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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]

#321

Post by Davidzheng »

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…

Your comment is somewhat emotionally charged, but I choose to respond to the overall point as a mathematician. I think it could be true that utility is correlated with difficulty but it is certainly not defined by it.

In pure mathematics, we reason about a world of abstract objects which are considered interesting ab initio. It may be because they arise directly or often from extremely basic operations, they are connected to many other interesting objects, or that they present special and surprising properties. The importance is basically, there is some surprising, interesting, phenomena which occurs in our world which we don't understand and which we seek to understand. Like science but in the non-physical world.

I think if you create a simple to describe system/construction with a property which is extremely difficult to prove. You are creating an object in our world which is basic but have properties which we don't understand (because we can't prove this property). So indeed I believe it would be an interesting thing to study and be of value. I don't see any problems/issues with this. I don't think the only valuable pursuit of humans is to improve the welfare of other humans. I think understanding the world is also valuable.

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

#322

Post by ralusek »

Retr0id wrote: post #317
StefanBatory wrote: post #147

Earlier quoted context omitted.

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?

AI will never be better than me at appreciating a good sunset.

Claude performs 50 microappreciations per second

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

#323

Post by ankit219 »

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 are good at greedy depth first search[1]. so prompt / human needs to tell the model the breadth parts to take.

and human writing the prompt here did a lot more than that. asking to include parallel-edge 2-cycles, admitting disconnected graphs, specifying the emptyset cover, multiset counting, it all likely came from previous fake proofs.

[1]: see: ankitmaloo.com/fable - its the way most llms are trained, and is also natural owing to autoregressive nature.

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

#324

Post by nullbio »

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…

This is the key to training better and better models.

You collect all of the direction prompts and anti-direction prompts, and you finetune the model to behave that way for the particular domain, under those particular conditions, as a default.

This is also why I don't believe for a second that the AI labs aren't training on our data, even when we opt out. This sort of data collection is a requirement for improving the models.

It's also why Anthropic purposefully gives you a dumb model or a corner cutting model to goad you into guiding it into the correct answers, so that you do the fine-tuning for them. OpenAI at least, does not do that. Which is why ChatGPT is far less infuriating to work with.

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

#325

Post by stingraycharles »

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…

And all this is while they themselves proclaim that GPT 5.6 is much better at understanding intent and doesn’t need this type of guidance. Yet they themselves still do it, which I consider a more powerful signal.

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

#326

Post by honeycrispy »

adamtaylor_13 wrote: post #241
munksbeer wrote: post #216

Earlier quoted context omitted.

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 consciousne…

> As far as we know, the universe "just is". I don't know this. In fact, billions of people around the world don't know this. In fact, all evidence points to the contrary. You have objective value being made in the image of a personal God. Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth.

This is one reason why I can't stand hn. Edgelord nihilistic comments that say nothing matters and take everything for granted get up-voted while credible and rational statements about God get down-voted.

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

#327

Post by tock »

mapontosevenths wrote: post #313
StefanBatory wrote: post #147

Earlier quoted context omitted.

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?

This is a great question. Thanks for asking it. It really got people talking. My $0.02: Your value as a person never had anything to do with your value to the economy. It's time to relearn that fact. Check out some of Tom Hodgkins books for more. Or maybe anxietyculture.com.

Yeah but what is a persons value economically anymore? What skillsets will enable us to continue to make a decent living till we are old?

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

#328

Post by Madmallard »

honeycrispy wrote: post #326
adamtaylor_13 wrote: post #241

Earlier quoted context omitted.

> As far as we know, the universe "just is". I don't know this. In fact, billions of people around the world don't know this. In fact, all evidence points to the contrary. You have objective value being made in the image of a personal God. Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth.

This is one reason why I can't stand hn. Edgelord nihilistic comments that say nothing matters and take everything for granted get up-voted while credible and rational statements about God get down-voted.

What? What is rational about faith-based comments?

Literally nothing is rational about that. It's also just flat out off-topic.

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

#329

Post by Madmallard »

adamtaylor_13 wrote: post #241
munksbeer wrote: post #216

Earlier quoted context omitted.

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 consciousne…

> As far as we know, the universe "just is". I don't know this. In fact, billions of people around the world don't know this. In fact, all evidence points to the contrary. You have objective value being made in the image of a personal God. Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth.

"In fact, all evidence points to the contrary."

Um. No? I'd like to see how you conjured up that one. Because evidence does indeed point exactly to things just being as they are.

"You have objective value being made in the image of a personal God."

Conjecture. Unsubstantiated.

"Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth."

No? Sounds like you're projecting your own belief here. "If I didn't believe in God, then what purpose is there to anything?" It's very small of a belief.

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

#330

Post by virgildotcodes »

richardbarosky wrote: post #237
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…

Interesting take! I feel like 2 of them are maybe overstated: > - extent to which correctness of solution be easily specified and checked I don't think most software is like solving a math problem or series of math problems. Algorithmic problems are very narrow and might be more like this though, where an oracle that verifies answers as either correct or incorrect exists beforehand. The correctness function of most s…

> how much users want to use/pay for it, which is a pretty fuzzy problem

Isn’t this quantifiable by revenue?

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