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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)

#121

Post by dogcomplex »

Syzygies wrote: post #34

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 statem…

Dream up ideas for proofs by Dawn, prove them by Day, decide which ones matter by Dusk, and disprove them by Night.

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

#122

Post by scarmig »

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…

I know multiple Physics PhDs who are living rich, happy lives.

Though, mostly by abandoning the field and moving to Anthropic.

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

#123

Post by scarmig »

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…

The beernet conjecture: all conjectures have both messy, ugly proofs, as well as a elegant clean proof lurking behind the scenes.

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

#124

Post by xboxnolifes »

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…

[dead]

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

#125

Post by scheme271 »

CamperBob2 wrote: post #87
ModernMech wrote: post #69

Earlier quoted context omitted.

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 e…

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.

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

#126

Post by lioeters »

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.

> 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.

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

#127

Post by ainch »

don_esteban wrote: post #98
ainch wrote: post #32

Earlier quoted context omitted.

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.

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

#128

Post by don_esteban »

ainch wrote: post #127
don_esteban wrote: post #98

Earlier quoted context omitted.

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.

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

#129

Post by don_esteban »

scarmig wrote: post #123
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…

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. ;-)

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

#130

Post by chermi »

StableAlkyne wrote: post #114
chermi wrote: post #81

Earlier quoted context omitted.

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.
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