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.
Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
Academic math is a bit like basic research. You come up with funny ways to look at numbers or prove weird statements about this thing you came up with and call a "group", and a couple years or decades or centuries later it turns out that this solves real problems in electrical engineering or biology
Or it ends up never becoming useful. But you can't know that in advance
Disagree, basic research, not matter how dull, is an observation of a measured reality. Math Theory are patterns of abstraction that may never be useful at all or representative of reality.
See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.
I find them useful bellweathers of genuinely out of domain performance and capability, regardless of their theoretical importance. What I see is that performance trends are remarkably stable both upstream (miraculous scaling laws of pretraining on validation loss) and downstream performance (epoch capability index). We get the equivalent of a GPT4->GPT5 performance leap every ~16-18 months, and we are not hitting ceilings nor do we see any deceleration.
Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.
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.
> 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.
It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types.
Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
Maybe masters will become more popular. I have no problem with bars being raised on PhDs, but it's crazy history that math may be the first one to have it raised (or brought back to old levels).
I don't believe this is raising the bar for minting fresh PhDs.
We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research.
They are not competing against AI to publish a counterexample to a known conjecture.
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.
My university had a great Physics department (Nobel laureate level), but every year you’d have a bunch of junior year (year 3) physics students desperately trying to switch into engineering as they realized that making a go of a pure physics degree was going to be more difficult. It was a consistently repeating pattern.
That's funny because so many of us in my PhD philosophy program would say: I should've been a physicist, philosophy ruined my life. I had a physics prof who told me it was foolish to go into philosophy when I could've been a physicist and I thought I knew better. Oh well, I probably would've been a software engineer in the end no matter what.
Exactly. These fields demand all you've got. Otherwise, you'll fall behind those who are willing to sacrifice everything else for the quest, so to speak. There will always be a long-tailed distribution populated largely by smart people with personal regrets.
Experimental physics is worrisome because as education becomes less-valued by society, there will be less funding available for research in general. Costly and elaborate 'big science' projects -- the kind that have to span multiple Presidential administrations in the US -- will be among the first to be classified as "waste," and performatively killed by legislators who earn votes by emulating the feeble-minded cultists who elected them.
I think the way forward, at least in the US, is going to involve leveraging AI to build better models to reduce our dependence on experiment. This will be true of both biology (where's the next HeLa line going to come from, anyway, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
As someone with a BS in physics, my advice to anyone who wants to study applied physics is to do robotics. Plenty of applied physics work to go around.
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.
The term “currently” is the sticking point here. As far as I know, there is still no one who can provide evidence as to why it should remain that way. And the "by luck with RL" is soon not by luck anymore. The more stupid and not so stupid ideas are discussed with the LLMs, the chances are growing that it is not a question about luck but by growing probability.
Frankly, such paradigm shifts are almost impossible for humans as well. If a mathematician proposes a truly radical paradigm shift, they're either a once-in-a-decade genius or a crackpot.
It is quite typical that once in a decade genius is treated like a crackpot... (what was the name of the Austrian doctor that suggested washing you hands before surgery?)
... of course, the number of crackpots is overwhelming
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.
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.
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.
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.
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'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.
That’s not a counterexample to P != NP, it’s a proof that P = NP. You can’t prove that two sets are the same by counterexample. What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
At best, a polynomial algorithm for SAT would be a counterexample to the claim that no NP-complete problem is in P.
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.
"They" don't "care" about anything. It is a stateless computational run across thousands of semiconductors. There is no objective this software has other than the computational function completing. To care would mean the model would have a level of discernment that goes along with sentience.
> 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?
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.
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.
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.
Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
All proofs are a form of tautology, you have to end up back at the point your theorem proposed. Math is games of logic. That's what it is.
Or it ends up never becoming useful. But you can't know that in advance
See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.
Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.
* 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.
It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types.
Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research.
They are not competing against AI to publish a counterexample to a known conjecture.
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.
Experimental physics is worrisome because as education becomes less-valued by society, there will be less funding available for research in general. Costly and elaborate 'big science' projects -- the kind that have to span multiple Presidential administrations in the US -- will be among the first to be classified as "waste," and performatively killed by legislators who earn votes by emulating the feeble-minded cultists who elected them.
I think the way forward, at least in the US, is going to involve leveraging AI to build better models to reduce our dependence on experiment. This will be true of both biology (where's the next HeLa line going to come from, anyway, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
... of course, the number of crackpots is overwhelming
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.
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.
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.
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.
And I look forward to a single example where this happened....
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.
How do you know they're correct if they're super messy and chaotic?
> AI "Proves" Collatz Conjecture with Lean 4 Bug
https://news.ycombinator.com/item?id=49101465
[0] https://en.wikipedia.org/wiki/P_versus_NP_problem#NP-complet...
At best, a polynomial algorithm for SAT would be a counterexample to the claim that no NP-complete problem is in P.
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?
(Wikipedia redirects Maxwell's conjecture to 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.
Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
Having the title "The Maxwell Conjecture Is False (GPT 5.6 Sol)" instead of "The Maxwell Conjecture Is False" is editorializing
Does this work like a bug bounty program, where OpenAI pays you if you find a nice application for ChatGPT?
EDIT: Guys! Sarcasm!