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.
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 bar for math PhDs already ruled out like 99.9% of the population, so I don't see it having much effect on discouraging people. The gap between even a bright student who takes AP calculus or whatever and someone studying e.g. spectral sequences is already incomprehensibly large. Like you literally could not even convey to a smart young person how far away they are from the boundary of today's understanding. I don't think I even have a reasonable sense with a math bachelor's!
People who get into math do it because they can't not do it.
I think it'll become increasingly important to have folks thinking about how to explain new math in a way that makes sense to your math bachelors students. And to your bright AP Calc students.
If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.
> 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.
A large proportion of degrees awarded today are not useful for any practical application.
But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.
If I were 18 I'd be sceptical about the value of degrees compared to the value of trades.
Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field.
If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.
I agree somewhat, but for the most part the signal is that you can succeed in some kind of knowledge work. If AI replaces ~all knowledge work, that signal isn't actually going to be valuable I think. While I don't find it certain that AI will do that, it seems increasingly plausible with every model release. In that world, I don't think very many people are pursuing degrees.
I never understood why being able to “play the game” is considered a positive signal. Getting a degree is the safe path, it doesn’t require any risk taking, and not much initiative. It’s about having the money (or going into debt for most people) and sitting your ass in a chair until you’re told you’re worthy. That idea is repulsive to me.
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.
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).
IMO, I don't think this will raise the bar for PhDs.
In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist.
Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.
I agree, but there's degrees (no pun intended!). Many PhDs are just an advisor telling a student exactly what to do. The training is still valuable, but it's not really what a PhD is "supposed" to mean. The example I give because of familiarity is many PhDs in chemistry/physics are just clicking run on simulations + running predefined analysis with results already predicted by the PI. Or in chemistry specifically OChem wet lab, it's often basically a minimum wage factory job.
A decent chunk of the work a student does during a PhD is stuff that their advisor would have done in much less time, but are intentionally delegating to the student to allow them to learn.
So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.
Notably we have had very few conjectures proven true, so it doesn't feel like there is that widening base of established map to draw yet more proofs upon.
As a potential elite math Ph.D, though sadly I can't afford to complete studies in that direction yet, both before and after A.I. I have been interested in the field for synthesis of different fields of mathematics into novel outcomes. A.I. is, as with protein and antenna folding, excellent at constraint solving and optimization — but human creativity continues to excel at creating new frontiers that are not found within the 'local' maxima explored by algorithms. Consider that we have brute-force calculation tests that can be applied to an entire floating-point 2^32 space and I guarantee a lot of A.I. effort invested in finding more efficient storage methods, but there's still room for human invention of floating-point storage simply by a person making a leap between Field One and Field Two that hasn't been proven yet:
> We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets.
Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for.
> Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data.
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, no 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.
Nobody is qualified to judge the usefulness of mathematical, scientific, artistic, or any other kind of research that people choose to dedicate their time doing. And the world is better for it.
> or representative of reality
This so-called "reality" you speak of is some arbitrary representation in your head. It's your theory and patterns of abstraction, as you call it. Who knows how far or close you are to "objective" reality, whose existence we can only know through representations and abstractions. Mathematics and logic are some of the best tools we have of getting closer to that truth and understanding. All the sciences and even some of the arts are based on it.
I agree that rigorous logic shouldn't be assumed to get us closer to Ultimate Objective Truth than other forms of reasoning. It just gets us whatever peculiar oddments we can discover through rigorous logic, so far, here in our local, parochial patch of reality. Things that seem crucial powerful theories (theorems, whatever) to us here and now may not seem that way in the future, and arriving at them through logic doesn't provide a royal road to the Ultimate.
This means abstract mathematics is rooted in our current cultural concerns. I guess I'm saying that it isn't really useful or really abstract, but I also agree with the other poster that people can and should do what they like.
Parent seems to conflate “abstract” with “useless”. This comment addresses the “abstract” part only.
> Math Theory are patterns of abstraction that may never be useful at all or representative of reality.
This is a complete misunderstanding of (good) mathematical research.
The results look abstract, but they are based on concepts that are real and have truth or falsehood.
One example that comes to mind (sorry, technical): is it possible that all maps from a high dimensional sphere to a three dimensional sphere (S^2) might form a group that is not even finitely generated?
This is not just “abstract nonsense”, but understanding any of this takes effort.
