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GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

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Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#211

Post by BeetleB »

s3p wrote: post #149
BeetleB wrote: post #58

Earlier quoted context omitted.

> I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field. > I was promptly pilloried, and shunned. Heh. In my day I may have participated in the pillorying. I do think that there is value/merit in professors mentioning real world applications, where they exist . What they'…

Hear me out on this one: For a lot of math departments, that is exactly why they teach this. Education is rooted in application. We have entire careers that depend on certain aspects of mathematics, so most companies gatekeep that career by a degree. The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the appli…

> For a lot of math departments, that is exactly why they teach this.

Depends on the course. That's why some departments have separate calculus courses for math majors - because otherwise the whole class will be full of non-math majors (engineers, etc) and focusing on their needs does a disservice to the students in their own department.

> The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications.

If I'm a CS major, and the degree is requiring a class outside of the CS department, you shouldn't expect the professor of the class to know why the CS department is requiring it. It's on the CS department and its faculty to explain it.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#212

Post by phatfish »

npinsker wrote: post #85
WhitneyLand wrote: post #75

If all checks out this is a huge milestone. AI has now solved one of the most famous open problems in graph theory, using an off the shelf model, in one hour. It might be a better mathematician than most humans at this point. Kind of like when chess software started beating everyone except grandmasters. What’s left? Proposing and building out entirely new theories and frameworks? Then better than any human? Then alie…

You say those things like they're a short step away, but that might not be how it works out. For example, AI has made zero progress in the last few years in surpassing professionals at art or writing. Its prompt-following skill is much better, and sure, it can render hands and text now, but its artistic sensibility is completely stagnant.

Art is so entwined with the human condition that generative AI which conjures output from a statistical pass over background noise is bound to be stagnant.

Obviously it can impersonate art, but where creativity and the human story matter artists need not worry.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#213

Post by sigbottle »

roncesvalles wrote: post #162
hyperpape wrote: post #154

Earlier quoted context omitted.

> rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. I disagree. Mathematicians care about the utility of a result. It is just that they regard mathematical understanding as a valid type of utility, and that can be arbitrarily far removed from practical utility. But a proof that doesn't help anyone unde…

It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason). Mathematics is largely just smart people working on pointless puzzles, and only…

There was no useful math after 1964? What? Or do you not count the entire field of computer science as math (arguably FFT belongs to computer science too)?

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#214

Post by FabHK »

andai wrote: post #38
zerobees wrote: post #22

This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal. We attach basically zero value to writing a new program that hasn't existed before, or a piece of text that hasn't existed before. It's boring, or even a net negative, unless you can show that the result benefits the world in some way. We'd find it weird if OpenAI put out a relea…

>mathematics is basically the only scientific discipline that rejected any notion of utility I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator). I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fie…

As my Linear Algebra prof used to say, basically everything is applied Linear Algebra.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#215

Post by stavros »

noname120 wrote: post #184

ChatGPT 5.6 Sol Pro believes that the proof is sound. Usually it’s very good at determining if proofs are correct and their mistakes (a friend of mine is a top mathematician researcher and confirmed): https://chatgpt.com/share/6a515ead-b464-83ed-b85c-c8674f56ea... Personally this gives me additional confidence that this is the real deal.

Of course it believes the proof is sound, it wrote it. If you want to check an LLM's output, you should use a different LLM.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#216

Post by munksbeer »

StefanBatory wrote: post #147
WhitneyLand wrote: post #75

If all checks out this is a huge milestone. AI has now solved one of the most famous open problems in graph theory, using an off the shelf model, in one hour. It might be a better mathematician than most humans at this point. Kind of like when chess software started beating everyone except grandmasters. What’s left? Proposing and building out entirely new theories and frameworks? Then better than any human? Then alie…

It's hard for me not to think what's the point. I am a very average, even below average person in times of intelligence. What is even my value or reason to be if I know anything I can do, LLMs can do better? What is even my value both on job market and as a human?

Sorry to be nihilist, but you never had any objective value if you're thinking in these terms.

As far as we know, the universe "just is". There is no universal objective value of human beings, at all, any one of us.

You have to make or find your own value in the universe. I try not to think too hard about the nihilist side and try to appreciate that for some unfathomable reason, I seem to have what I call consciousness - the ability to observe the present and have it superimposed on the past, and what may be the future, leading me to "experience" things. I don't understand it, no-one does (some people suffering from the Dunning-Kruger effect think they do, but they don't), and yet, here we are.

So it doesn't matter to me if machines perform better than I do, because already lots of other people do. Just try to find your own joy or meaning, somehow.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#217

Post by IsTom »

esafak wrote: post #93
WhitneyLand wrote: post #75

If all checks out this is a huge milestone. AI has now solved one of the most famous open problems in graph theory, using an off the shelf model, in one hour. It might be a better mathematician than most humans at this point. Kind of like when chess software started beating everyone except grandmasters. What’s left? Proposing and building out entirely new theories and frameworks? Then better than any human? Then alie…

> What's left? I think humans will be left to propose new conjectures while machines fill out the proofs. I don't know if there are enough interesting conjectures to go round to build new careers, though.

To be able to propose interesting conjectures you need to walk the walk of trying to prove things yourself. That's not great for future generations.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#218

Post by noname120 »

stavros wrote: post #215
noname120 wrote: post #184

ChatGPT 5.6 Sol Pro believes that the proof is sound. Usually it’s very good at determining if proofs are correct and their mistakes (a friend of mine is a top mathematician researcher and confirmed): https://chatgpt.com/share/6a515ead-b464-83ed-b85c-c8674f56ea... Personally this gives me additional confidence that this is the real deal.

Of course it believes the proof is sound, it wrote it. If you want to check an LLM's output, you should use a different LLM.

Your comment is not substantiated at all.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#219

Post by blovescoffee »

CSMastermind wrote: post #140
not-a-llm wrote: post #25

Earlier quoted context omitted.

pretty sure already millions of dollars (in inference costs) were already thrown at the Riehmann hypothesis as the models get stronger, larger amounts will be thrown at it imagine paying "just $1 bil" to go down in history as the company who's model solved the hardest/most famous open problem in mathematics. imagine the worldwide press headlines. as they say, the Riehmann Hypothesis is the hardest way to earn a milli…

I mean if there's something I'd bet against being solved by LLMs in my lifetime it's that one. We truly do not have line of sight into what a proof would even look like.

Why would you bet against it being solved by LLMs? Isn't this very post proof that LLMs in an agentic harness are capable of doing real math? If you just keep cranking away at the tokens I don't see a principled argument against that leading to more solutions to unsolved math, even the hardest problems.

Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

#220

Post by esafak »

IsTom wrote: post #217
esafak wrote: post #93

Earlier quoted context omitted.

> What's left? I think humans will be left to propose new conjectures while machines fill out the proofs. I don't know if there are enough interesting conjectures to go round to build new careers, though.

To be able to propose interesting conjectures you need to walk the walk of trying to prove things yourself. That's not great for future generations.

It is the same with anything technical; you need taste and wisdom, which are borne of experience. I think society will want to subsidize this learning lest it deskills en masse.
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