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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#791

Post by FridgeSeal »

tzone wrote: post #742
concinds wrote: post #508

Earlier quoted context omitted.

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.

What do you mean “we won’t need mathematics as it is today”?

To further human understanding is itself a goal that single-handedly justifies our efforts.

Jumping straight to the “answer” and therefore missing both the understanding of the actual problem, and any useful discoveries along the way is a waste at best, and actively harmful at worst.

Re: On the Navier–Stokes Millennium Prize Problem

#792

Post by TomJansen »

dyauspitr wrote: post #526
frotaur wrote: post #177

Earlier quoted context omitted.

Not sure about the dystopia... Had a similar thought when covid was beginning 'wow pretty exciting, just like in the movies'. Turns out actually living some terrible catastrophe is only fun in the movies.

I had a lot of fun during Covid. I loved the working from home. The fact that most outdoor places were sparsely populated, jobs were plentiful and prices were low. Covid was awesome.

Yeah Covid was awesome, say that to the people who died from it, or who committed suicide because of the lockdown depression

Re: On the Navier–Stokes Millennium Prize Problem

#793

Post by FridgeSeal »

stefap2 wrote: post #280
munificent wrote: post #217

Earlier quoted context omitted.

> This just pushes knowledge work further up the ladder, toward larger and more complex problems. You really think it makes sense for you to be higher on the "solving complex problems ladder" than the machines that solved fucking Navier-Stokes? I envy your self-confidence.

Maybe I should have been clearer. My point is that solving something like Navier–Stokes just pushes knowledge work further ahead, onto a new set of bigger and more complex problems. Navier–Stokes is a Millennium problem today, but once problems like that become solvable, they can open the door to entirely new classes of problems we haven’t even thought of yet.

Building on them without fundamentally understanding is akin to putting on robes, calling yourself a Tech-Priest, worshipping a machine god and doing your best Warhammer 40k impression.

Re: On the Navier–Stokes Millennium Prize Problem

#794

Post by cyberax »

mcfry wrote: post #784
cyberax wrote: post #353

Earlier quoted context omitted.

> But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering? Nothing, really. This mirrors other examples of blowups from the classical physics. It's possible to create a system with just gravitating bodies that exhibits a blowup to infinite speeds in a finite time. The root cause is that, in classical physics, the speed of gravity is instant. In th…

Great comment! I've been trying to understand it more and was hoping to find more people discussing the result, or the implications of the result, and this was the missing piece for me after watching a few videos.

Here's the paper about the blowup for the gravitating bodies: https://www.jstor.org/stable/2946572?origin=crossref

There's a Wiki article about it: https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture

Re: On the Navier–Stokes Millennium Prize Problem

#795

Post by drpixie »

lanthissa wrote: post #14

5 million messages, 300b output tokens, done in 5 days, and achieving something humans couldn't. the first "Country of geniuses in a datacenter" moment.

That is how the PR reads, but is not at all what happened.

A team of highly trained and skilled people used an AI tool, through many many instructions (prompts), to produce a specific mathematical theorem. The tool is impressive, the result (possibly/probably) interesting, but the PR skips the vital role of the humans (for the usual PR reasons).

Re: On the Navier–Stokes Millennium Prize Problem

#796

Post by covidwasfake »

TomJansen wrote: post #792
dyauspitr wrote: post #526

Earlier quoted context omitted.

I had a lot of fun during Covid. I loved the working from home. The fact that most outdoor places were sparsely populated, jobs were plentiful and prices were low. Covid was awesome.

Yeah Covid was awesome, say that to the people who died from it, or who committed suicide because of the lockdown depression

[dead]

Re: On the Navier–Stokes Millennium Prize Problem

#797

Post by [deleted] »

tedsanders wrote: post #181
lambda wrote: post #130

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? This is one of the major problems with these enormous closed models, and even most open-weights models, which don't disclose their training process or training data. You can never be sure what went into its training. Did it come up with an idea originally, or is it just plagiarising its training data? Are there ma…

To truly prove some incidental usage data made no difference we'd have to (a) identify any of their de-identified data that came from their usage of ChatGPT, (b) train a bunch of expensive giant models, and (c) ask them all to solve the Navier-Stokes Millenium problem until hitting some level of statistical significance. It's just not feasible to run experiments like this to prove whether a piece of data has an effec…

[deleted]

Re: On the Navier–Stokes Millennium Prize Problem

#798

Post by Aaron1011 »

QuesnayJr wrote: post #240
cv5005 wrote: post #105

Maybe a naive question, but how does one know that a particular lean proof is actually a proof of what one thinks? Like, ok the logic checks out and it proves something , but there's still the problem of does this logical result actually prove the initial question that was asked?

Someone has to actually check this. I'm guessing OpenAI had someone check it internally, but it's possible to get it wrong.

In this case, there was already an existing Lean statement of the problem in the formal-conjectures repository, which they re-used: https://github.com/openai/NavierStokesAndEuler/blob/8937a8f4...

Re: On the Navier–Stokes Millennium Prize Problem

#800

Post by biophysboy »

tristanj wrote: post #761
biophysboy wrote: post #109

Earlier quoted context omitted.

Why is it unlikely?

Because the models are trained on hundreds of billions of user conversations, across more than a billion different humans. The conversations are anonymized and not easily traceable back to a specific user. It's unknowable and not possible to prove if any one specific conversation contained the insights for solving Navier–Stokes. We also don't know if the authors unintentionally provided data to OpenAI through alterna…

I understand that AI is not just cut and paste, but some documents will have more influence than others w/ power law scaling. I would be very surprised if this distribution were not extremely steep for arcane math
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