QuesnayJr

There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...

It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.

I remember back when Gemini looked like it was the best model that this comment section was full of confident predictions that Google had "won" and that no one would ever catch up with them again. The most embarassing part is that I kinda believed them.

If the poster's (is it Kevin Buzzard?) suggestion works out and AI finds a counterexample to the Hodge conjecture, that would be a really big deal. It's one of the Millenium problems, for example.

One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.

That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.

That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.

No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.

This seems like the next step, but the smallest counterexample in n=2 is degree greater than 100. (This is a paper of Moh. Wikipedia has details.)

The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true.

Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.

If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.

Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.

Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.

We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.

It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.

I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)

At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.

Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.

The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)

I haven't seen these claims, but all of the open problems that have been solved so far have been in the category of "humans could have solved them, but didn't". This doesn't diminish the significance of the results, but it does mean there's still work for mathematicians to do other than glorified prompt engineers.

This specific counterexample really is trivial. There's nothing to cite. People have wasted hours and hours on a question whose answer you could give as a homework problem in Calc II.

There's a certain (unfair) stereotype of Russian mathematicians that this fits.

If governments introduced a whataboutism sin tax, they could solve budget deficits overnight.

Both systems are at end-of-life. If anything, China is closer to the end. The total fertility rate in China is down to 1, while the US is 1.62.

At this point you're just openly shilling for the CCP. The CCP killed millions of people through incompetence, and then they had the brilliant idea of copying the exact development model of the Asian Tigers. It's their previous incompetence that's the reason that they haven't already caught up with South Korea.

Philosophy majors do surprisingly well in getting jobs outside academia, so I think the market demand for confusion is bigger than you might guess.

They get paid to produce confusion.

If it was the students, then students can have things they think are cool or uncool.

If it was the professor, then that would be very embarassing on his or her part.

It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts.

The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.

According to Wikipedia the term basically wasn't used by Marx at all, and was coined by other anti-capitalists.

There's an old paper about the mathematics of nomograms that I found interested when I stumbled across it: https://doi.org/10.1016/0001-8708(65)90042-3

The first time I saw the red squiggles is one of the few times I thought "Good job, Microsoft".

Someone said to me yesterday "I'm thinking about building my own cyberdeck" and I boggled, because the only meaning I knew was the Neuromancer one.

GA is Clifford algebra plus a bunch of new terminology plus advocacy that it should replace linear algebra.

FWIW, I liked the way you expressed it through a dichotomy.

Isn't that a straightforward argument for preserving the status quo as much as possible in learning? We know how to get people to learn without AI, so we should keep doing it until someone figures out how to use AI effectively.

Dude, just use it to learn something. It's obviously true.

Everyone uses AI all the time now. People's impressions are not mediated by marketing.

This is not at all what happened. They did deliver, in the form of the "Attention Is All You Need" paper, which Google made public. They took nothing from Google that wasn't already public.

Unless you think that employees are like indentured servants, and Novo Nordisk owns not only Wegovy but the people who work on it.

Switzerland has nuclear reactors. They just stopped building new ones.

Semiclassical gravity is the best we can currently do for a theory of gravity without invoking speculative ideas that are currently untestable. If the paper holds up (I haven't read it), then there are several possibilities:

1. Maybe P = NP, and semiclassical gravity isn't special. 2. Maybe P = NP, and the way we'll prove it is finding an efficient way to simulate semiclassical gravity. 3. P = NP is a hypothesis about traditional theories of computation, but they don't rule out that we can build a special machine that solves them. There's a stronger hypothesis, the extended Church-Turing thesis (ECTT), that says this is impossible. Maybe the extended Church-Turing thesis is wrong, and this is how we'll show it. 4. If ECTT thesis is right, then maybe we can conduct an experiment where semiclassical gravity fails. This gives us a clue to new physics. 5. If we can't eventually conduct an experiment, then at least we learn about a new angle on complexity -- problems that can be efficiently solved this way but not by a deterministic Turing machine.

Both quantum mechanics and general relativity are thought to satisfy the ECTT, so the fact that our most experimentally successful combination of the two doesn't is of some interest. (Semiclassical gravity is thought to fail eventually, but in a way that's out of reach of current experiments.)