Earlier quoted context omitted.
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 Maxwell Conjecture Is False (GPT 5.6 Sol)
91–100 of 149 posts
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#92I can't wait to see AI disprove the DN conjecture soon.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#93Earlier quoted context omitted.
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
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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#94Earlier 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…
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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#95Earlier quoted context omitted.
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 c…
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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#96Maxwell's Silver Bullet?
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#97On 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 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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#98Earlier quoted context omitted.
> 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.
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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#99Tip 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…
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.
Re: The Maxwell Conjecture Is False (GPT 5.6 Sol)
#100Earlier quoted context omitted.
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 dis prove 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.