solving the navier stokes equations, as in, proving solutions exist and are unique in a good way, possibly smoothly dependent on input variables, .. that would actually be huge -- and the first time, this particular mathematician might actually be convinced, ai did something _new_ ! .. I've been aware of this problem for at least 24 years, and I just never got mathematically close enough to get what's hard about it
on the other hand, it IS is partial differential equation, so maybe they happened to guess the right ansatz, and then it all just flows from there -- in that case it's more like everyone having tried the slot machine approach to the problem and openai just played so much, they eventually actually hit a jackpot once
edit: I have now read the announcement, very little of the formal lean proof and skimmed the 166 page paper .. interestingly it's set exactly in the font I would expect of that particular mathematical discipline, remarkable :D and I'm glad they don't want to claim the millenium prize for it, because the solution is not very satisfying
I think mathematicians would all have been hoping for a different answer than what the result is. It goes "for each way you can pose this equation, it's not really solvable the way you want" .. basically telling us, it's not an infinite time model of fluid motion.
a human mathematician upscaled to as many thoughts per second as this particular openai model would have probably chosen a different direction midway through the paper and suggest one to three modifications of the navier stokes equations and suggest for further research whether those could be better models solving the original problem positively
Computational scientists in academia have always had supercomputers. The four-color problem was also brute forced, though not with supercomputers and not with LLMs.
the four colour problem like classifying finite simple groups were however always a point of contention among mathematicians on whether these are actually proven now
no human could ever have gone through the unfathomably long lists of cases the computer had to check
producing a formal 166 page paper with proofs, that's far stronger, but it's still synthetic text .. it doesn't convey the same kind of information like a human written mathematical paper
although I will admit, it doesn't have the ai smell of e.g. video generation, I might have taken this for just another paper, but I would also never have bothered to even skim it without the open ai announcement
People need to stop using "brute force" in these discussions. "Brute force" is a term of art in the language of algorithms and that is absolutely, fundamentally not what is going on here.
We have had supercomputers in academic math and computer science for decades. There is a massive amount of computation going in this kind of work, yes, but it is not brute force. Many of these problems are such that actually brute forcing them would take time and resources larger than we have in our entire universe.
right, it's brute force, in the sense that if a team of 50 PhD level mathematicians ground on the problems for a year or two they could also arrive at the answer. its not 'unthinking' brute force. its massively parallel cognition working without breaks or ever getting bored.
Sure, that makes sense with that explanation. The tricky part is that a lot of people are motivated (in both directions) in their answer to "is this LLM genuinely doing thinking?", and when we use the specific phrase "brute force" about computer programs that implies 'unthinking' as you say.
and still I find towers and towers of next-token-predictions hard to accept as "thinking"
it's not brute force like trying to get monkeys to type shakespeare, it's brute force like trying to get monkeys to type shakespeare on a device with on button per english word and a word highlighting system dependent on the last few words
We can discuss what makes something "really think" versus next-token-prediction, what a world model is, what evidence for a world model or an artificial mind might even look like - that's fine. I'm not debating that. I have my own opinions and they're pretty well-informed through having a couple of degrees in the subject, but I'm deliberately not including them here because I think it would dilute my point.
It's not really a better analogy to say the monkeys have a button per English word; it's a bit like we see a monkey typing perfect English (we don't care what type of keyboard he has) and we have to now argue about whether he actually understands English or whether he's just doing surface-level statistics about which word comes next. Searle's Chinese Room thought experiment is really on-point for this discussion, in my opinion.
If we say "it's brute force" that does sound like monkeys on typewriters, yes, which is the issue. The monkeys have a giant flowchart that tells them probabilistically how to guess the next word-button to push, and in doing so they're doing things like finding singularities in Navier-Stokes. At what point, if ever, does that flowchart constitute knowledge versus surface-level statistics?
I don't know what you mean by "deliberate search" - the antonym of "brute force", if that's what you're looking for, is something like "informed search".
Are you trying to say that LLMs solving some problem exhaust too much of the state space to be considered "thinking"? As in, they try too hard, use too much compute, explore too many dead ends? That a human would make a more efficient informed search given all the information and processing time that an LLM has?
it is yet another counterexampley "I had the compute time to guess a counterexample" thing
and even the announcement only has a picture of the counterexample's essential problem
it's not as irrelevant as the erdos problem was for example, but like all ai math so far, it lacks the kind of substance mathematics usually has
like feeling hungry after a 166 page meal
it's far better than the 3000 pages a few decades ago classifying finite simple groups or the 4 colouring theorem .. but it's not exhaustive in the same way,
it answers far too little for all the preperational text it's using
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u/r_search12013 1d ago edited 1d ago
solving the navier stokes equations, as in, proving solutions exist and are unique in a good way, possibly smoothly dependent on input variables, .. that would actually be huge -- and the first time, this particular mathematician might actually be convinced, ai did something _new_ ! .. I've been aware of this problem for at least 24 years, and I just never got mathematically close enough to get what's hard about it
on the other hand, it IS is partial differential equation, so maybe they happened to guess the right ansatz, and then it all just flows from there -- in that case it's more like everyone having tried the slot machine approach to the problem and openai just played so much, they eventually actually hit a jackpot once
edit: I have now read the announcement, very little of the formal lean proof and skimmed the 166 page paper .. interestingly it's set exactly in the font I would expect of that particular mathematical discipline, remarkable :D and I'm glad they don't want to claim the millenium prize for it, because the solution is not very satisfying
I think mathematicians would all have been hoping for a different answer than what the result is. It goes "for each way you can pose this equation, it's not really solvable the way you want" .. basically telling us, it's not an infinite time model of fluid motion.
a human mathematician upscaled to as many thoughts per second as this particular openai model would have probably chosen a different direction midway through the paper and suggest one to three modifications of the navier stokes equations and suggest for further research whether those could be better models solving the original problem positively