ComplexSystems

What situation? The development of an algorithm that can prove mathematical theorems?

You don't think OpenAI has any interest in having agents identify new interesting research terrain? They just want to solve existing important millennium problems and then they're done?

> The main practical complaint is "what do I do now", especially from PhD students who just lost their thesis topic. They sound like the Uber drivers fighting Waymo.

I wish mathematicians would just focus on this. This really is the problem, at the end of the day.

Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.

We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.

But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.

I think that it'd be funny if the polynomial time SAT solver was just "send your SAT instance to Astra and have it try to solve it for a few hours."

I know, it's awful. How do we get OpenAI to stop before they ruin math forever? /s

The idea that a US company is allowed to publicly talk about and release these bugs is a very new thing.

I agree he wants that, but also don't think it's that far fetched that there exist unpatchable hardware bugs that AI can learn to exploit.

Roughly speaking, O(n log n) means "grows no faster than k(n log n) for some k", whereas o(n log n) means grows strictly more slowly* than k(n log n) for all* choices of k.

"The psych had told me nothing of the nightmare that dependence on those meds can create"

I hear this all the time and never get it. Didn't these people go to school? People talk about this every day on the internet. It's right there on Wikipedia, even.

None of it is ridiculous in principle, but where do you draw the line? Is Google an AI company whose products I shouldn't use? What about Microsoft, where Vi worked for many years? Or is it specifically OpenAI and Anthropic? What about open source companies?

> History judges researchers who work with the bad guys ... To them I say: learn from history and get out now. You’ll still be ahead of the curve. The current backlash against AI may seem huge, but it’s nothing compared to what I think the backlash will be in coming years.

> I chose 7 years because I see certain parallels between the situation with researchers and Big AI today and the relationship between researchers and Epstein in 2019.

Look, I absolutely love Vi Hart's stuff. And I mean love love, and I have for almost 20 years.

But do people really believe this? That associating with AI companies, using their product, etc, is similar to taking money from a child sex trafficker?

I thought the original Mathathon sounded like a great idea. They shut that down, and I think the revised one seems like a half decent idea. But if I get involved in it, in 7 years it'll be the same as if I had been involved with Jeffrey Epstein?

I feel like the discourse on this is starting to get way out of control. I say that with no less love for Vi Hart's ideas than I have always had. But surely this technology can be used for some good?

This is a very loaded blog post. A unit-cost RAM is a fairly standard model of computation when working with algorithms. When we say that comparison sorting of strings takes O(n log n) time, for instance, we are also implicitly assuming a unit cost for comparing two strings. In real life, comparing two strings scales linearly with string length. Maybe that's silly, but it's par for the course. The original FFT is also assuming unit cost multiplication when we say it is O(n log n) time - we mean O(n log n) multiplies.

Similarly, yes, it's a galactic algorithm, but these things are still viewed as meaningful. The fact that it is even possible at all with a unit cost RAM asymptotically is an incredible result that nobody expected.

There are plenty of mathematicians who think the results are interesting. Not only do they have the "patience" you speak of, some even seem to be enjoying exploring the new results.

Here's a guy who's made a "beyond n log n" tracker: https://x.com/aurel_pr/status/2108214135179944096

He's tracking the community progress on sub-n log n multiplication. OpenAI started with 1 - 1.63e-55. The result has been now improved on 115 times, and the current record is "rohanarun"'s 1 - 9.87e-5. I'm sure by tomorrow it'll have improved again.

Does this look like people aren't having fun? Does it look like they aren't discovering stuff? It looks like it's spurred a cascade of interesting community activity. It doesn't really seem much different from what happened with the twin primes conjecture. Isn't that supposed to be the point of all this?

Have you looked around recently? Because there are plenty of mathematicians that are excited to read and learn about all of the new results. They've improved on the sub-n log n result and have even made a web site to track progress on it: https://beyond-n-log-n.netlify.app/

It looks like people are enjoying themselves, having fun with the new results, and generally doing all of the things you say "science" is supposed to be about. So what's the problem?

Yes, they should. They have invented a magic button that can tell you the long-awaited answers to the burning mathematical questions that you've spent your life researching. The caveat is that the technology is still new, so the explanations "are not fun to read" like set theory papers usually are (lol). If you don't think that's a worthwhile tradeoff, that's your call, but it sure as hell isn't everyone's.

I can't relate to this at all. AI models will surely get better at writing "enjoyable proofs," but for now the situation is what it is. You're passionate about this problem, right? But you don't want to do the work to understand the result? Fine. There's a new generation of younger, hungry mathematicians that are highly interested in figuring out why the result is true and I am sure they'd be happy to wade through it and spoon-feed you the answer instead. Maybe they should be running things.

It would have stayed broken assuming AI models never improve.

> I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.

Who do we demand this from? The AI companies? Or the mathematicians who are worried they will have nothing left to do?

Aside from the usual squabbling about AI, it seems the bombshell claim is this:

"In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations."

So these authors seem to be claiming that OpenAI has not really proven Navier-Stokes at all. If I get their idea correctly, they are claiming that the LLM has not formalized the original "natural language" idea of Navier-Stokes correctly. If true, it would mean that their purported Lean proof is not actually a proof of Navier-Stokes at all, but something that is an incorrect translation of the original natural language idea. If correct, this is a really bold claim and I would like to see if other researchers agree.

It can still benefit humans, but there are plenty of situations where humans prioritize guarantees of formal correctness over the ease of navigating granular details.

It seems that, despite spending decades obsessing over their various niche problems, they have little interest in a magic button that can immediately produce the answer. At best they view this button as a mild nuisance they need to contain somehow.

Someday I'd like to see someone do a similar parody of Linux where your job is to get a printer working in Kubuntu. Preferably on a laptop with a HiDPI monitor.

Turns out what you call "intelligence" was never needed to do math.

> But for math: what is the point of a proof if no one will read it and no one uses its result?

What is the point of writing software if nobody will run it?

> So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened.

So they ask the AI to explain the proof.

I admit I just don't understand this attitude.

People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on.

Similarly now we're getting AI doing math. The proofs compile but are a mess. So just make the models better at writing clean proofs and explaining what they're doing to humans. That's the end of it.

Rather than just go on and on about how it's the end of the world if we don't do this, why don't we just do it?

I agree. I found it refreshing to get away from all of the AI drama and just enjoy the advancement in math.

If it's unethical, then there is reason to call it out as Tristan is doing. I see no reason to blame the victim.

It's absolutely related.

Ridiculous. Air gap the agents, end this nonsense, and stop hacking unsuspecting websites to hype your product.