Can anyone find (or draw) a picture of the construction?
Yeah, unfortunately, they just proved there existed a better solution, they didn't construct it.
(Though in some ways that's actually more impressive.)
An OpenAI model has disproved a central conjecture in discrete geometry
271–280 of 1001 posts
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#272For 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…
There is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...
> The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.
I just wanted to highlight this very correct human-centric thought about the purpose of intellection.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#273To 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…
I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz…
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Re: An OpenAI model has disproved a central conjecture in discrete geometry
#274To 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…
If anything, this is more illustration of how llms are not useful to us... They will do their own thing, don't need us. In fact, we will be in the way... We can choose to study them and their output, but they don't make us better mathematicians...
I see where you are coming from.
However, in the role of personal teachers they may allow especially our young generations to reach a deeper understanding of maths (and also other topics) much quicker than before. If everyone can have a personal explanation machine to very efficiently satisfy their thirst for knowledge this may well lead to more good mathematicians.
Of course this heavily depends on whether we can get LLMs‘ outputs to be accurate enough.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#275Earlier quoted context omitted.
what basis do you have for assuming an LLM is fundamentally incapable of doing this?
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…
"I don't see them be making the next great song"
Meanwhile, songs are hitting number one on some charts on Spotify that people think are humans and are actually AI. And Spotify has to start labelling them as such. One AI "band" had an entire album of hits.
Also - music is a subjective. Mathematics isn't.
And in this case, an LLM discovered a new way to reason about a conjecture. I don't know how much proof is needed - since that is literally proof that it can be done.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#276This is impressive, no question. Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain origi…
Seems like a very tin-foil-hat-take to me
I'm not letting the government read my brainwaves.
In all seriousness though: My suggestion is that those shepherding the frontier of AI start acting with more transparency, and stop acting in ways that encourage conspiratorial thinking. Especially if the technology is as powerful as they market it as.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#277See 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.
The difference is that math answers (can answer) specific questions.
Like, "does the Riemann zeta function have zeroes that don't have real part 1/2," or "is there a better solution to the Erdős Unit Distance Problem."
The selection of question is matter of taste, but once selected, there is a definitive precise answer.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#278To 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…
I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz…
> 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?
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#279To all AI skeptics: What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?
Maybe after decades. 2022 models were microscopic compared to latest models.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#280I 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.)