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On the Navier–Stokes Millennium Prize Problem

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Re: On the Navier–Stokes Millennium Prize Problem

#491

Post by superxpro12 »

piker wrote: post #447
intenex wrote: post #441

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…

Sure, even a 20% chance at 1 million payday after 5-6 years of fulltime work on a project with zero practical application doesn't touch the, say, 200k/year guaranteed our best mathematicians would have to forgo to devote their intellect to the problem.

i wonder how many tokens it takes to run 10,000 agents? One could argue this is simply a problem of appropriations. I find myself wondering if a corporation could spend $5M on mathmeticians and arrive at the same end result.

Re: On the Navier–Stokes Millennium Prize Problem

#492

Post by fooker »

munificent wrote: post #217
stefap2 wrote: post #180

Earlier quoted context omitted.

This just pushes knowledge work further up the ladder, toward larger and more complex problems. If there are no knowledge workers, who is going to interpret these results, validate them, decide what matters, and put them into practical use? Rather than eliminating knowledge work, advances like this could create entirely new layers of problems to solve and opportunities to pursue, which will create even more jobs and…

> 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.

Yes, this is how science and engineering has worked for millennia.

For example there are no engineering implications of this solution yet.

For the next several decades, we'll have engineers (presumably with AI) optimize things like rocket engines and turbines and AC compressors to work a few percent better because the numerical approximations might have caused us to be overly conservative.

AI is not going to magically solve all random problems. Pick a career where you are in the driver seat.

Re: On the Navier–Stokes Millennium Prize Problem

#493

Post by jampekka »

piker wrote: post #447
intenex wrote: post #441

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…

Sure, even a 20% chance at 1 million payday after 5-6 years of fulltime work on a project with zero practical application doesn't touch the, say, 200k/year guaranteed our best mathematicians would have to forgo to devote their intellect to the problem.

Thousands of some of the brightest minds have worked on this problem for over a century. The million bucks is not the big deal here.

Re: On the Navier–Stokes Millennium Prize Problem

#494

Post by xhevahir »

karmakurtisaani wrote: post #484
Eridrus wrote: post #467

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?

Where was Tao unhappy? I thought he was sort of anticipating this.

Maybe he was dreading it.

Re: On the Navier–Stokes Millennium Prize Problem

#495

Post by baq »

echelon wrote: post #53
tiborsaas wrote: post #16

> We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean. WOW?

This is going to be dramatic in so many different ways. - First off, to reiterate, WOW. - Second of all, when does this end? Are we at the dawn of the singularity now? - People are saying OpenAI "stole" this from the work of an OpenAI user. If so, that's pretty fucked - how can we trust them? - Time to think about retiring from any knowledge work or business? This could be winner-take-all where a leading lab can butt…

> - First off, to reiterate, WOW.

> - Second of all, when does this end? Are we at the dawn of the singularity now?

normalcy overhang n. /NOR-muhl-see OH-ver-hang/

The uncanny period during the Singularity when superintelligence is already accomplishing feats that seem like magic, yet everyday life still looks mostly the same.

https://x.com/alexwg/status/2096214373001785794

Re: On the Navier–Stokes Millennium Prize Problem

#496

Post by Fordec »

HenrikPontoppid wrote: post #488
Fordec wrote: post #378

Earlier quoted context omitted.

If humans can figure out to optimize to circumvent bottlenecks, I have no doubt each new bottleneck will also get routed around, just now automated.

Yes. In other words: the singularity. I'll only believe it when I see it though.

I'm coming around to not liking the term singularity, it implies an endpoint or finish line rather than something that just keeps continuing and evolving.

Re: On the Navier–Stokes Millennium Prize Problem

#497

Post by free_bip »

rfgplk wrote: post #121
harhargange wrote: post #115

Just so everyone knows, although openAI pretends that the model generated solution and wrote the paper by itself ""with very little human input"" as Buckmaster himself mentioned in his statement. In reality they have team of researchers guiding the system, along with, probably training on user data, probably Buckmaster in this case, in order to come up with the proof.

This isn't really true.

Do you have anything to back that up?

Re: On the Navier–Stokes Millennium Prize Problem

#498

Post by StatsAreFun »

Can't help but shake an unsettling feeling about all this, frankly. I engage in some limited mathematical research and will often use any one of the latest frontier models to check some ideas. Lately, only the OpenAI models have been giving me a temporary message that says something like (paraphrasing from memory), "We're thinking extra hard about your request before we answer. You can choose another model to answer now or click here to learn more about why." When I click to read why it's doing this "extra thinking", the help page says that for cybersecurity and biosecurity-related information, it will review the answer and could refuse.

Now, keep in mind, I'm only asking strictly pure mathematical questions - nothing at all related to cyber or protein creation or biohacking or anything like that... And, like I said, only the OpenAI models are doing this. To be fair, all of the prompts have always eventually returned a satisfactory answer, as far as I can tell, and haven't used a weaker model to answer them. Maybe? I dunno, it has just struck me as odd every time it has given me that message to pure math prompts.

Re: On the Navier–Stokes Millennium Prize Problem

#499

Post by cyberax »

munificent wrote: post #217
stefap2 wrote: post #180

Earlier quoted context omitted.

This just pushes knowledge work further up the ladder, toward larger and more complex problems. If there are no knowledge workers, who is going to interpret these results, validate them, decide what matters, and put them into practical use? Rather than eliminating knowledge work, advances like this could create entirely new layers of problems to solve and opportunities to pursue, which will create even more jobs and…

> 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.

It did not solve Navier-Stokes. We still will need to use the bad old numeric methods to simulate the fluid behavior.

But it did find a long-suspected smooth solution with a singularity.

Re: On the Navier–Stokes Millennium Prize Problem

#500

Post by zingababba »

andrepd wrote: post #424
apical_dendrite wrote: post #271

Earlier quoted context omitted.

Their own tweets are also pretty eyebrow-raising: > One option we discussed was that Tristan could be the lead author on a rewrite of OpenAI’s Navier-Stokes proof. It is in that context that I said “it would be simpler if Levent was not an Anthropic employee” because I felt it would be inappropriate for an Anthropic employee to author OpenAI’s work. Why would you offer another researcher the lead authorship on your g…

Holy late capitalism. Everything revolves around line-go-up, and sociopaths rule the show. These people cannot even collaborate like civilised scientists on one of the most famous open problems in mathematics? “It would be simpler if Levent was not an Anthropic employee” I cannot believe this shit.

Soon Levent will just be turned into soylent and he will have never been an Anthropic employee. We still need some progress here though.
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