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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#741

Post by sp527 »

rakejake wrote: post #147
paxys wrote: post #37

Earlier quoted context omitted.

What else can they declare really? Yeah the model has training data from previous attempts. Alpöge and Buckmaster also similarly benefited from attempts before theirs.

I don't think OAI should be given the benefit of doubt. They are doing the research equivalent of front-running. Knowing where to look is one of the main challenges in research. Tristan's argument from his essay was that it is hard to brute force with a vanilla prompt (even for seasoned mathematicians) unless you knew very specifically what to mention i.e the search space would have been intractable even for OAI's co…

> Tristan's argument from his essay was that it is hard to brute force with a vanilla prompt (even for seasoned mathematicians) unless you knew very specifically what to mention i.e the search space would have been intractable even for OAI's compute budget.

This is a bad argument. This is clearly not how it works. And unless Tristan is some truly alien-like savant (and maybe he is), what's necessary to initiate the AI's work already exists in countless published research papers and not exclusively in his head or notes. AIs can survey the sum total of all prior work on a problem and discern reasonable paths for inquiry.

Tristan is acting as if he's working off of an outdated model of AI, similar to primitive chess-playing models that winnowed the search space much more deterministically. If someone this intelligent truly doesn't get that this is not at all what AI is anymore, then maybe there's no hope that we ever understand it.

But I think he does realize this and he's flailing about for counterarguments from a place of bitterness and dejection, accepting even those that are too weak to be defensible. And that is very human and even forgivable.

Re: On the Navier–Stokes Millennium Prize Problem

#742

Post by yauneyz »

intenex wrote: post #439

I think this is clear evidence that AI models are now at the far frontier of mathematics innovation and discovery and exceed human limits. This specific problem having had a $1 million bounty on its head and still remaining unsolved for 26 years after the bounty was placed is pretty clear evidence that many of the world's best human mathematicians would have solved this problem if they could have, and none were able…

I think the biggest hurdle remaining is that all these landmark results are generally counter-examples.

Proving something in the affirmative often requires the creation of an entire new sub-field of math, or new tools. Think of Fermat's Last Theorem or something like that.

These results, while impressive, are clever constructions using existing techniques. It isn't clear that AIs can build new machinery like this. But if/when they can, yeah it is probably game over.

Re: On the Navier–Stokes Millennium Prize Problem

#743

Post by tristanj »

vemacs wrote: post #715
Jtariiiii wrote: post #606

Earlier quoted context omitted.

>Just because something is legal and permitted by terms of service doesn't mean it's morally right. What are you expecting OpenAI to do exactly if these mathematicians voluntarily submitted their prompts into ChatGPT's training data? Are they supposed to manually review all their data to make sure competing mathematicians didn't accidentally leave the "submit prompts" toggle on? Or were they supposed to not try to so…

> Are they supposed to manually review all their data t Yes. They should determine if training data included this teams data. Consider the money they spent, the press release and the purpose of their publication. Since they failed to answer this question they shouldn't have published.

[dead]

Re: On the Navier–Stokes Millennium Prize Problem

#744

Post by ndriscoll »

lambda wrote: post #218
rfgplk wrote: post #156

Earlier quoted context omitted.

> Why can't they rule it out? Is even OpenAI unable to track the provenance of all of their training data? Probably? I have a few hundred TB of training data for various small scale models and I can attest that I have _no idea_ what's in them. As in, literally zero. Half is scraped from GitHub and other hosting sites, other than that, I couldn't tell you anything else. At OpenAI's scale their entire pipeline is likel…

Yeah, I'm sure it's completely automated. But that doesn't preclude being able to index and track what the sources of data are. For your data sets, I would hope you are including source information for where the data came frome. And at OpenAI's scale, I would presume they are doing some amount of rolling hashing or similar to weed out duplication, training on too much duplicate data can cause problems. AllenAI have a…

Attributing training data seems pointless for trustworthiness. The way you trust a model is the same way you trust a human; you ask it to:

  1. Provide a chain of reasoning from agreed premises. These days LLMs can even do this airtight with proof assistants.

  2. Cite data sources for non-agreed premises. I don't care where the model learned a fact. It might not have ever read a document directly from the primary source. I want it to link directly to either widely agreed facts (e.g. standard textbooks, and if necessary school syllabi demonstrating that the text is standard) or primary sources (e.g. datasets).
Training provenance is irrelevant. It's neither necessary nor sufficient to deal with truth.

