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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

#141

Post by zem »

cubefox wrote: post #116
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…

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...

wow, that was indeed a brilliant essay. i particularly liked the framing that "solving a big conjecture was a cryptographic proof that you had come up with a genuine conceptual innovation".

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

#142

Post by aspenmartin »

dvfjsdhgfv wrote: post #102
forgot_old_user wrote: post #80

Earlier quoted context omitted.

it will only get cheaper in the long run

for a sufficiently long definition of long

No for a very short definition of long, look at data on: how fast do prices decrease for a constant level of performance

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

#144

Post by red_admiral »

comboy wrote: post #24
phkahler wrote: post #21

I would have thought a triangular grid works better than a grid of squares. You get ~3n links vs ~2n for the square grid. Curious what the AI came up with.

Yes, not providing visualization of the solution seems criminal.

Unless it's a non-constructive proof.

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

#145

Post by CSMastermind »

0x5FC3 wrote: post #45
tonfa wrote: post #36

Earlier quoted context omitted.

Afaik this is because there is a community and database around them.

Interesting. OpenAI could also be trying to solve other problems, but Erdos problems maybe falling first?

No, Erdos problems were accepted as sort of a benchmark. There's a bunch of reasons they're favorable for this task:

1. They have a wide range of difficulties. 2. They were curated (Erdos didn't know at first glance how to solve them). 3. Humans already took the time to organize, formally state, add metadata to them. 4. There's a lot of them.

If you go around looking for a mathematics benchmark it's hard to do better than that.

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

#146

Post by segmondy »

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.

our local AI models are already capable of running McDonalds.

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

#147

Post by adam_arthur »

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…

Pretty much everything that appears novel in life is derivative of other works or concepts.

You can watch a rock roll down a hill and derive the concept for the wheel.

Seems pretty self evident to me

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

#148

Post by Karrot_Kream »

dvfjsdhgfv wrote: post #110
comboy wrote: post #41

Earlier quoted context omitted.

Not only it supercharged science it supercharges scientist. Research on any narrow topic is a different world now. Agents can read 50 papers for you and tell you what's where. This was impossible with pure text search. Looking up non-trivial stuff and having complex things explained to you is also amazing. I mean they don't even have to be complex, but can be for adjacent field where these are basics from the other f…

It's a very heavy hammer. I used it in the way you describe and after double-checking noticed some crucial details were missed and certain facts were subtly misrepresented. But I agree with you, especially in areas where they have a lot of training data, they can be very useful and save tons of time.

I don't think there's a substitute for reading the source material. You have to read the actual paper that's cited. You have to read the code that's being sourced/generated. But used as a reasoning search engine, it's a huge enabler. I mean so much of research literally is reasoning through piles of existing research. There's probably a large amount of good research (especially the kind that don't easily get grant funding) that can "easily" shake out through existing literature that humans just haven't been able to synthesize correctly.

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

#149

Post by protoplancton »

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.

One can argue that mathematical facts are discovered, but the tools that allow us to find, express them and prove them, are mostly invented. This goes up to the axioms, that we can deliberately choose and craft.

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

#150

Post by paulddraper »

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?

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