Not to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important.
Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.
Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.
AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on.
It seems to always be some form of this:
Mathematician:
“Propose conjecture a and conjecture b can’t be true simultaneously”
AI: “they can”
Everyone: “ok…”
I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in.
They might be low-hanging fruit but two things immediately come to mind:
* As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood.
* What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.
Tip for smart science-y young people: think about a career in experimental physics. Experimental data is the complement of theoretical power. Since theory can be provided cheaply by LLMs, experimental ability is now the bottleneck for progress in physics.
I expect to see frontier labs or startups hiring experimentalists to provide data for LLMs to analyze, pushing towards breakthroughs in areas like room-temperature superconductors and fusion.
Physics as a domain is a nightmare. Even the employment statistics are hard to understand because, like Philosophy, only the best of the best pursue it.
I've had countless friends throughout my PhD studies tell me that their decision to pursue a PhD in Physics ruined their lives. (Which is an exageration, but you get the point.)
The bottom line is that you should pusue Physics only if you still want to in the face of excessive media/reccomendations/statistics telling you not to.
Only theory that is a convex combination of existing theory. Any paradigm shift is currently unreachable to LLMs and can be only obtained by luck with RL due to the curse of dimensionality.
This sounds depressing. Imagine going to work every day and your boss is a computer telling you to do rote nonsense so it can barely-better-than-brute-force search for breakthroughs in whatever field. Then when it finds one we get another breathless news cycle like this while you get no credit at all. If you could understand what you were working on, you might be able to contribute more than a .csv of data, but the computer can't read you in because there is no understanding under the surface.
Barely better than brute force (I can't believe it's not brute force!™) aside, presuming we get super intelligence it will all be depressing when it comes to intellectual pursuits like this.
On the one-hand side, it's really impressive how LLMs drive mathematics forward, and this pace is only accelerating very quickly.
At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
> LLMs do not care about "elegance" the way human beings do, which is a big advantage.
It's just a matter of time before you can post train it for elegance too. Mathematical proofs in particular can be formally verified automatically which is a big advantage.
Why is it “just a matter of time”? Why do we assume and say this?
The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.
I'm not sure that elegance will be so easy to train for, the same way that writing skill has plateaued (or arguably declined) since earlier models. "Have you solved the problem" is verifiable, but questions of taste are harder to pin down.
I've actually been involved in annotation projects doing RLHF to train LLMs to do exactly that. It's not a matter of time, it's already happening - it's just seemingly lower priority than "profitable" projects like post-training LLMs to replace white collar workers.
Before LLMs, empire building was a very large incentive to hire. Teams tended to become larger than they needed to be so the boss feels good about their life choices.
LLMs do not fix this problem, they make it worse. Instead of the team being oversized, they’re now way oversized. It is still in everyone’s best interest to look busy anyways and LLMs do help a lot with that.
Isn't it a bit Catch 22 anyway? If someone finds a algorithm to reduce some NP task X to class P, then that just means X wasn't a true NP task and P!=NP is still undecided?
If it’s an NP-complete [0] problem like SAT, as many NP problems are, then we are done, because all NP problems can be reduced to it (in polynomial time).
You can prove something is in NP by providing a (polynomial) reduction from a known NP hard task, and vice versa. All the known NP problems (Knapsack, SAT, etc) are mutually reducable in this way, so solving one lets you solve the others. So if X was shown to be NP, then given a polynomial time solution to X, you can stack the polynomial time reduction from X to SAT to solve SAT in polynomial time too.
Sure they care about elegance, or at least brevity. Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems. It doesn't mean they are perfect at it though.
> Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems.
First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?
It is mathematical folklore that one should attempt to prove a conjecture by day, disprove it by night. Jordan Ellenberg recently popularized this in his 2014 book. He and I both heard this from Barry Mazur, but it dates at least to Bing, if not antiquity.
What is the purpose of mathematics? To be the architect of new conventions by seeing clearly past the old? If so, believing that the entire point is proving statements is a poor start. Bill Thurston was a visionary who happened to prove a great deal of what he saw, but his influence was his vision.
For those of us who like to understand every line of code we generate, and have labored for years to learn how to make best use of AI, a factor of two is a reasonable estimate for our productivity gain.
For those of us who believe mathematics is about achieving human understanding, having machines decide what's true and what isn't makes a night and day difference. Again, about a factor of two.