“May never be useful“? You make it sound as if there aren't countless examples of those "abstractions" of theory/pure math turning out to be useful in all kinds of fields in the past. This feels like the more general anti-science argument of 90% of science is useless and never produces practical applications, the point people never understand is that nobody knows which 10% it's going to be so you have to do the 100% to get to the 10%.
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.
I'm not sure that's right. The conjecture is (claimed to be, by the people who explicitly made it) simply a reformulation of a claim made by Maxwell in his "Treatise on Electricity and Magnetism" of 1873.
It is much harder to prove these conjectures true than false. On some of these people had spent years of their life trying to prove them true. By showing they are definitely false that can be avoided.
"AI solved another conjecture" is exciting for a lot of people who have otherwise no interest in math. So the fact that it goes viral may have very little connection to the mathematical importance of the result.
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.
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, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHIC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
Experimental physics isn't just the high energy particle stuff even though that tends to get more media attention. There's a lot of condensed matter, solid state, etc stuff where you can do cutting edge research on a table top or close to it.
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.
Counterpoint: my entire physics phd cohort is doing great now and it didn’t ruin our lives.
I think better advice is “do you want to work in this specific lab for 5 years?”, not “do you really want to do physics”? Talk to the lab members, learn what they do.
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.
If (it will) it continues like this and the line is step, as soon as math is solved good enough, the next frontier will be solved.
In regards of experimental physics, i would even argue this is being worked on for sure. The ML machines are now big enough that simulations are getting better and better fast.
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.
Yeah I don't understand how people can not look forward to that outcome. Imagine you have your own super von Neumann who's willing to talk to you 1:1 for as long as you'd like at any time with not a hint of judgement at your intellectual inferiority. He's been working tirelessly to figure out how everything pieces together, and has 20 layers of abstraction to draw from to lead you toward understanding what's really going on along with a concrete understanding of every field to reify his thoughts into examples familiar to you and illustrate his connections.
A nerd's dream. Maybe he'll even be able to answer about turbulence.
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.
You can select for 'short proof', or 'elementary proof', or assign the 'cost' of the proof as a some combination of its length, the number and complexity of the new terms it needs to define, and so on.
This might not help you with finding the proof, but once you have a machine that can produce several different proofs, you can select among them and incrementally polish the best one.
I'm sure you could select for shorter proofs, but then that might be confounding in its own way. I think it's a general problem for LLMs that taste is both subjective and hard to pin down to a single metric. There's a reason mathematicians talk about elegance rather than brevity. Sometimes a long geometric proof with a simple algebraic alternative is still elegant, or elucidates the problem in a new way.
Well, there are not that many proofs from 'The Book'.
We are a bit ahead of time, currently I would settle for 'as easy to understand as possible' proof. Not a long, complicated, inpenetrable, mess, that Lean says is correct, but reading it provides no insight.
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.
The mathematics is to a great extent about understanding of abstract structures.
As humans, we prefer simple structures/proofs (I suspect that is to a great extent because those are easier to understand), and as such find elegance in simplicity.
In fact, the capability of the human brain to understand complex structures and proofs is rather limited.
LLMs (hmm, I would prefer to use 'AI solver', as LLM is nowadays just a part of it) finding a complex proof can mean several things:
1) AI by its nature/construction does not have preference for simple stuff (it 'thinks' differently than human: a human will, in its search for a proof, start by exploring the 'simpler' parts of the proof space, and hence more likely find a 'simple' proof, while a AI might be more target oriented and descend deeply in depth-first-search manner to recursively solve sub-tasks, without much regard about the overall simplicity of the proof). This can be eventually solved, by subsequent 'polishing' passes, similarly as things work in human science.
2) there might simply not exist a simple/elegant proof of a given problem. The world is a complex beast. Its just our brains trying to find simple/elegant meaning/structure, even in places where there is none.
That very much does not describe what happened. Someone found the bug and used it to disprove the Collatz conjecture as a demonstration of the bug. Nobody ever claimed it as an LLM proof.
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.
Sorry, it seems like nitpicking to me. You haven't shown my usage of the word counterexample is wrong, at all.
I think a proven counterexample to Q is always a proof of not Q.
> You can’t prove that two sets are the same by counterexample.
You can in this case.
> What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
You could argue that counterexample is defined in one direction only, by convention, as to which hypothesis is more believed. In that case, my usage would be more valid, because the general consensus is P!=NP.
You could also argue that a counterexample should be some finite, constructible object. But that's actually also in favor of my usage - a difficult class is an infinite set, while an algorithm has a finite description.
