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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

#651

Post by jonahx »

midtake wrote: post #291
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…

You have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand…

Later Wittgenstein held the same view of mathematics, and wrote about it extensively. He was firmly in the "invention, not discovery" camp.

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

#652

Post by jonahx »

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…

> held that mathematical truths don’t report new facts about the world

I'm not as familiar with the early work, but later Wittgenstein held this belief too.

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

#653

Post by qnleigh »

isotypic wrote: post #608
ted_dunning wrote: post #505

Earlier quoted context omitted.

In Erdös idiosyncratic nomenclature, all the best proofs are "in the book" and it was always a joyful thing to not only find a proof, but to find the proof that is in the book. Who cares if it is God's book or the machine's Xeroxed copy?

I mean, my reaction to God coming down and saying they were bored of being God and instead they would just sit around and answer all of the mathematician's questions would largely be the same, so yes, who cares if its God's book or the machines Xeroxed copy? "The Book" is more interesting to me if I am the one coming up with the ideas to fill it in. Maybe this is a bit egotistical, but I'd like to think it is allowed…

> it'd be more fun to win a game if you were on the field than if you were benched

This is a good analogy for AI work displacement. Probably would resonate with some of the college students who boo'ed Eric Schmidt.

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

#654

Post by muhneesh »

recitedropper wrote: post #244

This is impressive, no question. Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain origi…

This type of discourse is just inane and more reflective of the author's sensibilities than anything it claims.

Congrats to the OpenAI team for one of the most significant breakthrough discoveries in AI history.

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

#655

Post by virgildotcodes »

doctorpangloss wrote: post #645
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…

> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though. Always, always always, the problem with research and development is leadership, not insuffi…

I think we’re looking at a new class of wonderful machines that can potentially make meaningful contributions to the sciences and maybe even humanity as a whole, in addition to far more insidious and destructive capabilities.

You are right to point out that the ones who fully own and pilot the machines all belong to the “fuck science and humanity as a whole” group. So the likely outcomes don’t look good.

Echoes the early promise of the internet vs the eventual state and consequences of it, although seemingly primed for far more dire and deeply penetrating consequences.

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

#656

Post by energy123 »

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…

Terence Tao gave a recent talk about this issue (lack of attribution). He called it the decoupling of implicit and explicit goals. AI is only good at solving the explicit goals for now, and humans don't have the bandwidth or the institutions to know how to integrate AI into the field.

https://youtu.be/Uc2zt198U_U?si=OkwO3xT8-zhSABwh

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

#657

Post by fleroviumna »

doctorpangloss wrote: post #645
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…

> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though. Always, always always, the problem with research and development is leadership, not insuffi…

[dead]

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

#658

Post by shalmanese »

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…

> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.

AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of theorems proven, it’s how proving theorems affects the working operational models of mathematicians. Only a rare few new theorems in mathematics nowadays have direct real world applicability.

If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.

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

#659

Post by energy123 »

isotypic wrote: post #608
ted_dunning wrote: post #505

Earlier quoted context omitted.

In Erdös idiosyncratic nomenclature, all the best proofs are "in the book" and it was always a joyful thing to not only find a proof, but to find the proof that is in the book. Who cares if it is God's book or the machine's Xeroxed copy?

I mean, my reaction to God coming down and saying they were bored of being God and instead they would just sit around and answer all of the mathematician's questions would largely be the same, so yes, who cares if its God's book or the machines Xeroxed copy? "The Book" is more interesting to me if I am the one coming up with the ideas to fill it in. Maybe this is a bit egotistical, but I'd like to think it is allowed…

If ASI does create an abundant future I think many are going to have that familiar listless feeling of enabling cheats on a computer game and all the mystery and fun is gone.

Technology in general (smartphones, social media, search) even without AI is creating this feeling, as it shrinks the world and makes it less mysterious.

It's worse than boredom it's more like nihilism.

Then when you strip purpose and meaning from a human you get something very bad, despondency being the best case outcome.

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

#660

Post by CamperBob2 »

xpct wrote: post #584
CamperBob2 wrote: post #377

Earlier quoted context omitted.

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 don't think change is inherently exciting.

Maybe plumbing, masonry, or mining would have been a better career fit, then. Tech isn't for everybody.
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