The Maxwell conjecture is a toy problem. The existence or nonexistence of a bound on the number of equilibrium points in an electrostatic arrangement of point charges doesn’t change much. I say that as an EE but not a specialist in electromagnetism.
"The idea behind this construction was suggested by an LLM (OpenAI’s GPT-
5.6 Sol). The authors have verified the mathematical details and have written
the argument in their own words. Computer algebra software (Mathematica,
Maple) was used to verify computations and produce visualisations"
Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
No but the Clay Mathematics Institute will give you $1,000,000 if you solve one of the 6 remaining Millennium Prize Problems, and if you solve certain Erdos problems you can get $10-10,000.
Not to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important.
Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.
Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.
Real talk.
AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on.
It seems to always be some form of this:
Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously”
AI: “they can”
Everyone: “ok…”
I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in.
"tautological games".
All proofs are a form of tautology, you have to end up at the point your theorem proposed. Math is games of logic. That's what it is.
Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.
They might be low-hanging fruit but two things immediately come to mind:
* As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood.
* What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.
Tip for smart science-y young people: think about a career in experimental physics. Experimental data is the complement of theoretical power. Since theory can be provided cheaply by LLMs, experimental ability is now the bottleneck for progress in physics.
I expect to see frontier labs or startups hiring experimentalists to provide data for LLMs to analyze, pushing towards breakthroughs in areas like room-temperature superconductors and fusion.
Politely, absolutely not.
Physics as a domain is a nightmare. Even the employment statistics are hard to understand because, like Philosophy, only the best of the best pursue it.
I've had countless friends throughout my PhD studies tell me that their decision to pursue a PhD in Physics ruined their lives. (Which is an exageration, but you get the point.)
The bottom line is that you should pusue Physics only if you still want to in the face of excessive media/reccomendations/statistics telling you not to.
At least in europe you can do a 3 year Eng Phys BSc and if it doesn't pan out you can do a master in EE or MEng.
i would make a similar argument for entrepreneurship and most things: do it only if you can’t bear the thought and reality of doing something else.
Only theory that is a convex combination of existing theory. Any paradigm shift is currently unreachable to LLMs and can be only obtained by luck with RL due to the curse of dimensionality.
Until we got robots doing that
This sounds depressing. Imagine going to work every day and your boss is a computer telling you to do rote nonsense so it can barely-better-than-brute-force search for breakthroughs in whatever field. Then when it finds one we get another breathless news cycle like this while you get no credit at all. If you could understand what you were working on, you might be able to contribute more than a .csv of data, but the computer can't read you in because there is no understanding under the surface.
Barely better than brute force (I can't believe it's not brute force!™) aside, presuming we get super intelligence it will all be depressing when it comes to intellectual pursuits like this.
How’s this different from just asking an llm to prompt you to perform experiments? You don’t need any expertise.
On the one-hand side, it's really impressive how LLMs drive mathematics forward, and this pace is only accelerating very quickly.
At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
> LLMs do not care about "elegance" the way human beings do, which is a big advantage.
It's just a matter of time before you can post train it for elegance too. Mathematical proofs in particular can be formally verified automatically which is a big advantage.
Why is it “just a matter of time”? Why do we assume and say this?
The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.
I'm not sure that elegance will be so easy to train for, the same way that writing skill has plateaued (or arguably declined) since earlier models. "Have you solved the problem" is verifiable, but questions of taste are harder to pin down.
This sounds kind of like unreadable code, though. So it's more than just taste.
I've actually been involved in annotation projects doing RLHF to train LLMs to do exactly that. It's not a matter of time, it's already happening - it's just seemingly lower priority than "profitable" projects like post-training LLMs to replace white collar workers.
>> post-training LLMs to replace white collar workers.
And I look forward to a single example where this happened....
Before LLMs, empire building was a very large incentive to hire. Teams tended to become larger than they needed to be so the boss feels good about their life choices.
LLMs do not fix this problem, they make it worse. Instead of the team being oversized, they’re now way oversized. It is still in everyone’s best interest to look busy anyways and LLMs do help a lot with that.
> most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter)
How do you know they're correct if they're super messy and chaotic?
formal verifiability e.g. vi Lean
Even Lean has bugs.
> AI "Proves" Collatz Conjecture with Lean 4 Bug
https://news.ycombinator.com/item?id=49101465
I've driven back roads in Ireland. Super messy and chaotic. I was still able to use a map to get to my destination.
