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An OpenAI model has disproved a central conjecture in discrete geometry

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Re: An OpenAI model has disproved a central conjecture in discrete geometry

#321

Post by truncate »

dist-epoch wrote: post #263
truncate wrote: post #124

Earlier quoted context omitted.

What's your basis for assuming LLM is capable of doing this? I honestly don't know personally either way. Based on my limited understanding of how LLMs work, I don't see them be making the next great song or next great book and based on that reasoning I'm betting that it probably wont be able to do whatever next "Descartes, Newton, Leibnitz, Gauss, Euler, Ramanujan, Galois" are going to do. Of course AI as a wider fi…

LLMs are already making the next great songs. Just check out the Billboard charts.

I'm sorry, I don't consider them "great songs". Obviously, different people have different taste.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#322

Post by cpard »

margorczynski wrote: post #313
cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

Yep. The thing is people (maybe because of our limited scope) just focus on the depth and not the breadth. Because this is a general purpose model - it also has PhD+ knowledge in Physics, Biology, History, etc. I think we still don't really comprehend how much can be achieved by a single "mind" that has internalized so much knowledge from so many areas.

there's so much opportunity on the breadth of things too! I think that you end up having different people focusing on different things though.

Personally I'm a more of a breadth person and I could never compete with peers who where more of the depth type of person at college.

But I get satisfaction from connecting things that feel irrelevant on first sight, that's what drives me.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#323

Post by davebren »

mooreat wrote: post #164

I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by…

> I think that's just a matter of having them able to work on longer and longer time horizons.

No this will never do the kind of math that humans did when coming up with complex numbers, or hell just regular numbers ex nihilo. No matter how long it's given to combine things in its training data.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#324

Post by HarHarVeryFunny »

sillysaurusx wrote: post #165
m-hodges wrote: post #40

To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…

It’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

They recombine and reuse the patterns in their training data, not the surface level training data itself.

An LLM generating Arc code is using the LISP patterns it learnt from training, maybe patterns from other programming languages too.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#325

Post by anthk »

sumeno wrote: post #282
kelseyfrog wrote: post #278

Earlier quoted context omitted.

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.) The experiment is feasible. If it were performed and produced a positive result, what would it imply/change about how you see LLMs?

How are you going to train a frontier level llm with no references to post 1700 mathematics?

Archimede was close.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#326

Post by lacewing »

dwroberts wrote: post #99

Would be interesting to know what kind of preparatory work actually went into this - how long did it take to construct an input that produced a real result, and how much input did they get from actual mathematicians to guide refining it

Why?

It's clearly not yet a tool that can deliver new math at a scale. I say this because otherwise, the headline would be that they proved / disproved a hundred conjectures, not one. This is what happened with Mythos. You want to be the AI company that "solved" math, just like Anthropic got the headlines for "solving" (or breaking?) security.

The fact they're announcing a single success story almost certainly means that they've thrown a lot of money at a lot of problems, had experts fine-tuning the prompts and verifying the results, and it came back with a single "hit". But that doesn't make the result less important. We now have a new "solver" for math that can solve at least some hard problems that weren't getting solved before.

Whether that spells the end of math as we know... I don't think so, but math is a bit weird. It's almost entirely non-commercial: it's practiced chiefly in the academia, subsidized from taxes or private endowments, and almost never meant to solve problems of obvious practical importance - so in that sense, it's closer to philosophy than, say, software engineering. No philosopher is seriously worried about LLMs taking philosopher jobs even though they a chatbot can write an essay, but mathematicians painted themselves into a different corner, I think.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#327

Post by npfries »

amelius wrote: post #223
throw-the-towel wrote: post #59

See the longstanding debate on whether new math is "invented" or "discovered". Most mathematicians I knew thought it's discovered.

This is like saying a sculpture always existed, the sculptor just had to remove the superfluous material. Or like a musical octave has only 12 semitones, so all music is just a selection from a finite set that already existed. Sure the insane computation we're throwing at this changes our perspective, but still there is an important distinction.

Bob Ross would like a word. He frequently talked about objects or features already existing, and using the tools at his disposal to “find” them.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#328

Post by gus_massa »

mooreat wrote: post #164

I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by…

Searching for a proof and disproof are sometimes not so different. In most cases, you nibble the borders to simplify the problem.

For example, to prove something is impossible let's say you first prove that there are only 5 families, and 4 of them are impossible. So now 80% of the problem is solved! :) If you are looking for counterexamples, the search is reduced 80% too. In both cases it may be useful

In counterexamples you can make guess and leaps and if it works it's fine. This is not possible for a proof.

On the other hand, once you have found a counterexample it's usual to hide the dead ends you discarded.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#329

Post by charlie90 »

paulddraper wrote: post #150
m-hodges wrote: post #40

To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…

"LLMs just interpolate their training data" Cracks me up. What exactly do we think that human brains do?

I agree. Humans are given a body that lets them "discover" things on accident, test out ideas, i.e. randomness.

As in, I would hazard a guess the discovery of the wheel wasn't "pure intelligence", it was humans accidentally viewing a rock roll down a hill and getting an idea.

If we give AI a "body", it will become as creative as humans are.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#330

Post by beepbooptheory »

midtake wrote: post #291
m-hodges wrote: post #40

To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…

You have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand…

I agree with you all around except it's somewhat up for debate actually that the PI is "contradicting" the Tractatus. That is, there is the so called "resolute reading" of the Tractatus that had some traction for a while.

But note this is more to say that the Tractatus is like PI, not the other way around. And in that, takes like GPs would be considered the "nonsense" we are supposed to "climb over" in the last proposition of Tractatus.

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