QuesnayJr

This is another rant about the author's bugaboo.

It would be great to prove that there is or isn't a non-galactic polynomial-time algorithm for NP-complete problems. But we can't even solve the easier problem of proving there is or isn't a galactic polynomial-time algorithm, so what hope is there for the harder problem?

If you could prove that any polynomial-time algorithm for NP problems had to be galactic, you would be the most celebrated computer scientist since Turing. It's just no one has any idea how to do that.

A general affine coordinate change in 3d is Ax + b, where A is a 3-by-3 matrix and b is a 3d vector. If you add a fourth coordinate that you set to 1, you can write it as a 4-by-4 matrix. That's homogeneous coordinates.

The problem with math is that there's more you need to know than there's class time to teach it to. Projective space is a good example. It comes up in several different ways, but you're probably not going to see them all.

Doesn't it? People are already experimenting with outsourcing your exact example of a quintessentially human activity to a machine.

Wait until you find about RizzGPT.

Given the capabilities of OpenAI's unreleased model, I think the model helps mathematicians more than it hurts them. Some of the open problems that were solved were "It would be nice to know that", but many of the others represented ways in which we had been stuck for many years. There had been very little progress on analytic number theory, particularly the Riemann hypothesis or its variants. (One result, "no Landau-Siegal zeroes", even had a 10-page elementary proof that everyone missed for a century.) Complexity theory had seemingly more open questions than results. Progress could meaningfully accelerate now that we have a quick test of which of our ideas are good, and which are bad.

The wildcard is that we have no idea what the model after that will be capable of, or the model after that.

I don't think it was a particularly well-reasoned essay (I thought the classification of finite simple groups is almost evidence for the opposite of what he says), but at the same time I don't think you understood his point.

If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.

The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".

They didn't discover a missing sporadic group. The big gap in the proof (the quasithin case) happened because it was covered in a 600-page unpublished manuscript that under further investigation turned out to be incomplete.

I think people assume that it would end up not being usable by humans without a lot of hand-holding. Though maybe that's wrong, and somebody should try it.

I'm surprised they only found one gap. I would have expected multiple (fixable) gaps would have been uncovered, especially since the result was spread over many papers by many different people.

It is weird. We'll have to wait and see if they are the same algorithm or not.

This isn't getting any attention, but Anthropic claims Claude independently proved that the Exponential Time Hypothesis was false back on October 16th, and had been working on a cleaned-up draft.

Polynomial is something like exp(n log x), where x is the problem size. Exponential is something like exp(x). Superpolynomial but subexponential is roughly when the exponent is between n log x and x. An exponent of (log x)^2 would do the trick.

The actual exponent OpenAI found was x/sqrt(x).

The article reports it in an odd way, making it sound like it singles out students from Nigeria, Kenya, India and Pakistan. The fee applies to any student who pays international tuition rates.

Knowing the answer is valuable, but it's not everything. We also want to understand why. The four-color theorem has a computer proof, so we know it's true, but we learned much less from that than we do from a human-understandable proof.

Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.

I suspect 6.1 Astra is the model that kept showing up in the news stories about an internal unreleased model trying to hack the Pentagon or whatever.

But you are living in the world after the temporary shock, and you are asking the people who are still in the middle of the shock "Why aren't you over the shock yet"?

The chess example is not an inspiring one. It was true for a decade that humans and chess programs working in tandem were better than programs alone, but since then humans only hold chess programs back. Carlsen plus Stockfish is a worse chess player than Stockfish alone.

Anyone who thinks they can predict the future of this technology more than a few months out is foolish, but the models got so good at answering specific questions in math because the frontier labs could use RLVR to train them. There is no RLVR for mathematical taste.

There was more progress on open problems in mathematics in the last six months (because of AI) than in the previous decade.

You're asking this of the people who are actually in shock. What do you think "shock" is?

It doesn't make sense to compare CPI to real wages. Real wages are wages divided by CPI or another inflation index.

The two most signficant number theory ones are considered solved: Quasi-Riemann and no Landau-Siegal zeros. Quanta has an article (https://www.quantamagazine.org/ai-proved-the-quasi-riemann-h...) reporting that experts agree that the Quasi-Riemann Hypothesis has been proven. The "no Landau-Siegal zeros" proof already has several manuscripts giving expositions of the proof.

I think the point of the "solution manuals" reference is that you have to do math to learn it, not just read the solution that someone else has written.

It's not a big ratings agency, and it's not investment grade.

Kahneman credits it to someone else. His book just popularized it.

For many people, the answer to 1 is yes. You maybe happy with our future where we putter around aimlessly as consumers, but some people want to live in a world where they can at least try to do something that matters. Virtually every professional mathematician is motivated by personally making a permanent contribution to human knowledge, no matter how small.

If anything, that makes the case that math is fucked even stronger. Nobody had any idea of how to prove quasi-Riemann. OpenAI produced three proofs, one of which was just a lemma in solving another problem. They have a machine that is better than every human being alive at math that solves problems humans cannot even when working inefficiently.

I think what makes it hard is that it's quick, but not sudden. I understand constant-rate improvement. I understand what it's like to be in a car crash. What I can't understand, emotionally, is the constant acceleration. Even now, if I look at an OpenAI solution in isolation, I can understand the big picture, so if things froze where they are I could adjust. Intellectually I know I'm not going to get that chance, but emotionally I don't.

There's no money in it. There's money in programming, and math is a good target to give the models superhuman reasoning abilities, but it already writes well enough for middle managers to write all of their emails.

It was really quick. I stumbled across a comment from KConrad on MathOverflow on how useless AI was for research. That comment was from 2024. When it disproved the unit distance conjecture in May, the solution involved a breath of knowledge of the literature that it made sense it took a computer to make the connection. It only took 5 months for it to advance to the point that it was better than humans at almost everything. Even if you expected it, the mind boggles at how fast it was.

Also, it's impossible to really know how you're going to react to something until it happens to you. When ChatGPT came out, I already thought "in the long run they're going to automate almost everything humans love to do". Despite having 4 years to brood about it, I did not really know what it would feel like until it happened to me directly.