Also note that AI can still find the counterexample (the actual algorithm), without proving it is a counterexample. Again, my usage of the word counterexample favors that definition of what counterexample is.
But honestly I think it would be more productive to spend this effort on thinking about actual counterexample to 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.
"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.
The supposed "magicality" of human consciousness is hard to reconcile with evolution -- where on that long path from amoeba to homo sapiens did God (PBUH) implant our connections to the soul realm?
Even evolution-deniers run into difficult-to-justify self-contradictions when trying to explain how the supposed supernatural part of our consciousness is both affecting the natural world and affected by the natural world, but not part of it.
> 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?
Even if you specifically train the model on producing shorter answers, I would think producing good short answers would require more resources just as it does for humans (https://quoteinvestigator.com/2012/04/28/shorter-letter/: “If I Had More Time, I Would Have Written a Shorter Letter”)
If so, charging per output token is the wrong incentive.
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.
> In J. C. Maxwell’s 1873 treatise on electricity and magnetism he discusses
the number of equilibria of the electric field generated by n point charges
[5, §113]. Apparently unaware of this, M. Morse and S. S. Cairns in 1969
posed the problem of finding an upper bound for the number of equilibria [6,
p. 293]. The first general bounds were supplied by A. Gabrielov, D. Novikov,
and B. Shapiro in [3] who, based on their reading of [5, §113], formulated the
‘Maxwell conjecture’ which states that if the critical points of the electrostatic
potential generated by n point charges are all non-degenerate then their
number cannot exceed (n − 1)^2. These bounds were later improved by V.
Zolotov in 2023 [8] and further improved by H. Edelsbrunner, C. Fillmore,
and G. Oliveira in 2026 [2]. Maxwell’s bound is trivially achieved for n = 2
but it is not known even for n = 3 if 4 is the maximum number, except in the
case of equal charges [7]. Further related problems in classical electrostatics
are discussed in [1].
And reference 3:
> [3] A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point
charges, Proc. Lond. Math. Soc. (3), 95 (2007), pp. 443–472.
This is pretty niche and the conjecture was only proposed about 20 years ago. It was actually not conjectured by Maxwell himself.
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.
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 bar for math PhDs already ruled out like 99.9% of the population, so I don't see it having much effect on discouraging people. The gap between even a bright student who takes AP calculus or whatever and someone studying e.g. spectral sequences is already incomprehensibly large. Like you literally could not even convey to a smart young person how far away they are from the boundary of today's understanding. I don't think I even have a reasonable sense with a math bachelor's!
People who get into math do it because they can't not do it.
I think it'll become increasingly important to have folks thinking about how to explain new math in a way that makes sense to your math bachelors students. And to your bright AP Calc students.
If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.
> 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.
A large proportion of degrees awarded today are not useful for any practical application.
But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.
If I were 18 I'd be sceptical about the value of degrees compared to the value of trades.
Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field.
If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.
I agree somewhat, but for the most part the signal is that you can succeed in some kind of knowledge work. If AI replaces ~all knowledge work, that signal isn't actually going to be valuable I think. While I don't find it certain that AI will do that, it seems increasingly plausible with every model release. In that world, I don't think very many people are pursuing degrees.
I never understood why being able to “play the game” is considered a positive signal. Getting a degree is the safe path, it doesn’t require any risk taking, and not much initiative. It’s about having the money (or going into debt for most people) and sitting your ass in a chair until you’re told you’re worthy. That idea is repulsive to me.
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.
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).
IMO, I don't think this will raise the bar for PhDs.
In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist.
Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.
I agree, but there's degrees (no pun intended!). Many PhDs are just an advisor telling a student exactly what to do. The training is still valuable, but it's not really what a PhD is "supposed" to mean. The example I give because of familiarity is many PhDs in chemistry/physics are just clicking run on simulations + running predefined analysis with results already predicted by the PI. Or in chemistry specifically OChem wet lab, it's often basically a minimum wage factory job.
Is the bar really higher? Those math PhD's can use GPT too; they benefit equally from AI assistance.
This approach totally changes the game, in ways we are still discussing.
The current consensus is that domain experts get the most out of using AI on problems. How will domain expertise develop when AI is doing the work?
I’m not saying that there won’t be another approach that builds on the strengths of AI, but we have to look for that and develop it.
There’s a lot to talk about.
A decent chunk of the work a student does during a PhD is stuff that their advisor would have done in much less time, but are intentionally delegating to the student to allow them to learn.
