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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

#971

Post by pojzon »

xbmcuser wrote: post #625
trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

This is the main thing that I keep harping about that human knowledge is too vast today for a person or even a group of people and llm will change that many discoveries that require serendipity in the past will be more likely than ever

There is one issue with this. When noone can prove or disprove what AI came up with.

Currently we can live with it because someone can review that work. Soon we wont be able.

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

#972

Post by necovek »

ori_b wrote: post #958
necovek wrote: post #957

Earlier quoted context omitted.

Yes, the photographic camera really destroyed the whole genre of photorealistic paintings and portraits!

This seems like a non-sequitur. The entire point of AI is that it's able to generalize and perform at or above human level. Let me know the next time your camera is able to simulate conceptualizing something. Or is your argument that AI is permanently doomed to not work?

It obviously does not follow since it precedes AI in "automating ... art" by 150 years.

What happened to art since?

We got artistic photography on top of paintings as well. It did not become widespread right away as people were mostly enamored by the simplicity of getting a realistic image first. But after that died away, people made art out of photography too.

Yes, those who did landscapes or portraits for hire were affected financially, but we ended up with more art, not less.

If I need to spell it out: genAI for image generation will also become an avenue for real artistic expression, as some pioneers are demonstrating, even if image generation is democratized, there will be a difference between art and non-art. It will also not kill conventional art either.

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

#973

Post by krackers »

lg5689 wrote: post #929
ccvannorman wrote: post #761

I looked at all linked articles and could not find an example of the points (they show a square grid of points with n~=100 but no other ordering of points to show the more optimal layout(s)). Is there anywhere an image example of a superior layout for example with n>={100,1000,10000}..? I would love to see it. I am imagining it would look somewhat like a sloppy pizza.

One of the authors said in a reddit comment (and I hope I am summarizing accurately) that it's impossible to show a diagram as the smallest instance of the technique gives like 10^1000000 points.

That was for this particular technique, but the comment suggested that a similar construction can (but is unproven) to beat erdos and at least gives some flavor of what this looks like

https://x.com/i/status/2057490144546927046

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

#976

Post by mejutoco »

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 is the old discovery vs invention in mathematics.

Or of you prefer philosophy: Parmenides (nothing changes) vs Heraclitus (you cannot bath twice in the same river aka everything changes all the time).

Postmodernism also claimed that everything has been done already. IMO these 2 are points of view that one can adopt, not truths based on fact. So the distinction is a matter of taste or perspective, not of truth, IMO.

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

#977

Post by anonymousDan »

adastra22 wrote: post #718
dyauspitr wrote: post #686

Earlier quoted context omitted.

> Only a rare few new theorems in mathematics nowadays have direct real world applicability. I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe. > If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative. I think the idea he…

The key word in that sentence is “new.” New math is typically explored without expectation of practical use. There are exceptions, but it is generally true. On the other hand, there are many applied mathematicians and theorists from other fields that mine new maths for applications to their fields. But they are almost always not the ones that come up with the new math. Historically, of course, mathematics was always…

Out of interest, what would you estimate the proportion of new maths that is used by other fields to be? Do you think much of this new maths is potentially underutilised as it were?

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

#978

Post by anonymousDan »

qnleigh wrote: post #798
trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

Can you describe what the reaction to these results has been like in your department? Obviously many people are excited, but what else? How do grad students feel about this? Are any professors getting worried about becoming obsolete?

One concern is that it will become more and more challenging to conduct cutting edge maths research without substantial resources only available at very rich institutions (to pay for state of the art AI assistants).

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

#980

Post by nathan_compton »

reactordev wrote: post #108
nathan_compton wrote: post #87

Earlier quoted context omitted.

Weird thing to say about a report which literally has the attached mathematical proof.

Except its not a proof. It's an existential proof of what? Projecting points and loosing density? Nah, it's wrong. At least with Edros you could solve f(x) or not solve it (inf). You can not with this. All they did was balance a really fancy quadratic equation. The projection from C^f to R² doesn't demonstrate geometric injectivity, so nⱼ = |X| isn't established, and the bound collapses.

You got downvoted, but I'm a bit curious - can you elaborate on these critiques?
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