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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

#171

Post by FrustratedMonky »

peterlk wrote: post #120
voooduuuuu wrote: post #98

Earlier quoted context omitted.

I think you are conflating composition and prediction. LLMs don't compose higher abstractions from the "axioms, symbols and rules", they simply predict the next token, like a really large spinning wheel.

Yes they do…? Who cares if they just predict the next token? The outcome is that they can invent new abstractions. You could claim that the invention of this new idea is a combination of an LLM and a harness, but that combination can solve logic puzzles and invent abstractions. If a really large spinning wheel could invent proofs that were previously unsolved, that would be a wildly amazing spinning wheel. I view LLM…

"Who cares if they just predict the next token?"

Exactly. I also only write one word at a time. Who knows what is going on in order to come up with that word.

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

#172

Post by sswatson »

blueone wrote: post #132
Tenobrus wrote: post #96

Earlier quoted context omitted.

what basis do you have for assuming an LLM is fundamentally incapable of doing this?

> what basis do you have for assuming an LLM is fundamentally incapable of doing this? because I have no basis for assuming an LLM is fundamentally capable of doing this.

Good on you for spelling out this reasoning, but it is manifestly unsound. For a wide variety of values of X, people a few years ago had no reason to expect that LLMs would be capable of X. Yet here we are.

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

#173

Post by peterlk »

voooduuuuu wrote: post #121
peterlk wrote: post #120

Earlier quoted context omitted.

Yes they do…? Who cares if they just predict the next token? The outcome is that they can invent new abstractions. You could claim that the invention of this new idea is a combination of an LLM and a harness, but that combination can solve logic puzzles and invent abstractions. If a really large spinning wheel could invent proofs that were previously unsolved, that would be a wildly amazing spinning wheel. I view LLM…

Ask an LLM to invent a new word and post it here, I will be waiting. You will see that it simply combines words already in the training data.

[dead]

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

#174

Post by bustermellotron »

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.

The grid of squares actually gets > Cn for any C. (More in fact… C can grow like n^a/loglog(n).) The AI proved > n^{1 + b} for some small b > 0, which a human (Will Sawin) has now proved can be about b = 0.014. The grid can be rescaled so the edges are not necessarily length 1, but other pairs will have length 1; that is necessary to get more than 2n unit distances.

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

#177

Post by ndiddy »

Silamoth wrote: post #140
edbaskerville wrote: post #92

Earlier quoted context omitted.

I just visited a McDonald's for the first time in a while. The self-order kiosk UI is quite bad. I think this is evidence in favor of the idea that an incompetent AI will soon be incompetently running a McDonald's.

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.

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

#178

Post by pulkitsh1234 »

dvt wrote: post #61
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 think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibniz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science. For example, many belive that to…

Creation is done by humans who have been trained on the data of their life experiences. Nothing new is being created, just changing forms.

A scientist has to extract the "Creation" from an abstract dimension using the tools of "human knowledge". The creativity is often selecting the best set of tools or recombining tools to access the platonic space. For instance a "telescope" is not a new creation, it is recombination of something which already existed: lenses.

How can we truly create something ? Everything is built upon something.

You could argue that even "numbers" are a creation, but are they ? Aren't they just a tool to access an abstract concept of counting ? ... Symbols.. abstractions.

Another angle to look at it, even in dreams do we really create something new ? or we dream about "things" (i.e. data) we have ingested in our waking life. Someone could argue that dream truly create something as the exact set of events never happened anywhere in the real world... but we all know that dreams are derived.. derived from brain chemistry, experiences and so on. We may not have the reduction of how each and every thing works.

Just like energy is conserved, IMO everything we call as "created" is just a changed form of "something". I fully believe LLMs (and humans) both can create tools to change the forms. Nothing new is being "created", just convenient tools which abstract upon some nature of reality.

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

#179

Post by volkercraig »

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.

> we'll see more specialized math AI resembling StockFish soon

Heuristically weighted directed graphs? Wow amazing I'm sure nobody has done that before.

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

#180

Post by ASalazarMX »

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.

Math is an abstraction of reality, it had to be invented, so more inventions or discoveries could be made within it.
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