms. Nothing new is being "created", just convenient tools which abstract upon some nature of reality.