You mean countries who had escaped Russia's grasp asked to join NATO so they wouldn't get invaded by Russia. Fear of Russia made the Baltics ask to join NATO. And of course, Russia then invaded Ukraine, a country not in NATO, showing the Baltics were right to be worried. On top of that, to invade Ukraine and then continue its car with Ukraine, Russia had to remove troops along the borders with NATO countries, showing that the Russian government, for all its claims otherwise, understands that NATO is not a threat to Russia except in so far as it stands in the way of the Russia government's imperialist ambitions.

This is a 30 year old module. It was originally made for MIR and has now been in space for 25 years. Even good engineering is in space is going to start having leaks at that point. Space is really hard.

Do you mean fusion power? If so, it is worth recognizing that we've made major progress on fusion power with the average predicted time to fusion power going down over time. For example, the triple product, which is an important measure of how effective a fusion system is, has been growing since the 1950s. with a brief pause slowed down in the early 2000s when almost all fusion research money went into ITER and is now increasing again https://www.fusionenergybase.c... . Additionally, usion research has been drastically underfunded compared to what predictions of fusion being soon would have assumed https://x.com/ben_j_todd/statu... . .But even given that, the predictions by experts of when we're going to have fusion power gone down over time https://link.springer.com/arti... at about 1 year every 3 to 5 years. To some extent, the question for fusion is not will we develop it, but given the timeline, will it ever be cost competitive in practice against very cheap wind and solar whose prices are dropping rapidly.

The state of quantum computing is pretty similar. There's ongoing progress in a bunch of ways. There's been not just improvement on the physical end, but there's been improvement on the algorithmic end on how quantum error correcting codes and other needed algorithms would function, reducing the quality of qubits and number of qubits needed for applications. See for example https://www.quantamagazine.org/thirty-years-later-a-speed-boost-for-quantum-factoring-20231017/. Microsoft's 2029 claim is likely overly optimistic, but it is a mistake to think we're never going to have these systems.

I'm not sure the Pope's opinion matters, but it is ironic that evidence suggests that sections were written via AI: https://www.lesswrong.com/posts/GbWwesBnetyiomxEH/many-portions-of-magnifica-humanitas-appear-to-be-ai-written.

As I've pointed out before, I have an actual PhD in number theory. I've explicitly discussed here specific results which are due to me and linked even to one of my papers before. You should be able to think that maybe, just maybe, people who are subject matter experts might know what they are talking about, and maybe know something you don't. But for some reason that possibility seems to be one you immediately and completely discount.

The proof structure here is somewhat simple, enough that any algebraic number theorist can follow most of the argument. But that isn't because it is a counterexample. There are occasions where a counterexample requires an extremely difficult, delicate, construction and there are times where a proof of the claim in question is surprisingly straightforward. This is the sort of mistake that one makes if one a) Doesn't know as much about number theory as one thinks one does and b) have a lot of motivated reasoning going on to discount the significance of the result.

I'm a mathematician, so there's a chance that I maybe, just maybe know what I'm talking about it here. You appear to be using an extremely narrow definition of "leverage" and then insisting anyone who doesn't use your definition is wrong. But whether one insists on an overly narrow definition of leverage or not, the central point remains: when people make mathematical discoveries, they are building on and using existing ideas. Whether you object to the word "leverage" there isn't relevant to that central point.

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.