Comment by tiborsaas

Pure mathematics, almost by definition, doesn't typically argue the field is always or even often "useful" (for some other purpose or application).

But, as mathematicians learn and push forward, occasionally something like elliptic curves will emerge as having useful applications, making all that previously "pointless" specialized knowledge newly valuable.

Or advances in physics, that suddenly have a need for a specific mathematical underpinning to develop a theoretical framework. Like how Einstein benefited from Minkowski's work on hyperboloids to create a coherent mathematical description of spacetime.

It was the AI labs themselves not mathematicians who were happy to conflate proofs for open math problems with some kind of tangible technological advancement in the real world. They would surely prefer to be able to claim a cure for cancer vs. a math problem but that loop requires a lot more time/money/test tubes/etc and they need headlines now not in a decade.

And so, thanks to OpenAI/Anthropic, we're now in a world where thousands of crypto bots on X breathlessly hype up each new problem being solved that previously wouldn't have any got any attention beyond academia and passionate fans of math.

Hopefully this won't lead to a trough of disillusionment as more people start to feel like you, with mathematicians getting the blame for inflating the value of their work even though the hype was coming entirely from the labs not them.

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.

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

He's not unhappy with it being solved, but the solution is less important than the learning you have to do to arrive at the solution. If they're just chucking compute at it and publishing the answer and hiding the path to get there, it sort of negates the whole point of posing such problems to begin with.

I suppose this is how Lee Sedol felt when AlphaGo beat him. But in the same vein, didn't it ultimately advance human understanding of the game?

Here's the point: when people solve problems, they come together and create a community to eventually use the new knowledge in positive ways, including inspiring younger mathematicians by sharing insights. The human element is key and it's not just about solving problems. People only think that because we've been conditioned by computers to value answers more than how we got to them.

But if AI can solve any problem and existing mathematicians just use AI to solve problems for the sake of solving them, the community itself with wither and so will the interest in mathematics and over a longer period of time, it will just become soul-less and uninteresting and the entire community powered by the fire of fascination will simply die.

Where did you even read that Tao is unhappy with “open problems being solved”? There was no indication of that in his Bluesky thread.

Why would you go out of your way to make a case of something being not useful when, ironically, so much advancement in human history has come from the discipline?

Your motive is more worrying than your straw man argument.

I think the point is that one of the biggest problems in the field has been solved and yet the excitement from this is nearly nil. Can you imagine if say breast cancer were cured under similar circumstances, or even worse (say OpenAI openly admitting it basically stole a bunch of other researcher's chatGPT conversations)? No one would care about these petty bickerings- or at least the headline "CURE FOR BREAST CANCER FOUND" would completely swamp anything else. This is embarrassing: it tells you almost no one- not even mathematicians themselves really care about their own problems- if they're not careful people will get the impression it's all a form of bean counting in a carefully constructed "safe space" where making sure people get the credit is more important than the work itself. That only happens in fields/problems where no one actually really cares about the output.

Drama aside, this solution would be a counterexample disproving the smoothness postulate, which means that it leads to nothing new unfortunately. We already had working solutions to navier stokes, the only thing we didn't know is if the equations possessed a technical property

Its a bit like solving p = np with a negative result. Its an incredibly difficult problem, but it doesn't lead to anything at all on its own. This is why people are talking about the fact that the solution methodology is much more interesting than the solution - the tools used to crack something like this may lead to solving more useful problems

To be quite clear, the solution to the _Navier Stokes problem_ is one in which you get a finite time blow up (i.e. infinite pressure). This is more meant to suggest that Navier Stokes is unphysical in some way which is not necessarily unexpected.

There's unlikely to be any engineering applications since even if the solution can be approximated, you still need to set up the initial conditions but at that point you can also drive pressure in other ways.

> But apart from AI and drama now that we have working solution to Navier-Stokes, what improvements can we expect in engineering?

Minor productivity boost in mathematics as people are no longer nerdsniped by the problem

> 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 the case of Navier-Stokes, the fluid is incompressible. So technically any force that you apply to it is supposed to instantly affect everything else. This can be exploited to create these blowups. In reality, no fluid is incompressible, and it takes time for any action to affect the material.

It's just that Navier-Stokes equations are so slippery that it's hard to pin their behavior down. They basically just restate the momentum conservation law for a continuous medium.

I am absolutely struggling to sell my workplace (which is entirely knowledge work) on the usefulness of LLMs for proofreading let alone on automation of hairy parts of our workflows. So yeah, even people who should be able to see what is coming are not looking.

> 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.

Practical?! The goalposts will keep moving until morale improves (narrator: it doesn't)

feels like moving the goalpost. is the achievement impressive or isn't it?

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

I suggest this article as background reading, "Losing control of AI is actually the plan"

https://substack.com/home/post/p-214653853

It is not reasonable to not believe anything unless there is "evidence" (narrowly construed as an observation incompatible with the negation of some state of affairs). Beliefs have a wide spectrum of characterizations, and not all belief must wait until publicly corroborated evidence is available. Some events defy evidence and we can and should use experience and reasoning to infer unobservable states of affairs.

...obviously it is evidence of the singularity. You're far more likely to see models solving Millenium Prize problems if a singularity is coming than if it's not. One'd have to be doing quite a lot of mental gymnastics to pretend otherwise.

Who is the AI doing the job for if no one is able to afford anything?

How many fucks should be given, in your estimation?

https://mastodon.social/@tristanbuckmaster/11723647135247030...

He very much is accusing them of stealing his work

The accusation is that they stole his approach of solving it. If OpenAI didn't bruteforce it with dozens of agents, he would have solved Navier-Stokes eventually since he evidently had the right approach. So knowing which approach to take makes all the difference, if they didn't know the approach they couldn't have solved it.

It's like he had a treasure map and was about to find the treasure, but they copied his treasure map and scooped him with a faster boat and found the treasure first. But he would have found it if it weren't for them.

> The said user (Tristan Buckmaster) didn't solve the millennium problem. He didn't really accuse that OpenAI stole his research either. The beef came from the fact OpenAI asked him to remove another mathematician, who works for Anthropic, from the credit.

Not quite accurate, Buckmaster was taking an approach that nobody else was, and this new proof uses this same approach just weeks after he saved those results to OpenAI workspaces. He asked OpenAI if they used chat logs for training the new model, and they did not confirm or deny.

Asking to remove his collaborator is also totally over the line though.

Edit: although this OpenAI post is not comforting: https://x.com/OpenAI/status/2097375276384567642

Quote: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. "

Have you read the actual statement https://cims.nyu.edu/~tristanb/statement.pdf ?

> I should say here why I interpreted their statement the way I did, the in- terpretation I will discuss below. The route to the Clay problem through a smooth force, options c and d in Fefferman’s statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack. Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement. When I heard “forced,” it was a bright red flag.

...

> I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI. > I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.

It's not a direct accusation, but it's not far off.

You shouldn't accuse other people of spreading misinformation when you haven't read the actual sources in question, it's possible that they might know more than you.