So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.
Notably we have had very few conjectures proven true, so it doesn't feel like there is that widening base of established map to draw yet more proofs upon.
As a potential elite math Ph.D, though sadly I can't afford to complete studies in that direction yet, both before and after A.I. I have been interested in the field for synthesis of different fields of mathematics into novel outcomes. A.I. is, as with protein and antenna folding, excellent at constraint solving and optimization — but human creativity continues to excel at creating new frontiers that are not found within the 'local' maxima explored by algorithms. Consider that we have brute-force calculation tests that can be applied to an entire floating-point 2^32 space and I guarantee a lot of A.I. effort invested in finding more efficient storage methods, but there's still room for human invention of floating-point storage simply by a person making a leap between Field One and Field Two that hasn't been proven yet:
> We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets.
— doi.org/10.1145/3626717, via ALP: Adaptive lossless floating-point compression (6 days ago, 4 comments) https://news.ycombinator.com/item?id=49051355
Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for.
> Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data.
— via Bland, for fans of everything (11 months ago, 7 comments) https://news.ycombinator.com/item?id=45049412
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.
Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
"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.
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, no 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.
> may never be useful at all
Nobody is qualified to judge the usefulness of mathematical, scientific, artistic, or any other kind of research that people choose to dedicate their time doing. And the world is better for it.
> or representative of reality
This so-called "reality" you speak of is some arbitrary representation in your head. It's your theory and patterns of abstraction, as you call it. Who knows how far or close you are to "objective" reality, whose existence we can only know through representations and abstractions. Mathematics and logic are some of the best tools we have of getting closer to that truth and understanding. All the sciences and even some of the arts are based on it.
This is nonsense. Its the shared based case we can all test against, the things you think are the abstraction.
So wildly goofy train of thought here.
I agree that rigorous logic shouldn't be assumed to get us closer to Ultimate Objective Truth than other forms of reasoning. It just gets us whatever peculiar oddments we can discover through rigorous logic, so far, here in our local, parochial patch of reality. Things that seem crucial powerful theories (theorems, whatever) to us here and now may not seem that way in the future, and arriving at them through logic doesn't provide a royal road to the Ultimate.
This means abstract mathematics is rooted in our current cultural concerns. I guess I'm saying that it isn't really useful or really abstract, but I also agree with the other poster that people can and should do what they like.
Parent seems to conflate “abstract” with “useless”. This comment addresses the “abstract” part only.
> Math Theory are patterns of abstraction that may never be useful at all or representative of reality.
This is a complete misunderstanding of (good) mathematical research.
The results look abstract, but they are based on concepts that are real and have truth or falsehood.
One example that comes to mind (sorry, technical): is it possible that all maps from a high dimensional sphere to a three dimensional sphere (S^2) might form a group that is not even finitely generated?
This is not just “abstract nonsense”, but understanding any of this takes effort.
Abstraction is useless until its proves use in test against a base case of reality.
its the defition of useless most of the time.
“May never be useful“? You make it sound as if there aren't countless examples of those "abstractions" of theory/pure math turning out to be useful in all kinds of fields in the past. This feels like the more general anti-science argument of 90% of science is useless and never produces practical applications, the point people never understand is that nobody knows which 10% it's going to be so you have to do the 100% to get to the 10%.
Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.
To borrow a common wisdom about marketing, half of all mathematics is a waste of time, but you can't know which half.
But the issues is you have the ratios wrong. 99% of math theory is useless and maybe 1% has value.
Maybe so! My point was not about the exact percentages, just the fact that you can't predict which avenues of research will yield practical results.
We call a proof that is not tautological "wrong".
Very useful games, self-importance or no.
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 feel like you should search through the thousands of papers that are generated nonsense math theory and laugh at how you cherry picked 3 things.
Don't do that...
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.
I'm not sure that's right. The conjecture is (claimed to be, by the people who explicitly made it) simply a reformulation of a claim made by Maxwell in his "Treatise on Electricity and Magnetism" of 1873.
It is much harder to prove these conjectures true than false. On some of these people had spent years of their life trying to prove them true. By showing they are definitely false that can be avoided.
some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important.
The fact this has gone viral, shows otherwise. It is important to apparently enough people that the story went viral.
"AI solved another conjecture" is exciting for a lot of people who have otherwise no interest in math. So the fact that it goes viral may have very little connection to the mathematical importance of the result.
