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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

#291

Post by midtake »

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 the full vastness of what would objectively be "math" and that it is possible that our perceived math is terribly wrong and a subset of a greater math. Whether that greater math has the same seemingly closed system properties is not something that can be known.

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

#293

Post by danielmarkbruce »

symfrog wrote: post #260
bbor wrote: post #246

Earlier quoted context omitted.

To keep my usual rant short: I think you’re assuming a categorical distinction between those two types of innovations that just doesn’t exist. Calculus certainly required some fundamental paradigm shifts, but there’s a reason that they didn’t have to make up many words wholesale to explain it! Also we shouldn’t be thinking about what LLMs are good at, but rather what any computer ever might be good at. LLMs are alrea…

We have had LLMs for much longer than 3 years.

No, we haven't, for any reasonable definition of L.

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

#294

Post by agentultra »

I’m curious about the “autonomous” claim. Usually these systems require a human to guide and verify steps, clarify problems, etc. are they claiming that the reinforcement model wasn’t given any inputs, tools, guidance, or training data from humans?

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

#295

Post by iwontberude »

nh23423fefe wrote: post #118
dvt wrote: post #105

Earlier quoted context omitted.

Because by definition LLMs are permutation machines, not creativity machines. (My premise, which you may disagree with, is that creativity/imagination/artistry is not merely permutation.)

god of the gaps

non overlapping magisteria

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

#296

Post by jiggawatts »

net01 wrote: post #288
Rover222 wrote: post #251

Earlier quoted context omitted.

Seems like a very tin-foil-hat-take to me

I’m quite certain that a few months ago, some problems were claimed to be solved by AI. However, those claims were actually false and were exactly that, solved erdos problems that were not marked as solved and the solution was "found" by AI. edit: >> https://techcrunch.com/2025/10/19/openais-embarrassing-math/

The corollary is that this is a very valuable capability of AI!

The ability to find incredibly obscure facts and recall them to solve "officially unsolved" problems in minutes is like Google Search on steroids. In some sense, it is one core component of "deep expertise", and humans rely on the same methodology regularly to solve "hard" problems. Many mathematicians have said that they all just use a "bag of tricks" they've picked up and apply them to problems to see if they work. The LLMs have a huge bag of very obscure tricks, and are starting to reach the point that they can effectively apply them also.

I suspect the threshold of AGI will be crossed when the AIs can invent novel "tricks" on their own, and memorise their own new approach for future use without explicitly having to have their weights updated with "offline" training runs.

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

#297

Post by tern »

daishi55 wrote: post #202
lubujackson wrote: post #60

For anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time. Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool…

> the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept. Isn't this just anthropocentrism? Why is understanding only valid if a human does it? Why is knowledge only for humans? If another species resolved the contradictions be…

Do the forms etched into stone by weather over millennia in Moab matter to the wind? Certainly yes, in one sense, but not in the same sense we mean when we say things matter to us, or to animals, or even bacteria.

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

#298

Post by auggierose »

vatsachak wrote: post #11

As I have stated before, AI will win a fields medal before it can manage a McDonald's A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation. LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.

> A difficult part was constructing a chess board on which to play math

We have that chess board for quite a while now, over 40 years. And no, there is nothing special about Lean here, it is just herd mentality. Also, we don't know how much training with Lean helped this particular model.

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

#299

Post by dawnerd »

raincole wrote: post #280

I like how everyone laughed when OpenAI said their models will have "PhD-Level Intelligence" and now the goalpost has been moved to if AI can create new math (i.e., not PhD-Level, but Leibniz/Euler/Galois level.)

Yet it still codes like a junior developer that memorized all of stack overflow.

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

#300

Post by KalMann »

CGMthrowaway wrote: post #199

How do you even get an LLM to try to solve one of these problems? When I ask it just comes back with the name of the problem and saying "it can't be done"

Maybe you need to phrase it better. Like with a more specific direction of thinking.
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