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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

#671

Post by freakynit »

cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

Many breakthroughs come from taking an idea from one field and applying it somewhere else. But, almost every serious field is now so deep/complex/huge that humans rarely get the time, or even have enough practically useable memory, to understand and correlate multiple unrelated areas properly.

And this is where machines, such as these reasoning LLMs, can help. Because they can remember patterns across many domains and try absolutely bonker weird connections and ideas.

We, the humans still have to verify the work (at least as of now). But, the "maybe this tool, or idea, or trick, from that completely unrelated field applies here" reasoning/experimentation could become much easier.

I have always said this and will say it again: reasoning is just experimentation with a feedback loop and continuous refinement.

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

#672

Post by sigmoid10 »

energy123 wrote: post #656
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

That is an odd summary of the talk. He was talking about how the explicit goal of solving a problem is kind of becoming trivialized, but the abundance of 100-page AI generated proofs will not help the implicit goal of furthering human understanding, because we lack the bandwidth to really digest them. Adhering to things like (human-focused) academic etiquette is a different problem and can probably easily be solved by just giving the model the right context. But having humanity keep up with AI insights into math and science is something we might have to give up eventually. Or at least whoever does will be far ahead of us as a society, because most people's lives will only be affected by the explicit results.

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

#673

Post by necovek »

black_knight wrote: post #354
dvt wrote: post #61

Earlier quoted context omitted.

> 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…

It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances! This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are di…

What frequently happens when we recombine axioms like that is that they end up leading to inconsistencies or contradictions.

Do you know if this topos with every total function on real numbers is continuous has been constructed and proven to be a viable set of axioms? If so, I am curious about the source.

My go to example still remains the one of hyperbolic geometry and axiom of parallel lines, so the more approachable examples I can get, the better.

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

#674

Post by nhinck2 »

FuckButtons wrote: post #644
wayeq wrote: post #478

Earlier quoted context omitted.

or put differently, you melted x cubic meters of polar ice

Based on some napkin math, that would be about ~100 watt hours of electricity on an H100 cluster, or, roughly the same amount of energy needed to boil a kettle for a cup of tea.

That's an exceptionally fast output you have there...

Mind showing your working out?

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

#675

Post by no-name-here »

stemchar wrote: post #620
fourseventy wrote: post #586

Earlier quoted context omitted.

What is the last model you used... lol. Linus Torvalds himself said the newest models are better than him at coding.

This doesn't sound correct. Source?

In recent months, Linus said it specifically about code for a personal side project of his. The quote was in the commit message. (I’m not the grandparent commenter, and I think grandparent commenter’s claims may be too broad or require context.)

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

#676

Post by no-name-here »

bigstrat2003 wrote: post #600
Supermancho wrote: post #572

Earlier quoted context omitted.

>That's literally my job... Since you’re not in a unique position, I can confidently state that your comparison of LLMs to jr developers seems unfounded. Today, LLMs produce code that is superior to junior developer code by an order of magnitude. Notably, they demonstrate consistent syntax, clear separation of concerns, strong test coverage, organizational rigor, idiomatic API usage, and the ability to generate and m…

LLMs absolutely do not exceed the abilities of junior devs. They don't even meet that bar, let alone exceed it. Junior devs are capable of getting syntax right without someone going "hey you messed that up". LLMs are not. Junior devs get basic logic right. LLMs do not. Comparing an LLM to a senior developer is an absolute joke.

1. Which LLM are you using that is “not capable of getting syntax right”?

2. Are you referring to without having a compiler or LSP check it? Although even then, the recent LLMs I've used still frequently get syntax right, whereas I'd expect juniors are often using a LSP or compiler to catch mistakes while writing code?

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

#678

Post by diordiderot »

virgildotcodes wrote: post #655
doctorpangloss wrote: post #645

Earlier quoted context omitted.

> 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 e…

Not in academia, but the amount of crying over rapid technological and intellectual progress because you're not getting credit validates everything critics say about you.

No interest in human advancement, just attribution.

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

#679

Post by energy123 »

sigmoid10 wrote: post #672
energy123 wrote: post #656

Earlier quoted context omitted.

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

That is an odd summary of the talk. He was talking about how the explicit goal of solving a problem is kind of becoming trivialized, but the abundance of 100-page AI generated proofs will not help the implicit goal of furthering human understanding, because we lack the bandwidth to really digest them. Adhering to things like (human-focused) academic etiquette is a different problem and can probably easily be solved b…

An odd summary of a talk you didn't even listen to? He explicitly mentioned references and attribution as a special case of implicit goals.

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

#680

Post by sigmoid10 »

dap wrote: post #474
sigmoid10 wrote: post #49

Earlier quoted context omitted.

Managing a McDonalds is a question of integration and modalities at this point. I don't think anyone still doubts that these models lack the reasoning capability or world knowledge needed for the job. So it's less of a fundamental technical problem and more of a process engineering issue.

Have you not seen: https://www.anthropic.com/research/project-vend-1 https://www.wsj.com/tech/ai/anthropic-claude-ai-vending-mach... (Two different examples of a similar idea)

Both links talk about the same thing? The first one just being more general. And yes, I would expect no less from a poorly constrained single agent that was instruction trained to be helpful and friendly. But if you look at how this has evolved as a benchmark [1] then the latest models show no doubt that can actually deal with this limited, simulated scenario given the correct setup.

[1] https://andonlabs.com/evals/vending-bench-2

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