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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)

#111

Post by ndriscoll »

fc417fc802 wrote: post #95
alasano wrote: post #48

Earlier quoted context omitted.

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.

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

#112

Post by jmalicki »

tcp_handshaker wrote: post #28
jmalicki wrote: post #26

Earlier quoted context omitted.

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....

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.

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

#113

Post by spullara »

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…

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.

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

#114

Post by StableAlkyne »

chermi wrote: post #81
ashleyn wrote: post #43

Earlier quoted context omitted.

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).

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.

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

#115

Post by dualvariable »

logicallee wrote: post #63

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.

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

#116

Post by paulpauper »

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…

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.

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

#117

Post by ian-g »

ndriscoll wrote: post #105
ashleyn wrote: post #43

Earlier quoted context omitted.

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…

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'…

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.

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

#118

Post by CWuestefeld »

throwaway0123_5 wrote: post #68
ashleyn wrote: post #43

Earlier quoted context omitted.

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…

> 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'…

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.

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

#119

Post by fn-mote »

julianeon wrote: post #72
ashleyn wrote: post #43

Earlier quoted context omitted.

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…

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.

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

#120

Post by fn-mote »

tsunamifury wrote: post #93
wongarsu wrote: post #83

Earlier quoted context omitted.

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.

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.

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