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The Maxwell Conjecture Is False (GPT 5.6 Sol)

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Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#81

Post by chermi »

ashleyn wrote: post #43
mellosouls wrote: post #29

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…

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 i…

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).

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#82

Post by kmeisthax »

beernet wrote: post #5

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…

Keep in mind the last big LLM maths proof (disproving the Collatz conjecture) turned out to just be exploiting five different bugs in LEAN

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#83

Post by wongarsu »

tsunamifury wrote: post #60
moomin wrote: post #54

Earlier quoted context omitted.

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 simultaneousl…

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

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#84

Post by throw-the-towel »

tsunamifury wrote: post #60
moomin wrote: post #54

Earlier quoted context omitted.

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 simultaneousl…

To borrow a common wisdom about marketing, half of all mathematics is a waste of time, but you can't know which half.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#85

Post by efficax »

ComputerPerson wrote: post #47
d_burfoot wrote: post #25

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…

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 st…

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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#86

Post by hgoel »

ashleyn wrote: post #43
mellosouls wrote: post #29

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…

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 i…

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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#87

Post by CamperBob2 »

ModernMech wrote: post #69
ComputerPerson wrote: post #47

Earlier quoted context omitted.

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 st…

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.

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#88

Post by don_esteban »

StilesCrisis wrote: post #70
storus wrote: post #59

Earlier quoted context omitted.

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.

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

Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)

#90

Post by astro1234 »

mellosouls wrote: post #29

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…

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.

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