This is a *famous* unsolved math problem. It was already highly unlikely that there was a solution hiding in the literature for Problem 1196. The Unit Distance Problem is so much more famous, with so much more work, it is genuinely hard to express how fantastically unlikely it was for this solution to be somehow hidden in the literature.

This seems to be somewhat incorrect. They also released the rewritten "cleaned" chain of thought here https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf(pdf). That isn't everything, and has been cleaned in some respects, but shows a massive number of dead-ends, unnecessary complications, and everything else you expect to see in a working mathematician's initial attempt at a proof. As far as I can tell, the primary thing they've done here is just compile the text LaTeX into a PDF form but given that this is a proprietary model output it wouldn't surprise me if it also had some things they don't want to leak scrubbed from it.

Lemma 2.2 struck me as a type of bound on an extension with complex multiplication that I had not seen before and seemed clever. I was also struck by even as the Lemma itself was clever, that the proof of that Lemma was pretty straightforward. The overall approach is in many respects pretty similar to existing work and feels in some respects in the same spirit as Erdos's own lower bound construction, but having a tower of fields which seemed clever to me, but the writeup notes three prior papers where a tower was used to produce a combinatorial object of some type. That said, I still find this choice of the class field tower to be clever and intricate in this context.

I am mostly in agreement. Disagreement here:

We're not at that point yet. Right now, we're not seeing it solve the genuinely hardest problems, like say the Riemann Hypothesis, or P ?= NP. What is true is that these systems are at least as good as a beginning grad student in all subfields and are outputting results equivalent to a top-notch mathematician on some problems. But it is also true that these systems are improving rapidly. So while your statement is false right now, it looks likely your statement is going to be true within just a few short years.

Have you read the paper? I have, and it is very much not the case of what is going on here. There are multiple deeply clever bits in this argument. If this were written by a human, it would be recognized as highly insightful. Moreover, you are also missing how much what human mathematicians often do really does look like what you are dismissing. I've worked on hundreds of problems, and gotten successful results in maybe 5 or 6 of them. If someone dismissed humans under that basis, you'd recognize the problem.

And if you read the raw output of the AI, it looks a lot like what human mathematicians do. We try one thing. It fails. We try to look at a related theorem; doesn't generalize. We go check a few cases; doesn't give much insight, so we rope an undergrad into writing some code for us to go up a big more. Then, we're sitting in a seminar on a completely different topic, and trying to pay attention while the speaker does a really poor job explaining their research, we're like "Hmm, what if I tried to combine it with that other thing we saw 2 years ago." That still doesn't work. But then six months later, you bash your head against the problem a bit more trying to use some sophisticated representation theory results, and then you are falling asleep and you realize that other thing from now 2.5 years ago combines with a pattern the undergrad mentioned in the data that you didn't think was important, and you get a result. Mathematicians work by trying lots of different things on lots of different problems.

All mathematicians build math by leveraging what other mathematicians have done. When Andrew Wiles proved enough of the Modularity Theorem to prove Fermat's Last Theorem, he was leveraging ideas from all over, from algebraic geometry, from representation theory, from complex analysis, from Galois theory, from elliptic curves, etc. Combining that was the big thing. When Peter Scholze and Dustin Clausen recently did their work on "condensed mathematics" (which may get Clausen a Fields Medal- not Scholze since he already has one), they were seeing deep patterns in a whole bunch of existing work, and then used existing work to leverage together that those patterns were real. All math builds on existing math going back to people in caves wearing bear skins who were thinking about how to count beyond using their fingers and toes.

More than any other AI use yet to solve an open problem, this one cannot be dismissed without just being completely irrational. Even Erdos 1196 people could use maybe was somewhere hidden in the training data (which as a mathematician in a closely related area seemed extremely for a whole bunch of reasons I'm happy to expand on) or that the problem just hadn't gotten a lot of attention (which was arguable there even as it was a well known enough problem that I had heard of it). But the Erdos unit distance problem is a genuinely famous problem. There's no way that there was a lack of attention to the problem, and there's no way to say some solution was in an obscure journal no one noticed. This is a problem which literally gets discussed in some undergrad classes.

The Annals of Mathematics is the most prestigious math journal in the world, and most mathematicians will never get a paper published there at all (I certainly don't expect to). I talked with another mathematician whose work is closer to this problem and asked "So is this the time when an AI first gets a result that should be essentially in the Annals?" and his response was "delete essentially from that sentence and the answer is yes." I have a bet with another mathematician that there would be no papers in either the Annals, Inventiones, or Crelle where the result was discovered by an AI before 2028. 72 hours ago I thought I had a decent chance at winning that bet. Now, I'm seeing what is likely the result that is going to make me lose.

This system did not use Lean. But note that systems which do use Lean as a direct verification shouldn't be dismissed either. The fact that LLMs with symbolic verifiers are powerful doesn't get to be less true because it seems like a really clunky architecture to some people. We don't know the exact way this system functions since it is an internal model used by OpenAI that they have not released to general use or given a lot of details about.

How often do we encounter endangered bird embryos that lack eggshells?

I can see how this technology might be used to revive extinct species, but the claim that it helps endangered species is nonsense.

That people are using examples from three years ago is pretty good evidence.

Republicans have always been hypocritical about states' rights.

Abortion was a states' rights issue until RvW was overturned, and suddenly they wanted a national ban.

They want the Feds to overturn state-level pot legalization, ban sanctuary cities, etc.

That was three years ago, which is an eon in AI terms.

There have been vast advances since then.

That is nominal GDP based on a very weak currency.

In PPP terms, Iran's GDP is about $1.2T, about the same as that of Illinois or the Netherlands.

Then the future will belong to China.

Better start learning Mandarin.

This is wrong. Phonics works really well. We have actual data. https://www.jstor.org/stable/3516004 Nor is phonics a new thing; it was how we taught using phonics in most of the 19th century and early 20th century. Other methods such as three cues are actively bad and teach children how to read using the methods poor readers use to read. See https://www.apmreports.org/episode/2019/08/22/whats-wrong-how-schools-teach-reading for an excellent article on this.