https://www.claymath.org/millennium-problems
1 was solved - the Poincare conjecture - by Grigori Perelman. There are 6 remaining.
There are 17ish years left before this market closes.
Apr 4, 1:51am: Will at least 4 of the 7 Millennium Problems by solved by 2040? → Will at least 4 of the 7 Millennium Problems be solved by 2040?
1,000
3.00@zsig Of course, however AI is likely to have advanced significantly as whole if it reaches such a level. Compute costs are also likely to be lower due to new innovation and AI will be more widespread. It doesn't need to be a all powerful predicter if it just constantly searches the web and updates its estimates based on new information like most traders.
@SentientTree I'm agnostic. My analysis centers on the effective amount of top-tier mathematical research effort that will be directed at the problems in the next 15 years, versus the past few decades/centuries since they were posed. I think it's easy to conclude that we will do the equivalent of several orders of magnitude more top-tier serial mathematical thinking in the next 15 years than perhaps all of previous human history combined.
@AdamK I think that sweeps too much informative detail under the rug.
Presumably you're referring to AI, but what has AI actually contributed to mathematics so far?
It's definitely not nothing, but it's also only to a very specific type of problem, where one wants to construct one specific object satisfying some criteria, for which we have a notion of being close to or far from the criteria, and a notion of combining candidate objects into better candidates. All the DeepMind results are of this kind, and I think only 1 of the 6 remaining MPs are amenable to this (Navier-Stokes). And that's only if the true answer is blowup is possible. If the true answer is smooth initial conditions never lead to singularity, the DeepMind approach is powerless.
The other 5 MPs are not amenable to this kind of approach, so what else do you have? The IMO approach of generating 1000s of proof fragments and hoping you can glue them together into something valid? That only works when there's a valid proof that's only 1-2 pages long, and it is known in advance what the relevant techniques are. ("This is a functional equation, let's use functional equation techniques!") Otherwise the search space is too large.
@pietrokc I don't think the right question to ask is what current AIs can contribute to the leading edge of math. Five years ago the answer was obviously nothing; now the answer is very little but not nothing; in five years the answer could be "quite a lot" if not everything; fifteen years is long enough in AI years for the result to be overdetermined.
It has often been the case in the history of AI forecasting that an overemphasis on object-level methods and challenges for a class of capabilities will severely underestimate the gains that come from general-purpose scaling. It is such an uncomfortable intuition to develop that even most AI researchers struggle with it despite seeing it repeated in domain after domain, to say nothing of the protests object-level experts make when they encounter AI for the first time, encroaching slowly at first. That is why it's called the Bitter Lesson.
@AdamK I think all of that is incorrect.
First of all, AI did not start five years ago. In fact AI contributed new theorems as far back as the 50s [1]. An open problem that Alfred Tarski failed to solve was solved by an automated theorem prover in the 90s [2]. So, by your reasoning, AI should have solved the millenium problems already -- perhaps even before they were made MPs!
Second, your reasoning includes nothing about the problems themselves, and is therefore trivially disprovable. It is known [3] that arbitrary Diophantine equations are as hard as the halting problem, so if one of the MPs happened to be a Diophatine equation equivalent to ZFC being inconsistent, no AI would ever solve it.
Third, scaling discourse is severely misguided and lots of revisionism is happening. It used to mean "just train a model with more nodes and more data and it will solve everything". Except that, even restricting ourselves to this latest incarnation of AI, GPT-4.5 is scaled all to hell and it still cannot solve very simple math problems. (There are in fact theorems to this effect [4].) You have to use "reasoning mode", where as far as anyone can tell, it just spends tokens searching for proofs using simple heuristics, same as the theorem provers of 70 years ago.
I rather think people refer to a Bitter Lesson more because LessWrong likes Dramatic Slogans than due to anything based in fact. Relatedly, I did Good Judgment Open for several years and I encountered a lot of people who forecast based on grand generalities instead of detailed facts. It was not very hard to beat them.
[1] https://dl.acm.org/doi/10.1145/1460361.1460381
[2] https://en.wikipedia.org/wiki/Robbins_algebra
[3] https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
assuming no AGI/ASI, almost certainly P vs NP won't be solved by 2040. RH probably has the most thought put into it, but hasn't really budged and still lacks lots of crucial ingredients to attempt a proof.
Birch and Swinnerton-Dyer and Navier-Stokes have good partial results and probably the closest to be solved, still terribly difficult. Hodge and YM are also very unlikely
IMO unless AI deeply, deeply improves pedagogy in the next few years that gives everyone the ability to be some timelord super mathematician i dont see 3 of these being solved, and I'm not too convinced that current paradigm AIs will lead to a solution to any of these (admittedly 15 years is a long time for someone to find some new paradigm and make some supermachine)
P vs NP is almost universally considered the hardest of the 7. Next to zero chance it's solved by 2040. That means we need 3 out of the remaining 5
Navier-Stokes: most likely candidate. If it's false as many physicists suspect, I would expect we're able to find a counterexample before 2040. Otherwise it may remain unsolved
Birch and Swinnerton-Dyer: mathematicians have built up a good bit of the framework here, so it's possible but I'd give it a <50% chance.
Yang-Mills: the harder of the physics problems. Physicists have been at it for 50 years with little progress. Unlikely.
Riemann: the most famous of the 7. Results have come along the way which could be useful, but the workhorse of the proof has not been established. Unlikely
Hodge: I don't have the mathematics background to even understand the problem statement but I've heard it's the hardest of the pure math problems.
I know an online betting site is going to full of people that think AGI is coming tomorrow but AI is not going to be our savior here. It can do casework quite well but the millennium problems are not that type of problem. My credence is ~8% that this resolves YES. I'd put 2 out of 7 around ~50% and 7 out of 7 at essentially 0%
@SentientTree So what I'm seeing is:
- 1 already solved + 2 on average solved "naturally"
- I think AGI is coming >50% by ~2029/2031
- AGI can make ASI and/or improve math/physics progress a LOT in almost a decade
- AGI/ASI only needs to solve 1 "unnaturally" in that decade-ish
- This only whole chain only needs to occur slightly more than 50% of the time for a bet to be worthwhile