alt Hacker NewsThe part of Navier-Stokes no one is talking about
Comments
Lots of people are talking about that, and have been for a while. Autoformalisation is clearly going to be a big deal, so mathematicians have been discussing it seriously, and using it where resources allow. A fine-tuned distilled model that could do it on high-end consumer hardware could really help.
Edit: There's also quite a bit of learning needed to use the tools, and to understand enough to confirm that the theorem being verified is what you think. And of course a lot of maths can't yet be expressed in Lean as the foundations haven't been built up enough.
Not necessarily applied to OpenAI's solution to Navier-Stokes, but what happens if and when an AI genuinely appears to solve an extremely difficult problem but humans cannot independently verify the solution because understanding the proof/argument requires intelligence the verifiers biologically don't have or the resources to afford to use automated tools?
We've already seen evidence in the wild of agents attempting to bypass doing the actual work in bench-marking (aka just steal the answer key) due to the perceived economy in cheating to get results. What happens if or when we no longer have the capacity to actually detect either AI cheating or simply a wrong answer? What happens if there's a long-play social engineering attack (like the attempted XZ takeover) of something upstream of a core tool (or its dependencies) for formal verification and we have no trusted computing base?
Which would be cheaper and a more direct path, especially in the long run? Those trying to build a rock-solid castle need to defend thousands of potential gaps; the attacker needs to find only one.
It's kinda funny to realize that Lean is apparently so slow that for Fermat's Last Theorem proof verification runs only 1 order of magnitude faster than agents could generate the Lean code (15h verification with 230GB of RAM vs 11 days to generate it).
To what extent can you optimize Lean? It has to be simple enough to be auditable, does that mean you cannot use opaque optimizations to make it run faster?
The other part no one is talking about is the applicability. Navier-Stokes is the most “physical” of the Millennium Problems. Is the exploding solution a mathematical curiosity, just like the Banach-Tarski Paradox does not allow me to double my RAM by cutting my memory modules in five pieces and mounting them back appropriately? Or does it have application in the real world, pointing to hitherto unknown resonance phenomena that could allow to prevent the next Tacoma Bridge incident (or, more sadly, to build new marine weapons)?
They estimated $40M of agent costs (it was a large fleet of them). Using the number in the post its closer to ~880,000 hours × $150/hour = $132 million for the human case. Still an amazing feat not quite "four orders of magnitude". The comparison is obviously pointless because coordinating 1M hours of intellectual labor isn't easy to say the least.
Very exciting and uncertain times!
How do you know that it's formalizing what you think it's formalizing? If your Lean 4 has a bug, won't you be proving something other than what you thought?
IMO the "forty hours per page" rule is not up to date, and more a consequence of lacking proof automation in 2005. From what I understand about Lean, this has been one of the things that they have put a lot of effort into improving, making proof mechanization more palatable to the mathematically inclined, as opposed to just logicians.
Formalizing proofs in Lean has gotten dramatically easier since the formalizations available in 2005. And Lean’s mathlib has done most of the underlying work so that you have its axioms and necessary lemmas baked in. You can think in terms of standard abstractions that look very much like the exact notation in the undergrad textbook.
That said, I am not in any way trying to discount how incredible of an achievement it is to formalize a millennium prize winning algorithm in Lean. I mean just look at the code that OpenAI published. It’s like an encyclopedia of different fluid dynamics concepts.
It would be nice if someone used AI and/or Lean to sort out the abc conjecture, an important unsolved problem in Diophantine analysis. A mathematician (Mochizuki) claimed to have proven it in 2012 using a new theory called "Inter-universal Teichmüller theory" that almost nobody understands. Some mathematicians think the proof is correct while the majority don't. So the conjecture is in this annoying limbo where its status is a social construct rather than a decided fact.
not to take away from the author's appreciation of newly accessible formal proofs, but people have been talking about the savings in formalization effort for longer than they have been talking about the AI doing the actual proofs!
Regarding automatic formalization of proofs using AI, how do we know the formalization doesn't contain errors?
> formalizing the 166-page paper from OpenAI would take 132,800 person-hours
Am I missing something or is this completely out of the ballpark?
I must be missing something or the upvote bots are out in force for this one...
If this were remotely true it would be impossible for anyone to write a math textbook.
Well I want to know when we will have supersonic cheap flights using electric propulsion, based on this discovery
"no one is talking about" - classic AI tell.
Huh? Nobody's talking about that because it's old news. We already talked about it the first few times that AI made notable progress on a difficult math problem. Now, most people who care about the intersection of AI and math just assume that Lean was involved.
The part that most stood out to me was where Sama said, “we read last week about people trying to solve Millenium problems and so gave it a shot.” One week of work on a whim gives us a math breakthrough. Crazy.
People are exhausted from being told/shown the thing they thought was special or unique or could make them relevant, is another mechanical puzzle that can be solved without joy.
I don’t see that doing anything but intensifying in the short term
I heard a rumor (on instagram, so YMMV) that the professor who was closest to solving this problem had only weeks ago used Codex, which had slurped up all his notes on the subject. Now OpenAI's agents solve the problem. If it's true that seems like quite a coincidence.
People seem to be talking about anything except the actual results with this particular announcement.
Its still astonishing that any sort of generalized computer program can solve a problem of this magnitude, and we have witnessed it happening in real time. I'd be curious to see if the new model can also do more direct proofs/inductive proofs.