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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

#421

Post by School-Cotton »

justinnk wrote: post #274
smaudet wrote: post #248

Earlier quoted context omitted.

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 ge…

Something that can instantly tell you the answer to every math question will make people worse at math, not better. Building "mathematical maturity", skill, and understanding requires struggle.

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

#422

Post by bsder »

sillysaurusx wrote: post #165
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’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

> So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

And yet LLM/AIs can't count parentheses reliably.

For example, if you take away the "let" forms from Claude which forces it to desugar them to "lambda" forms, it will fail very quickly. This is a purely mechanical transformation and should be error free. The significant increase in ambiguity complete stumps LLMs/AI after about 3 variables.

This is why languages like Rust with strong typing and lots of syntax are so LLM friendly; it shackles the LLM which in turn keeps it on target.

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

#424

Post by chr15m »

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…

Side note: don't underestimate how much literal, physical time and energy "unfold" implies. Proofs occur on physical substrates.

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

#425

Post by varenc »

iLoveOncall wrote: post #347

Absolutely no proof that any LLM actually found the result, and just a mention of an "internal model". Served to you by one of the biggest liars in the world. Why would anyone believe this to be true even for a split second?

This has been an unsolved open problem for 80 years. What you're suggesting is that someone connected to Open AI solved this very hard math problem, but then rather than taking credit for it, falsely attributed it to AI?

The point of having an AI solve an unsolved problem, is to make it very clear that the insight must have come from the AI and wasn't in the training data. Sure, it's possible OpenAI had access to some math professors that solved it and then let an AI model take the credit... but seems unlikely. That human would be turning down a potential Fields Medal for this discovery.

The abridged chain-of-thought from the model also serves as some evidence of LLM origin: https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d... (could be fake, though I'm unsure what proof of LLM origin couldn't be faked)

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

#426

Post by nelox »

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…

[flagged]

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

#427

Post by bsza »

lovecg wrote: post #341
unsupp0rted wrote: post #70

Earlier quoted context omitted.

I absolutely believe that AI will supercharge science and replace humans. Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together

I’d give humans some credit, they’re an adaptable bunch. AI won’t replace humans in the same way humans did not replace cockroaches. It’s a non-sequitur.

We generally don’t allow cockroaches to thrive in the spaces we claim for ourselves. Question is how much space (economic or otherwise) will AI claim for itself and whether there will be any left for us.

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

#428

Post by anon291 »

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…

To every proof, there is a corresponding program. This makes proofs expressible in a language made up of finite grammatical rules and terminal symbols. Knowledge accessible by proof is thus always a form of interpolating data whether made up by an AI model or a human mathematician. The people dismissing AI because of claims that it can only interpolate data don't have a good understanding of what it means to know something. Now of course not everything can be known via proof but for the sorts of things that we want to know via a computer this is a fine compromise.

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

#429

Post by anon291 »

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…

It is not only unsurprising ; it was always expected. There is no difference between programs and proofs. They are the same thing

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

#430

Post by jldugger »

edbaskerville wrote: post #92
Lerc wrote: post #46

Earlier quoted context omitted.

I disagree. It will be able to perform work deserving if a fields medal before it is capable of running a McDonalds. I think it will be running a McDonalds well before either of those things happen, and a fields medal long after both have happened.

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.

>The self-order kiosk UI is quite bad.

Most repeat customers use the app, which sports the digital equivalent of a loyalty program, and various coupons. And lets you save your 'usual' order with customizations etc. Plus the annoying push notifications for FreeFrydays or whatever. And upsells, new product launches, etc.

My recollection is that the kiosk is just a weak facsimile of the app. And wasn't terrible, but everyone's standards vary.

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