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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

#431

Post by horhay »

missyougowers wrote: post #348
ks2048 wrote: post #133

Timothy Gowers' tweet about this: "If you are a mathematician, then you may want to make sure you are sitting down before reading futher.". woah.

Unfortunately Gowers has taken Tao's lead on this one. Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs It is disheartening to see him jump into this GenAI puffery. I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work wi…

I'm not sure your characterization of Tao is accurate lol. In that companion paper, only Gowers seems to extensively show no pragmatism in the implications of this accomplishment. Even the younger math experts in that paper were a lot more cautious with their statements. Tao seems to follow that same tune most of the time even though he uses AI for first-pass inspections of solutions brought to his attention.

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

#432

Post by fragmede »

oh_my_goodness wrote: post #384
paulddraper wrote: post #150

Earlier quoted context omitted.

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

We don't know what human brains do.

We have some idea.

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

#434

Post by asdfasgasdgasdg »

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.

When people say this what they mean is that we've had plausibly useful LLMs for around three years, and I would say that is basically true.

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

#435

Post by jkhdigital »

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…

At this point I think the category theorists hit the foundational idea squarely on the mark:

1. Start with a few simple but non-trivial terms and axioms

2. Define "universal constructions" as procedures for building uniquely identifiable structures on top of that substrate

3. Prove that various assemblages of these universal constructions satisfy the axioms of the substrate itself

4. "Lift" every theorem proven from the substrate alone into the more sophisticated construction

I'm not a mathematician (I just play one at my job) so the language I've used is probably imprecise but close enough.

It may be true that you can't prove the axioms of a system from within the system itself, but that just means that you need to make sure you start from a minimal set of axioms that, in some sense, simply says "this is what it means to exist and to interact with other things that exist". Axioms that merely give you enough to do any kind of mathematics in the first place, that is. If those axioms allow you to cleanly "bootstrap" your way to higher and higher levels up the tower of abstraction by mapping complex things back on to the simple axiomatic things, then you have an "open" or infinitely extensible system.

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

#436

Post by asdfasgasdgasdg »

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.

When people say this what they mean is that we've had plausibly useful LLMs for around three years, and I would say that is basically true. The stuff before 2023 could barely be classified above the level of an interesting toy.

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

#437

Post by autoexec »

llbbdd wrote: post #402
beering wrote: post #353

Earlier quoted context omitted.

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 wa…

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, testabl…

If we managed to create very fast monkeys with typewriters and software that can review their output quickly enough that we end up with a result that's worth reading we'd still have people insisting that we've created intelligence. The monkeys however remain monkeys.

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

#438

Post by horhay »

alansaber wrote: post #7

AI isn't going to supercharge science but I wouldn't be as dismissive as other posters here.

It's a very complicated matter honestly. This is a new height that AI has reached, even though it follows the usual methods of success that it has had.

What strikes me as unusual though is that they do make a point of saying things like "this is a general purpose model that wasn't trained on the problem" among a few other things as if that's new. The last bountied problem they accomplished used a public model that ALSO didn't rely on specialized training. And that didn't make their blog.

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

#439

Post by mrdependable »

dmix wrote: post #413
mrdependable wrote: post #283

Earlier quoted context omitted.

How is that a "tin-foil-hat" take? It's not a secret, and in fact widely reported, that these companies are spending billions on creating training data.

So you think that OpenAI paid some mathematicians to either solve this conjecture problem, or a bunch of related unpublished math related to it, then fed it into an LLM model so they could announce it as being solved by the model? How is that not a conspiracy theory?

It is just a theory, the conspiracy part is not really applicable. I don't see what is controversial about it. Are you implying the machine taught itself the mathematics to do all this?
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