I agree, counterexample to P!=NP would be great. I tried but it's a mess.
I’m pretty sure “counterexample” is the wrong word here.
Why? A counterexample to P!=NP would be a polynomial algorithm for SAT. If it exists, it might be a constructible object.
That’s not a counterexample to P != NP, it’s a proof that P = NP.
At best, it would be a counterexample to the claim that no NP-complete problem is in P.
Isn't it a bit Catch 22 anyway? If someone finds a algorithm to reduce some NP task X to class P, then that just means X wasn't a true NP task and P!=NP is still undecided?
If it’s an NP-complete [0] problem like SAT, as many NP problems are, then we are done, because all NP problems can be reduced to it (in polynomial time).
[0] https://en.wikipedia.org/wiki/P_versus_NP_problem#NP-complet...
You can prove something is in NP by providing a (polynomial) reduction from a known NP hard task, and vice versa. All the known NP problems (Knapsack, SAT, etc) are mutually reducable in this way, so solving one lets you solve the others. So if X was shown to be NP, then given a polynomial time solution to X, you can stack the polynomial time reduction from X to SAT to solve SAT in polynomial time too.
AIUI if you have an (polynomial-time) algorithm to reduce some NP-complete task to P then you have indeed shown that P=NP.
Sure they care about elegance, or at least brevity. Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems. It doesn't mean they are perfect at it though.
> Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems.
First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?
So they can charge more per token and decrease the pressure on their infra.
You clearly haven't used Claude to generate code or documentation.
"First rule in government spending: why build one when you can have two at twice the price" - S.R. Hadden
Personally, I've found city roads to be more reliable than the private roads where I live.
It is mathematical folklore that one should attempt to prove a conjecture by day, disprove it by night. Jordan Ellenberg recently popularized this in his 2014 book. He and I both heard this from Barry Mazur, but it dates at least to Bing, if not antiquity.
What is the purpose of mathematics? To be the architect of new conventions by seeing clearly past the old? If so, believing that the entire point is proving statements is a poor start. Bill Thurston was a visionary who happened to prove a great deal of what he saw, but his influence was his vision.
For those of us who like to understand every line of code we generate, and have labored for years to learn how to make best use of AI, a factor of two is a reasonable estimate for our productivity gain.
For those of us who believe mathematics is about achieving human understanding, having machines decide what's true and what isn't makes a night and day difference. Again, about a factor of two.
Could you expend on what you mean? I don’t have a math background and don’t really understand your comment
Does anyone have any idea why there's no Wikipedia article (or redirect) for Maxwell Conjecture: https://en.wikipedia.org/wiki/Maxwell_Conjecture
Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
Please, what does that mean for Maxwell equations? For electromagnetism?
(Wikipedia redirects Maxwell's conjecture to Maxwell equations).
The Maxwell conjecture is a toy problem. The existence or nonexistence of a bound on the number of equilibrium points in an electrostatic arrangement of point charges doesn’t change much. I say that as an EE but not a specialist in electromagnetism.
> what does that mean for Maxwell equations?
Nothing. They're still just as valid as they were before.
> For electromagnetism?
In practical terms, nothing significant. It's not going to change how anyone builds devices that use electromagnetism.
Seems like that anyone can now prove math conjecture. Maybe someone already prove some math problem and is not even aware of it.
Disprove, you mean.
This is so inelegant I can't tell if it's accurate or not. ...On the other hand, I can't solve it myself.
"The idea behind this construction was suggested by an LLM (OpenAI’s GPT- 5.6 Sol). The authors have verified the mathematical details and have written the argument in their own words. Computer algebra software (Mathematica, Maple) was used to verify computations and produce visualisations"
Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
Looks like the figures are cut off?
The experimental HTML view is messed up, but the actual PDF is fine.
Ok, who gets the credit?
Does this work like a bug bounty program, where OpenAI pays you if you find a nice application for ChatGPT?
No but the Clay Mathematics Institute will give you $1,000,000 if you solve one of the 6 remaining Millennium Prize Problems, and if you solve certain Erdos problems you can get $10-10,000.
Even if you use AI tools?
Yes. But you’ll spend more in tokens than you’ll get back from prize money.
Some people might be on the ChatGPT Pro subscription plan or consuming their employer’s tokens.
This is not true
Lies, obviously. AI is worthless.
EDIT: Guys! Sarcasm!