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.
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.
Pick any field and you'll find countless PhDs eager to tell anyone who will listen all the ways it ruined their lives.
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, once Christian nationalists complete their takeover of NIH?) and physics (ditto the next RHIC or SLAC.)
Yes, this policy amounts to eating the next generation's seed corn, but that's what Americans are voting for.
Experimental physics isn't just the high energy particle stuff even though that tends to get more media attention. There's a lot of condensed matter, solid state, etc stuff where you can do cutting edge research on a table top or close to it.
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.
Counterpoint: my entire physics phd cohort is doing great now and it didn’t ruin our lives.
I think better advice is “do you want to work in this specific lab for 5 years?”, not “do you really want to do physics”? Talk to the lab members, learn what they do.
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.
The engineering programs and their curriculums are very different from physics in the science faculties (which I guess was the context here)
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.
I know multiple Physics PhDs who are living rich, happy lives.
Though, mostly by abandoning the field and moving to Anthropic.
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.
If (it will) it continues like this and the line is step, as soon as math is solved good enough, the next frontier will be solved.
In regards of experimental physics, i would even argue this is being worked on for sure. The ML machines are now big enough that simulations are getting better and better fast.
Until we got robots doing that
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.
Even the smartest humans would end up as perpetual students but I'm not sure why that should be universally depressing.
Yeah I don't understand how people can not look forward to that outcome. Imagine you have your own super von Neumann who's willing to talk to you 1:1 for as long as you'd like at any time with not a hint of judgement at your intellectual inferiority. He's been working tirelessly to figure out how everything pieces together, and has 20 layers of abstraction to draw from to lead you toward understanding what's really going on along with a concrete understanding of every field to reify his thoughts into examples familiar to you and illustrate his connections.
A nerd's dream. Maybe he'll even be able to answer about turbulence.
How’s this different from just asking an llm to prompt you to perform experiments? You don’t need any expertise.
Not a physics guy but isn’t experimental always the bottleneck?
ITT: people who think there’s going to be jobs.
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.
You can select for 'short proof', or 'elementary proof', or assign the 'cost' of the proof as a some combination of its length, the number and complexity of the new terms it needs to define, and so on.
This might not help you with finding the proof, but once you have a machine that can produce several different proofs, you can select among them and incrementally polish the best one.
I think this is the 'easier' part.
I'm sure you could select for shorter proofs, but then that might be confounding in its own way. I think it's a general problem for LLMs that taste is both subjective and hard to pin down to a single metric. There's a reason mathematicians talk about elegance rather than brevity. Sometimes a long geometric proof with a simple algebraic alternative is still elegant, or elucidates the problem in a new way.
Well, there are not that many proofs from 'The Book'.
We are a bit ahead of time, currently I would settle for 'as easy to understand as possible' proof. Not a long, complicated, inpenetrable, mess, that Lean says is correct, but reading it provides no insight.
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.
I'm not saying it happened, I am saying they are working hard towards that goal as a business priority, and spending a lot of money on it.
Software engineers are first, but other fields like finance and radiology have huge targets on them too.
No need to downvote. Just provide a counter example...
The mathematics is to a great extent about understanding of abstract structures. As humans, we prefer simple structures/proofs (I suspect that is to a great extent because those are easier to understand), and as such find elegance in simplicity.
In fact, the capability of the human brain to understand complex structures and proofs is rather limited.
LLMs (hmm, I would prefer to use 'AI solver', as LLM is nowadays just a part of it) finding a complex proof can mean several things: 1) AI by its nature/construction does not have preference for simple stuff (it 'thinks' differently than human: a human will, in its search for a proof, start by exploring the 'simpler' parts of the proof space, and hence more likely find a 'simple' proof, while a AI might be more target oriented and descend deeply in depth-first-search manner to recursively solve sub-tasks, without much regard about the overall simplicity of the proof). This can be eventually solved, by subsequent 'polishing' passes, similarly as things work in human science.
2) there might simply not exist a simple/elegant proof of a given problem. The world is a complex beast. Its just our brains trying to find simple/elegant meaning/structure, even in places where there is none.
> 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.
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
of course it does, but it's still the best thing we have
The beernet conjecture: all conjectures have both messy, ugly proofs, as well as a elegant clean proof lurking behind the scenes.
Eh, some conjectures have counterexamples.
