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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

#411

Post by Sohcahtoa82 »

ndiddy wrote: post #177
Silamoth wrote: post #140

Earlier quoted context omitted.

Out of curiosity, what issue did you have with the McDonald’s self-order kiosk? I actually think McDonald’s has the best kiosk I’ve ever encountered. The little animation that plays when you add an item to your cart is a little annoying (but I think they’ve sped that up). But otherwise, it’s everything I’d want. It shows you all the items, tells you every ingredient, and lets you add or remove ingredients. I have a b…

It takes longer than ordering with a cashier, it keeps trying to upsell you, and it's always out of receipt paper because unsurprisingly the company that isn't willing to pay a person to take orders is also not willing to pay a person to maintain the kiosks.

> It takes longer than ordering with a cashier

Depends on what you're ordering and who the cashier is.

If your order is the happy path of no customizations of a combo with an experienced cashier, it can be done in seconds, for sure. "Medium #4 with a Diet Coke", pay, done.

But if you customize your burger or ordering a lot of items a la carte and you're dealing with a new cashier that has weak English skills, good fucking luck. You'll likely need to wait for them to figure out they need to call someone over to help, have to repeat your order, and you end up spending far more time.

> it keeps trying to upsell you

Yeah, I'll agree that's obnoxious, especially when it's trying to upsell you something that's already on your order. I ordered a combo. I don't need you to add another fry.

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

#413

Post by dmix »

mrdependable wrote: post #283
Rover222 wrote: post #251

Earlier quoted context omitted.

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

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So you think that OpenAI paid some mathematicians to either solve this conjecture problem, or a bunch of related unpublished math related to it, then fed it into an LLM model so they could announce it as being solved by the model?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?

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

#414

Post by Yizahi »

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…

Because it is, for now. For a while at least. You can prove that LLM doesn't understand what it does and it is surprisingly simple. Request it to add two integers and then ask it to explain how it arrived at that result. The answer will be completely unrelated to the actual process LLM used because both results were generated independently and without understanding their meaning and connection.

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

#415

Post by School-Cotton »

CamperBob2 wrote: post #377
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

What's happening is the verbal/linguistic equivalent of the invention of calculus. No intellectual field will ever be the same again. Who wouldn't find that exciting, and want to experience it?

I'm not sure I grasp the analogy to the invention of calculus. Calculus helped us solve new and interesting math/physics problems. Repeated for emphasis: helped *us* solve.

This technology is solving interesting math/physics problems for us, which is completely different.

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

#416

Post by JonathanMerklin »

pseudocomposer wrote: post #222
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…

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 agree with almost all of what you have stated, save for a minor nitpick: I frankly don't think most functional adults think about the Fields Medal, similar annual prizes, or the qualities of the innovations of their candidate pools. I also think that that's totally okay. I think among a certain learned cohort of adults it's okay to hope that, and I think it's okay to imagine an idealized world where having an opinion on this sort of matter is a baseline, but I don't think it's realistic or fair to imply that (what I believe handwavily to be a majority of) adults are nonfunctional for not sharing this understanding.

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

#417

Post by ryandrake »

Silamoth wrote: post #342
ndiddy wrote: post #177

Earlier quoted context omitted.

It takes longer than ordering with a cashier, it keeps trying to upsell you, and it's always out of receipt paper because unsurprisingly the company that isn't willing to pay a person to take orders is also not willing to pay a person to maintain the kiosks.

Hmm. I’ve never really had those issues. It’s also much faster and easier than ordering with a human. I guess it does try to upsell you, but humans often do, too. And to me, it’s worth it to just click “No” in exchange for the added convenience (mostly in getting my order right). I have had them run out of receipts, but it’s never mattered for me. If I’m dining in, the plastic number you carry to your table makes sur…

> It’s also much faster and easier than ordering with a human.

I'm not sure how that could be. I can walk up to the counter and say "Big Mac Large Fry Small Coke" faster than you can navigate the first screen of the kiosk, and a skilled counter worker can key that in and be done before I even get my credit card out.

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

#418

Post by davebren »

beering wrote: post #353
ykl wrote: post #334

Earlier quoted context omitted.

I like to think of it as: Imagine every bit of human knowledge as a discrete point within some large high dimensional space of knowledge. You can draw a big convex hull around every single point of human knowledge in a space. A LLM, being trained within this convex hull, can interpolate between any set of existing discrete points in this hull to arrive at a point which is new, but still inside of the hull. Then there…

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…

> There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs

This doesn't make any sense, by their nature they can't "guess-and-check" things outside their training set.

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

#419

Post by ferris-booler »

What strikes me in this case (and I haven't seen in other comments) is that it's a _disproof_ of a conjecture put forth by Erdős and supported (at least according to OpenAI) by other professional mathematicians. Erdős, one of the greats, thought that the limit was O(n^{1 + o(1)}), which GPT disproved.

We can argue about recombination/interpolation of training data in LLMs, but even if this was an interpolation, the result was contrarian rather than a confirmation. Any system that can identify an error in Erdős's thinking seems very useful to me (though perhaps he did not spend much time thinking about or checking this particular conjecture).

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

#420

Post by FloorEgg »

cute_boi wrote: post #39
FloorEgg wrote: post #13

Earlier quoted context omitted.

Gemini seems better trained for learning and I think Google has made a more deliberate effort to optimize for pedagoical best practices. (E.g. tutoring, formative feedback, cognitive load optimization) As far as academic research is concerned (e.g. this threads topic), I can't say.

Gemini is like someone with short-term memory loss; after the first response, it forgets everything. That being said, I have checked multiple model and gemini can sometime give accurate answer.

Gemini is a series with a lot of individual models.

What you are describing doesn't match my experience at all with Gemini 3 or 3.1, especially the pro version.

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