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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]

#431

Post by andai »

throwup238 wrote: post #170
s3p wrote: post #149

Earlier quoted context omitted.

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…

I think for many people (myself included) understanding mathematics is rooted in application because it helps bridge the divide between intuition and rote memorization. Without the application, IMO instructors are doing a disservice to their students and pedagogy of mathematics itself. They’re intentionally ignoring a significant fraction of the class, unless they’re teaching some esoteric grad level pure math.

Yeah this was my position. I don't mean "Sally has six apples" necessarily, but my brain needs some concrete handles to grab onto so it can actually manipulate this stuff intuitively.

See also: Using spaced repetition to see through a piece of mathematics

https://news.ycombinator.com/item?id=18895613

Where the author describes his mind eventually turning the mathematical objects (symbols) into some kind of mental objects he was able to actually manipulate "directly". As a result of this facility, his mind spontaneously produced many new insights into the matter.

That level of fluency obviously requires a great deal of work, but in my experience it can be approximated and accelerated with analogies. (e.g. see Feynman's teaching.)

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

#432

Post by andai »

madaxe_again wrote: post #61
andai wrote: post #38

Earlier quoted context omitted.

>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…

I’m a physicist, so I’m biased, but my experience of pure maths was about the same. We had to do it, but at no point was any utility actually demonstrated - that was left to the physics professors. It was all just “look at this thing I can do with these symbols” without any actual tangible relationship to anything. Then again, I remember how we were taught calculus at high school - we were taught how to mechanistical…

Well, in at least some cases, they do understand what they're doing, on a deep intuitive level!

Using spaced repetition to see through a piece of mathematics

https://news.ycombinator.com/item?id=18895613

The title is a slight misnomer in my opinion. The author did make heavy use of spaced repetition of course, but deemed it neither necessary nor sufficient for the result he described.

The actual active ingredient is curiosity to the degree of obsession. Actually wanting to understand what's going on.

You can do that with pure math just as much as with applied.

But it may be harder for many people for such a desire to arise if there is no obvious connection to real-world matters. (I certainly had such thoughts, sometimes! What is this for? What's the point?)

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

#433

Post by tripzilch »

wyager wrote: post #333
plaidfuji wrote: post #230

It seems like a solid set of criteria for how easily a task can be automated by AI agents is: - extent to which correctness of solution be easily specified and checked - extent to which new potential solutions can be implemented as text - extent to which prior art exists online This basically maps to software engineering and math. I think a fair bit of AI hype comes from the fact that the very architects of AI are th…

> the very architects of AI are the people whose jobs are most easily automated by AI Think very hard about what this implies for the future pace of AI R&D

It's going to turn into slop, exponentially fast, any moment now.

No but seriously it could imply grinding to a halt, turning into a laughing stock of slop, or, imho much less likely given the current state of AI (barring huge technological breakthrough and a couple of other assumptions) you were probably implying an intelligence explosion.

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

#434

Post by andai »

madaxe_again wrote: post #61
andai wrote: post #38

Earlier quoted context omitted.

>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…

I’m a physicist, so I’m biased, but my experience of pure maths was about the same. We had to do it, but at no point was any utility actually demonstrated - that was left to the physics professors. It was all just “look at this thing I can do with these symbols” without any actual tangible relationship to anything. Then again, I remember how we were taught calculus at high school - we were taught how to mechanistical…

Re: obvious intuitive explanations are ignored.

I remember squinting at the definitions in my calculus textbook for a full 10 minutes before realizing... oh my God, if this had been a 5 second animation, I would have understood it instantly.

I remember being rather upset! "If it's so obvious and intuitive, why didn't they just say so?"

I want to say, "well, unfortunately you cannot put an animation in a textbook." But that's actually false!

Even as a child I made flip book animations. And there was plenty of room here in the margins for several of those :)

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

#435

Post by andai »

stymaar wrote: post #65
andai wrote: post #38

Earlier quoted context omitted.

>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 a friend of mine who also happens to be a math professor once said: mathematicians are like sculptors who marvel about the beauty of their creation, and are kind of disgusted when a physicist comes nearby and says “that's a cool hammer you got there, may I borrow it?”.

John: "I've just gotten word. They found an application for my theorem. They say they've used it to cure cancer."

Frank: "I'm so sorry."

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

#436

Post by BeetleB »

andai wrote: post #430
samus wrote: post #352

Earlier quoted context omitted.

The ultimate goal of university education is to raise researchers , who are the people that investigate the knowledge frontier of their field and then advancing it. To do that they have to understand a large part of the existing field so they can communicate with their peers, avoid investigating things that have already been throroughly explored, and draw useful connections to other fields. Even in more applied field…

> The ultimate goal of university education is to raise researchers The head of my CS department was quoted as saying, "our job is to produce researchers!" This was in the context of a conversation about why the program wasn't better suited for industry. It turned out that in our department 7% of students ended up going into research. So he was openly declaring his intention to disregard the needs of 93% of his stude…

CS is special in that it is about the science, and not the application. They'll teach you mostly theory. That's why some universities/countries have a separate degree called "SW Engineering".

Other engineering departments are not as fussy about the distinction.

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

#438

Post by grantcas »

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…

[dead]

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

#439

Post by hoppp »

deebosong wrote: post #428
hoppp wrote: post #362

Earlier quoted context omitted.

It is a little bit suspicious that the big replacement didn't happen, yet. If it was so valuable then AI companies would directly create valuable things, not just sell API tokens. A proof like this is a good effort at trying to provide value directly, but still far away from real use. When OpenAI cures cancer I'll accept the fate, but till then I'm still seeing a lot of gambling going on.

Reminds me of crypto. A lotta tokens. A lotta gambling. Very little supplanting of crypto from fiat. Very few use-cases that revolutionize society, etc.

LLMs are genuinely useful but the cult surrounding it makes it super sketchy.

Tech needs no anti-humanist and low empathy cults. It's a disgrace to the technology.

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

#440

Post by fnordpiglet »

Garlef wrote: post #196
qarl2 wrote: post #193

Earlier quoted context omitted.

> However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed. Why is that a "however"? My reading is that it found a genuinely new solution that is both elegant and previously missed. Seems like exactly the kind of result a human mathematician would aspire to.

> a human mathematician would aspire to Some do. But there's also the notion that a clever trick is a bad explanation.

But a proof isn’t an explanation it’s a proof. Proof by assuming the opposite is true and demonstrating a contradiction is very indirect and not at all directly explanatory yet it’s a proof non the less. The goal of proofs is to demonstrate something to be provably true, not expository knowledge gathering.

In fact most mathematicians (myself included!) think the more clever the trick the better the proof! The trick itself being clever is interesting because it often yields a new way of tackling or thinking about your own proofs. A bland explanatory proof that elicits some conceptually “why” is only preferable if it has a reason for doing so - does understanding why yield a new avenue of research? Often then the “why” is quite a clever trick too.

I think it’s a bit the opposite of programming. There you want your solutions to demonstrably not be clever and the code be its own documentation. It’s a different discipline.

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