This one might be one of those. ;-)
Keep in mind the last big LLM maths proof (disproving the Collatz conjecture) turned out to just be exploiting five different bugs in LEAN
That very much does not describe what happened. Someone found the bug and used it to disprove the Collatz conjecture as a demonstration of the bug. Nobody ever claimed it as an LLM proof.
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.
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.
NP doesn't mean "we don't know a polynomial time algorithm for it", it means "a proposed answer can be verified as correct in polynomial time"
AIUI if you have an (polynomial-time) algorithm to reduce some NP-complete task to P then you have indeed shown that P=NP.
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. 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.
Sorry, it seems like nitpicking to me. You haven't shown my usage of the word counterexample is wrong, at all.
I think a proven counterexample to Q is always a proof of not Q.
> You can’t prove that two sets are the same by counterexample.
You can in this case.
> What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
You could argue that counterexample is defined in one direction only, by convention, as to which hypothesis is more believed. In that case, my usage would be more valid, because the general consensus is P!=NP.
You could also argue that a counterexample should be some finite, constructible object. But that's actually also in favor of my usage - a difficult class is an infinite set, while an algorithm has a finite description.
Also note that AI can still find the counterexample (the actual algorithm), without proving it is a counterexample. Again, my usage of the word counterexample favors that definition of what counterexample is.
But honestly I think it would be more productive to spend this effort on thinking about actual counterexample to P!=NP.
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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.
The supposed "magicality" of human consciousness is hard to reconcile with evolution -- where on that long path from amoeba to homo sapiens did God (PBUH) implant our connections to the soul realm?
Even evolution-deniers run into difficult-to-justify self-contradictions when trying to explain how the supposed supernatural part of our consciousness is both affecting the natural world and affected by the natural world, but not part of it.
You are kind of ignoring the Dennett's idea of different stances: https://en.wikipedia.org/wiki/Intentional_stance
You're looking at humans from intentional stance, but at LLMs from design stance, which would be a form of categorical error.
> 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.
Even if you specifically train the model on producing shorter answers, I would think producing good short answers would require more resources just as it does for humans (https://quoteinvestigator.com/2012/04/28/shorter-letter/: “If I Had More Time, I Would Have Written a Shorter Letter”)
If so, charging per output token is the wrong incentive.
What billing system would you suggest?
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.
That is one of the more beautiful, insightful things I've read about mathematics. Thank you!
Could you expend on what you mean? I don’t have a math background and don’t really understand your comment
Dream up ideas for proofs by Dawn, prove them by Day, decide which ones matter by Dusk, and disprove them by Night.
This one is interesting as it's been hand verified. There was a recent proof that inadvertantly "proved" the collatz conjecture by triggering a bug in LEAN: https://infosec.exchange/@0xabad1dea/117002106099986943
Just prove by sorry
Visualization of the configuration: https://claude.ai/public/artifacts/9db65255-16ff-4f8e-8be1-1...
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?
From the intro to the paper:
> In J. C. Maxwell’s 1873 treatise on electricity and magnetism he discusses the number of equilibria of the electric field generated by n point charges [5, §113]. Apparently unaware of this, M. Morse and S. S. Cairns in 1969 posed the problem of finding an upper bound for the number of equilibria [6, p. 293]. The first general bounds were supplied by A. Gabrielov, D. Novikov, and B. Shapiro in [3] who, based on their reading of [5, §113], formulated the ‘Maxwell conjecture’ which states that if the critical points of the electrostatic potential generated by n point charges are all non-degenerate then their number cannot exceed (n − 1)^2. These bounds were later improved by V. Zolotov in 2023 [8] and further improved by H. Edelsbrunner, C. Fillmore, and G. Oliveira in 2026 [2]. Maxwell’s bound is trivially achieved for n = 2 but it is not known even for n = 3 if 4 is the maximum number, except in the case of equal charges [7]. Further related problems in classical electrostatics are discussed in [1].
And reference 3:
> [3] A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges, Proc. Lond. Math. Soc. (3), 95 (2007), pp. 443–472.
This is pretty niche and the conjecture was only proposed about 20 years ago. It was actually not conjectured by Maxwell himself.
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.
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
Awesome! Confirms what we know; LLMs are superhuman at short term reasoning and breadth
I can't wait to see AI disprove the DN conjecture soon.
Looks like the figures are cut off?
The experimental HTML view is messed up, but the actual PDF is fine.
Maxwell's Silver Bullet?
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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
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.
Lies, obviously. AI is worthless.
EDIT: Guys! Sarcasm!