Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

cdn.openai.com

Post reply on HN
Print view View original

351–360 of 467 posts

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

#351

Post by emil-lp »

kertoip_1 wrote: post #347
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 job of a programmer isn't to write code, but to automate things. Code itself doesn't have any value unless it solves some real problem not related to coding. So if the work of a programmer can be automated then this means that any work can be automated. So no, it's not about software engineering only.

> if the work of a programmer can be automated then this means that any work can be automated

This is a very typical programmer thing to say.

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

#352

Post by samus »

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…

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 fields it can take decades before advances become practically relevant. Restricting teaching to topics that have immediate practical relevance would therefore do students a huge disservice and prevent them from approaching the knowledge frontier of the field.

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

#353

Post by samus »

torginus wrote: post #79
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 thought linear algebra was pretty much the poster child of applied mathematics - the entire field was invented to represent computations in a regularized form to feed into computers. Well not really, but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.

Many important advances in linear algebra happened in WWI to solve optimization problems for logistical and industrial planning. A lot of these applications boil down to high-dimensional systems of linear equations, which back then were solved by hand. Efficient algorithms translate to reduced labor costs.

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

#354

Post by gf000 »

amluto wrote: post #338
noname120 wrote: post #218

Earlier quoted context omitted.

Your comment is not substantiated at all.

No, the comment is right. The prompt had GPT-5.6 reviewing the proof, and the result, unsurprisingly, survives review by GPT-5.6.

Given a new context, why couldn't the same model have a decent shot at reviewing some results? It's not like they identify whether this output is from them and then go "yeah correct", that's not how they work.

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

#355

Post by samus »

FabHK wrote: post #214
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 my Linear Algebra prof used to say, basically everything is applied Linear Algebra.

Even when this really isn't the case, the math is preferably reduced to a linear problem whenever possible since dealing with the original complexity is usually so much more painful.

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

#356

Post by imdsm »

emil-lp wrote: post #351
kertoip_1 wrote: post #347

Earlier quoted context omitted.

The job of a programmer isn't to write code, but to automate things. Code itself doesn't have any value unless it solves some real problem not related to coding. So if the work of a programmer can be automated then this means that any work can be automated. So no, it's not about software engineering only.

> if the work of a programmer can be automated then this means that any work can be automated This is a very typical programmer thing to say.

i think most things can be automated, but not everything

i think AI will just replace jobs with new jobs, and humans will continue with open-ended goal setting, and probably jobs where being human is necessary: responsibility and authenticity, so judges, politics, leaders, chefs, etc.

so in a way, the above commenter was correct, anything a programmer can automate, AI can do

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

#357

Post by samus »

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…

Teaching only the practical side risks not teaching the subject with the appropriate theoretical depth and the ability to generalize it to other applications. Courses for purely applied fields utilize calculus to solve the current problem and then move on without teaching the finer points. Basing a calculus course on physics alone might be preferable in high school, but would be of disservice to students in university.

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

#358

Post by krona »

mNovak wrote: post #253

Unrelated to the accomplishment or proof itself, but it's interesting how much of the prompt, even in this latest-and-greatest model, is spent essentially telling the model to actually solve the problem. Things like "Reject status reports, vague optimism, and claims that an unproved global compatibility statement is 'routine'." Also a lot prompt spent feeding it strategies, which feel like they should/will eventually…

LLMs don't have the executive control to dynamically manage and compose mental sub-routines, when it has never seen those sub-routines in its training corpus. That requires fluid intelligence. ARAOC benchmark measures this specifically.

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

#359

Post by amluto »

Sniffnoy wrote: post #350
amluto wrote: post #340

Earlier quoted context omitted.

I was not a fan of the writing style of the proof. There seem to be some irrelevant details: Is the mention of 8-flow at all relevant? I, at least, found the definition of L on the first line of the proof of Lemma 2.2 to be needlessly inscrutable, and my thesis advisor would have likely stopped reading there and told me to fix it. Maybe someone should ask the model to make a more clearly written and thus easy to veri…

I was confused at first when you asked if the 8-flow is relevant, when like, the 8-flow is a key input that the cycle double cover is built out of. Then I realized, oh, I guess technically they're not using the 8-flow, they're using the Z_2^3-flow. But like. The existence of an 8-flow and the existence of a Z_2^3-flow are equivalent, and I gather most graph theorists are going to talk about it in terms of the existen…

My issues with the definition of L are mostly about the order in which things are written.

L(t, epsilon)_e breaks down the range of L onto its component values indexed by edge, but this only really makes sense when you know that t and epsilon are. They are sort of defined in the middle of a sentence in the proof of 2.1, which IMO is asking a lot of the reader, and this sort of sloppiness is a way that errors can hide in a proof. (Not that I see an error here. But a formalization in Lean or whatever would not get away with this.)

And, in the same definition of L, for some reason the e=uv part comes at the end only after u and v are used.

What would be wrong with stating, in the definition, what sorts of objects t and epsilon are and with omitting e entirely in favor of just calling the edge uv everywhere?

Post reply on HN