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.
moomin 24 minutes ago [-]
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.
tsunamifury 17 minutes ago [-]
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.
efficax 13 minutes ago [-]
"tautological games".
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.
dwaltrip 14 minutes ago [-]
Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.
ashleyn 32 minutes ago [-]
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.
throwaway0123_5 11 minutes ago [-]
> 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.
julianeon 7 minutes ago [-]
Is the bar really higher? Those math PhD's can use GPT too; they benefit equally from AI assistance.
37 minutes ago [-]
d_burfoot 53 minutes ago [-]
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.
ComputerPerson 29 minutes ago [-]
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.
drob518 4 minutes ago [-]
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.
pdhborges 24 minutes ago [-]
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.
l33tman 9 minutes ago [-]
The engineering programs and their curriculums are very different from physics in the science faculties (which I guess was the context here)
20 minutes ago [-]
ModernMech 10 minutes ago [-]
Pick any field and you'll find countless PhDs eager to tell anyone who will listen all the ways it ruined their lives.
coderatlarge 19 minutes ago [-]
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.
storus 17 minutes ago [-]
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.
StilesCrisis 10 minutes ago [-]
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.
stubbi 43 minutes ago [-]
Until we got robots doing that
bre1010 35 minutes ago [-]
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.
alasano 29 minutes ago [-]
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.
simianwords 30 minutes ago [-]
How’s this different from just asking an llm to prompt you to perform experiments? You don’t need any expertise.
Syzygies 41 minutes ago [-]
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.
dgellow 18 minutes ago [-]
Could you expend on what you mean? I don’t have a math background and don’t really understand your comment
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.
hawtads 58 minutes ago [-]
> 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.
travisgriggs 17 minutes ago [-]
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.
ainch 44 minutes ago [-]
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.
card_zero 35 minutes ago [-]
This sounds kind of like unreadable code, though. So it's more than just taste.
jmalicki 52 minutes ago [-]
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.
tcp_handshaker 47 minutes ago [-]
>> post-training LLMs to replace white collar workers.
And I look forward to a single example where this happened....
pitched 40 minutes ago [-]
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.
pdonis 1 hours ago [-]
> 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?
I've driven back roads in Ireland. Super messy and chaotic. I was still able to use a map to get to my destination.
js8 45 minutes ago [-]
I agree, counterexample to P!=NP would be great. I tried but it's a mess.
layer8 41 minutes ago [-]
I’m pretty sure “counterexample” is the wrong word here.
Good4boothee 38 minutes ago [-]
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?
layer8 26 minutes ago [-]
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.
SetTheorist 31 minutes ago [-]
AIUI if you have an (polynomial-time) algorithm to reduce some NP-complete task to P then you have indeed shown that P=NP.
js8 29 minutes ago [-]
Why? A counterexample to P!=NP would be a polynomial algorithm for SAT. If it exists, it might be a constructible object.
layer8 16 minutes ago [-]
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.
dcsommer 2 hours ago [-]
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.
windexh8er 12 minutes ago [-]
"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.
Someone 1 hours ago [-]
> 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?
yreg 1 hours ago [-]
So they can charge more per token and decrease the pressure on their infra.
senorrib 2 hours ago [-]
You clearly haven't used Claude to generate code or documentation.
nelox 1 hours ago [-]
"First rule in government spending: why build one when you can have two at twice the price" - S.R. Hadden
KPGv2 60 minutes ago [-]
Personally, I've found city roads to be more reliable than the private roads where I live.
vatsachak 7 minutes ago [-]
Awesome! Confirms what we know; LLMs are superhuman at short term reasoning and breadth
JPLeRouzic 1 hours ago [-]
Please, what does that mean for Maxwell equations? For electromagnetism?
(Wikipedia redirects Maxwell's conjecture to Maxwell equations).
gjskngnf 54 minutes ago [-]
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.
pdonis 1 hours ago [-]
> 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.
Most common names have redirects and Wikipedia is very complete. Was it just not commonly known by that name?
olirex99 55 minutes ago [-]
Seems like that anyone can now prove math conjecture. Maybe someone already prove some math problem and is not even aware of it.
layer8 39 minutes ago [-]
Disprove, you mean.
josefritzishere 1 hours ago [-]
This is so inelegant I can't tell if it's accurate or not. ...On the other hand, I can't solve it myself.
tcp_handshaker 44 minutes ago [-]
"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
jdc-pub 4 hours ago [-]
Looks like the figures are cut off?
smallerize 4 hours ago [-]
The experimental HTML view is messed up, but the actual PDF is fine.
4 hours ago [-]
faangguyindia 24 minutes ago [-]
[dead]
amelius 1 hours ago [-]
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?
chorsestudios 1 hours ago [-]
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.
echelon 57 minutes ago [-]
Even if you use AI tools?
muglug 55 minutes ago [-]
Yes. But you’ll spend more in tokens than you’ll get back from prize money.
beering 28 minutes ago [-]
Some people might be on the ChatGPT Pro subscription plan or consuming their employer’s tokens.
simianwords 29 minutes ago [-]
This is not true
qarl2 49 minutes ago [-]
Lies, obviously. AI is worthless.
EDIT: Guys! Sarcasm!
Rendered at 15:48:24 GMT+0000 (Coordinated Universal Time) with Vercel.
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.
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.
* 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.
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.
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!