qnleigh

This moat issue comes up a lot, but look at what's been happening so far; open models are consistently 6-12 months behind. What if their moat is simply staying ahead?

And once their models start to accelerate their progress, which may already be happening, this would further solidify their lead.

The introduction is very nicely written, you're right. I learned a lot from it. There's some detail on how impractical their algorithm is:

> Our result is purely asymptotic: for arbitrary 3-CNF formulas with m = 5n clauses, the bound we prove does not improve on the 2n of exhaustive search unless n > 10^10000,

So their algorithm beats standard algorithms only when you have more variables than there are atoms in the universe (10^80). Even if every atom contained a whole universe, and each atom in each universe itself contained a whole universe, iterated 100 times, your still not there.

I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.

The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.

There's already some precedence for this in machine learning; DeepMind spun out Isomorphic Labs after the success of AlphaFold in solving the protein folding problem, hoping to use it for drug discovery.

LLMs haven't yet made a contribution to science or technology at the level of AlphaFold, but it looks like at least Anthropic is gunning for something like this with their AI-driven wetlab.

Probably they have either tried using LLMs to crack most cryptographic systems, or they have strict rules against doing so. Either way, the US govt. will probably make sure it gets a chance to do so before this internal model is released. There's a good chance they have started already.

Given the internal models that they have access to, I'm more worried about software companies being out of touch with the next generation of models will enable. Anecdotally I've seen that it takes a while for an organization to adjust to new LLM capabilities, even if everyone had access.

This article has basically nothing to do with the OpenAI paper; it's a critique of the entire field of computer science, rehashed to pick at one of their papers.

Wake me up when someone has understood the proof.

> This brings us to a big underlying problem with the way AI is being sold to us: it's being exaggerated to sound like way more than it is, and it's being sold to people/governments as something that will be capable of exceeding human "intelligence" in the near future when it's not capable of intelligence at all.

How can you possibly hold this belief after Tuesday? An LLM solved a significant number of major open problems in many subfields of mathematics, including a major breakthrough on the Riemann Hypothesis. Get your head out of the sand.

Terence Tao:

> And I mourn the path not taken, and the opportunities lost in the frantic race to develop this technology. Labs submitting their frontier models to independent researchers for proper scientific evaluation. Coordination with the research community to ensure these tools are applied to complement and enhance the abilities and activities of human researchers, rather than compete with them. Use of these tools to foster collaboration and sharing, rather than competition and secrecy. Opening new doors, without closing old ones.

But that is not the path we now find ourselves in. Instead, the community needs to come together more than ever. To clearly declare our own standards and values, to build our own tools and practices, to support our most vulnerable members, and to chart our own path forward. Let’s get to work.

Also this, from Peter Scholze:

We should remember that we are all in this together; mathematics is a marathon, not a sprint; and the goal is and always will be the human understanding of mathematics, which will invariably take time.

What is worrying me the most, at the moment, is actually the (in)security of cryptography. Finding algorithms breaking standard cryptographic protocols is a number theory problem whose difficulty does, to my non-expert eyes, not significantly exceed what these systems are now capable of. And it would have disastrous consequences on society if such an algorithm is found.

This is pretty insightful:

> One thing to keep in mind about this drop of papers is that they were not created by OpenAI with the main goal of making progress in mathematical research, or even primarily for PR (there were some PR posts, but e.g. the final announcement was comparatively restrained). The main reason these papers exist is that OpenAI used unsolved math problems to evaluate their internal models for the purpose of improving their capabilities. They had to use hard open problems because these are the only ones still posing a challenge to frontier AI models. I believe that, from OpenAI’s point of view, the papers are essentially a byproduct of their internal development.

I choose option 1. It's in the name! OpenAI should give mathematicians the chance to do at least some of this work themselves. It would be a lot easier for the community to adjust if they could use these same tools to explore how math research can adapt to the new circumstances. Instead, OpenAI is racing ahead to get as many results as possible with a proprietary model that the mathematical community cannot access. If these companies are building AI to benefit humanity as they assure us, then shouldn't they share their resources rather than vacuum up everything for themselves? Especially for something with as little direct monetary value as pure math.

I think there is way too much focus on how they released the results; it was going to be a huge shock no matter what. We should be talking about the massive concentration of "mathematical wealth" occurring here, and how it is a microscosm for what may happen on a much larger scale as AI outperforms humans on more and more domains.

> That doesn't take away from the fact that Math just turned 180° on its head and there's no going back.

100% agree. I'm not saying that it does.

These were all related to the Hodge conjecture, one of the Millennium Prize problems. There are still several other manuscripts related to the Hodge conjecture.

We'll see if there are any other gaps found in the coming months.