My interpretation is they don't care about scooping mathematicians they were just trying to scoop a competitor. They had a short window in which to complicate Anthropic's priority, when Anthropic announced be able to say "ok nice but we did that too". Upon realizing they'd misunderstood, incredibly they said to this academic (who did not resolve NS) okay well just go ahead and claim the prize, so long as you're not Anthropic let's make this a good story.

If you spend $20M working out a Millennium Prize problem, in what universe would you offer that an unrelated effort should take credit? This is a branding game rather over whether software engineers are going to use codex or claude. In that light $20M (or whatever it was) might be worth it to squash even the rumor that claude code is more capable. Engineers look up to mathematics, while at the same time business and probably most engineers think the problem was to solve the problem. GPTs solved one the hardest known problems so they can solve my company's problem.

Some go further looking at these people, very on-the-nosely likened by one commenter here to ants, talking about education and responsibility and "core values" etc and just don't care. There was a major problem at the beginning of the week that is not a problem now and that is uncomplicated progress.

It's not wrong for OpenAI/Anthropic to do math for product development or even just branding but seemingly at no cost now they could work in an arena real mathematicians aren't interested in, versus just mowing the field. Everyone involved on their side should admit the purpose of these demonstrations is not to engage mathematical ideas it's about Claude/Codex. there's no shame in that. Which is better at solving random hard Diophantine systems? That would seem to tell me as much as I need to know insofar as a model's value is represented by raw mathematical power - then, take my money just as well!

Assuming the worst accusations are not true I think there are ways forward going to be acceptable for all. The labs themselves do not represent Terrance Tao as some kind of gate-keeping dinosaur in this. They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes are paid 7-8 figure salaries ultimately for the product, they solve a Millennium Prize problem then pretty quickly seem to move past it.

>they weren't trying to scoop a mathematician they were just trying to scoop a competitor

Well. Obviously, yes. But in doing so they still DID scoop out a mathematician in a very unethical way.

In doing so they showed that they basically have a huge gun they can point at X work you care about and develop, and can cut you across the finish line. And take credit for it. Obviously this already _existed_, but is just much more significant because even a rumor can be converted into a complete takeover of a discovery.

> They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes take home 7-8 figure salaries, they solve a Millennium Prize problem then pretty quickly move past it.

I get your point, they don't really care about solving all the maths problems. But they're still going to solve them for clout and profit motives. Up until it stops wow-ing people... at which point they will have likely decimated the frontier of the field.

And this is kind of the root of the entire concern. They will move into the forest and completely steamroll all the problems, then declare victory and move on, leaving only pavement and asphalt behind.

This is "just" an attribution and credit assignment problem. OpenAI could have done vastly better than they did at attribution. They should have spent another $10M just on attribution/credit research to annotate the contributions to the lean and paper and their blog.

I guess by their own account they were doing competitive analysis on the product and incidentally solved NS. Probably the two companies should agree to benchmark on random diophantine systems, which are harder and more applicable anyway. It wasn't "brute force" but they did spend $10-20M over the course of a few days based on a rumor, which is ... something. The fact they're not actually taking credit, and published no insights other than "it's a vortex" shows the company really has no business in mathematics. They're not interested (the guy said as much) and it obviously won't pay the bill - rather, they were able to pre-empt an announcement they thought Anthropic may have been preparing to make.

Big leaps in insight I don't think are received as elegant, generally. Fourier's contemporaries actually thought he was wrong (which he was), about both the series and heat equations, and even with the whole thing worked out it's not really straightforward. It fits into integral transforms/linear operators but in a way more complicated than can explained in those terms alone. Plus truncation error is commonly unbounded! Complex numbers and "fluxions" were also objectionable on their face although now one can frame them more or less elegantly.

> I could never imagine an LLM in its current form inventing something as elegant as the Fourier transform.

This is the crux imo. It doesn't really matter what absolute capability AI has at the moment, but whether we are on track with respect to architecture and training approaches. In my mind the only test for AGI is, if the thing trained up to the cut-off of Fourier's time (or Descartes, Newton etc), it should arrive at or exceed their insight.

^ As a measure of intelligence. But it's probably the case either way that LLM is more valuable in terms of the coordinated grunt work we'll put it to, than leaps of insight.

A proof may be impossible to fully grok, since about Leibniz - but problems will be solved now without even understanding the problem statement.

I mean, it is. Textiles are a technology. Extracting usable fiber from plants or animals was something we invented, so is making yarn. Knitting was invented, and shaping cloth to a foot was too. And then there's the modern side of automation. There's a lot of sophistication that goes into it.

I don't disagree yet it's also clear "hard science fiction" is meaningless if it's just, the people in this universe wear clothing, make campfires etc.

CoT evidently helps but a bias towards both correctness and innovation has to come from somewhere. The article implies OpenAI's proofs may be supported by Lean but regardless, who knows how many people are trying to disprove the next conjecture each day throwing away nonanswers. Fundamentally these systems are more powerful with better training and sampling methods, or better prompting. Tokens matter but you can rewrite many prompts to get a much better, faster answer using fewer tokens vs allowing "thinking" to go on and on.

Hey cool project! I built a solar system visualizer some years ago but just using the ephemerides data, no physics. This is ambitious. Noting your age I would say, invest some time in the basics - Euler, Improved Euler, finite differences, explicit vs implicit, multi-step methods etc.

Eg I see you've got a powerful adaptive Runge-Kutta method implemented in integrator.js. While that will do really well, for the sake of study you might make the solver implementation swappable and experiment with basic techniques. Some are very slow. Some maybe unstable and blow up the solar system. Why? Numeric methods are not one size fits all - see what the different tradeoffs are and how they respond to fiddling parameters. Understand the fundamentals.

Thank you! this is exactly the kind of feedback I was hoping for. Making the solver swappable is a genuinely great idea I hadn't considered. Watching Euler blow up the solar system while RK4 stays stable would make the tradeoffs immediately visible. Adding that to the roadmap.

A key thing about the undecidability problem wrt physics is preparation of the initial state. In math and computer science it is relatively straightforward to prepare such problems now (though this represented an enormous leap conceptually), but the "undecidability" of all physical problems relies on construction of materials that are clearly unconstructable - systems of infinite negentropy (eg Turing machines), infinite mass (the lattice), bespoke local interactions etc. Problems standing in the way of physics decidability are typically chaos, far from equilibrium mechanics, elementary SNR considerations and so forth, not problems of logic.

In physics we don't talk about decidability, but solvability.

The infinite lattice doesn't represent a "real" physical processes, it's just mathematical technique for closing a (fundamentally) quantized combinatorial sum over millions of interacting elements. The gap problem exists in the limit. For real systems the spectrum can be measured (in principle) by probing the ground state. The computational paradigm is incredibly general but only within what's apparently a pretty atypical thermodynamic regime (the ordered universe).

In this case quantum thermo.