Re: On the Navier–Stokes Millennium Prize Problem

#745

Post by tzone »

concinds wrote: post #510
Eridrus wrote: post #465

Earlier quoted context omitted.

Seeing mathematicians such as Terry Tao being unhappy with open problems being solved makes me sort of question the usefulness of any of this pure mathematics. If we're not happy that the problems are being solved, why care about this field at all?

His issue is more nuanced than that. Most of the value was in humans reaching new insights or new math during failed attempts to solve these problems, whereas AI is basically "too efficient" in beelining to the goal and discards potential new insights reached along the way. I assume this is solvable.

Well, if in future we do end up with a magical tool that can solve any formal mathematical problem on a whim, we really won’t need field of mathematics anymore as it is today.

There would be no need to deliver new mathematical insights by solving problems. You would just have a magical math problem solving machine and that’s it.

Re: On the Navier–Stokes Millennium Prize Problem

#746

Post by ninjahawk1 »

The problem is the precedent this creates. For non-famous people using public APIs like this it could mean AI companies sucking up the information and throwing millions in compute at it.

The sequence for Navier-Stokes was that these researcher spent a year working on it, then they published a possible breakthrough, OpenAI then spends $15M within a couple days to finish it.

This was incredibly opportunistic.

Re: On the Navier–Stokes Millennium Prize Problem

#747

Post by FabCH »

tantalor wrote: post #368
echelon wrote: post #362

Earlier quoted context omitted.

There's an event horizon and we're maybe past it?

Heh. Wrong "singularity"

They meant that even the happens-very-fast AI singularity isn’t sub-picosecond. It still exists in time, and has a duration.

The event horizon would then be the time period between the singularity becoming inevitable and it actually happening.

Re: On the Navier–Stokes Millennium Prize Problem

#748

Post by kypro »

reducesuffering wrote: post #358
kypro wrote: post #320

Earlier quoted context omitted.

As someone with a background in AI and who has been playing around with neural nets for decades at this point, it's been genuinely amazing watching extremely intelligent people make confident predictions about AI capabilities and progress, then be so completely wrong. There's a kind of theory of mind for AI (specifically neural nets) which I now realise I seem to have which is very hard to explain to people who haven…

> extremely intelligent people make confident predictions about AI capabilities and progress, then be so completely wrong. The ability for the human mind to rationalize conclusions to maintain denial in the face of a very scary future is immense. Genuinely grappling with the implication of where we're headed is usually very crushing. It's not easy to engage with the possibility, and very intelligent people will use t…

> Genuinely grappling with the implication of where we're headed is usually very crushing.

As someone currently prepping for various AI doom scenarios and who has been dealing with AI-related nightmares for years this is very relateable.

Although, I don't personally think it's this. In my experience the opposite is more true – the majority of high probability doomers seem rather laid back about considering what they believe will happen to the people they love in a few years. Equally I don't get the sense those who don't have such extreme predictions are worried at all. If anything there's not enough emotion.

In my opinion people just don't reason well when it comes to exponentials and are ignorant about things they don't have good mental models of. At least I know I struggle with this.

Re: On the Navier–Stokes Millennium Prize Problem

#749

Post by azan_ »

diehunde wrote: post #610
azan_ wrote: post #327

Earlier quoted context omitted.

Existence of AI capable of solving millennium problem has enormous bearing on everything though.

What kind of enormous bearing? Computers have been able to do things humans can't for decades now.

Are you being flippant?

Re: On the Navier–Stokes Millennium Prize Problem

#750

Post by [deleted] »

enraged_camel wrote: post #631
tedsanders wrote: post #236

Earlier quoted context omitted.

All of those statements sound true, based on what I've heard. - "very little human" input feels ambiguous, and if someone spends a few days prompting a model to solve a super hairy problem requiring a 100-page proof, I can understand reasonable people interpreting that as both "very little" and "not very little" human input - it's all true that a team worked on this, a bunch of compute was burned, and the problem was…

>> I'm not sure how any of this provides evidence that OpenAI took any of their work. Sorry, but the burden of proof lies in the other direction: OpenAI needs to definitively prove that their agents did not look at the existing work that was about to be published. Otherwise OpenAI simply stole the glory and the spotlight (and I'm being charitable here).

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