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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]

#151

Post by hks0 »

charcircuit wrote: post #141
madaxe_again wrote: post #61

Earlier quoted context omitted.

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…

>At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change In my experience you get taught the definition of a derivative of a function at a point is equal to the instantaneous rate of change and that integrals are defined as a Reimann Sum, the sum of the area under the curve. Everything in the class comes from building on top of those definitions.

That you think this way (and if like me, it makes you excited!) I think it's because it has clicked for you.

For many that light bulb above their head doesn't flash on, hence they get to dislike the subject or forget it after they are done with their studies. I was lucky enough to appreciate math that much to redo it in my free time after high-school and make it click for me.

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

#152

Post by cfiggers »

throwoutway wrote: post #80
_the_inflator wrote: post #69

Earlier quoted context omitted.

Wow, you couldn't be more wrong here. Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us. I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate. A good example is set theory. Cantor was not openly welcomed after he introduced his…

You're also wrong "Math is something humans invented" Majority of mathematicians are platonists and believe arithmetic was existed and was discovered and was not "invented". "There is no logic per se" There is logic to it! Most logicians are mathematicians at heart. See Russel, Godel, Hilbert, etc "no beauty of Mathematical Logic" Mathematicians do focus on beauty. Entire books have been written on this. G.H. Hardy i…

Consensus may give a hint to what is or isn't reality. But consensus—even expert consensus—does not determine reality. Experts can be wrong. Most of the experts, even, can be wrong simultaneously.

Philosophy is the exercise of testing ideas for oneself in the laboratory of one's own mind.

When I test the idea that math is discovered in my own mind, from my own perspective, with my own experience and education brought to bear, I find it unconvincing.

When you test the same idea in the laboratory of your mind, with your experience and your education applied, and get a different result, that is interesting. Your result is relevant information to me. If nothing else, it's a good prompt/trigger for me to revisit my earlier conclusion and see if it still holds.

But your disagreement—or indeed, the disagreement of a majority of trained mathematicians—does not constitute an automatic reason for me to conclusively determine that you/they are right and I am wrong.

I still have my own examination of the concept, with my own supporting and detracting arguments. And the result of my examination continues to be that math being invented is the significantly more persuasive view.

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

#153

Post by overgard »

I don't really like these articles, because they seem extremely hard to verify. OpenAI has published a lot of stuff in the past where, upon close inspection, what they're saying is technically true but a lot less interesting or impressive than the headline. Except by the time anyone looks into it, the hype has moved on. It seems like there's maybe a thousand people in the world that can even say if this is good or not?

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

#154

Post by hyperpape »

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…

> 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 understand anything interesting is not valued. I could go out and define some pointless construction and create proofs about it immediately. It would only matter if I connect it to some other subject of interest within math.

I would argue that mathematical understanding is valuable for extrinsic reasons, but it is true that by the time you're a math grad student, you're usually willing to pursue it for no external purpose.

Although not a mathematician, Daniel Dennett had a wonderful example about higher order truths of "chmess". https://personal.lse.ac.uk/robert49/teaching/ph445/notes/den...

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

#155

Post by varjag »

nilkn wrote: post #150
varjag wrote: post #144

Earlier quoted context omitted.

Human unreadable proofs have only limited value.

I disagree. It's the only way to scale AI mathematics far beyond human mathematics. Any interesting verified result would, obviously, be rewritten back into natural language for human understanding and consumption (as well as potentially for the benefit of AI conjecturers too). You are falsely assuming that advances in formal mathematics would not feed back into similar (potentially massive) advances into informal ma…

I didn't say they have no value. Just limited value. A novel readable proof that expands the horizons of human insight is certainly more valuable than a megabyte sized trychnobezoar of machine generated predicates.

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

#156

Post by moomin »

ak_111 wrote: post #26

Unlike the unit distance problem, the impressive thing here is that it is a proof rather than a counter-example. However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed. So not to dunk on this amazing result (or move the goal post), but it seems now the only achievement that AI hasn't managed in mathematics is presenting an autonomous "theo…

For comedy’s sake, I asked ChatGPT 5.5 about the significance of the problem and the chance that 5.6 would solve it with a three page solution. It said close to zero.

I invited it to search the internet and it remains extremely sceptical.

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

#157

Post by perching_aix »

charcircuit wrote: post #10

But is the proof accepted to be correct? That is what distinguishes this from being notable compared to any other AI slop proof.

I'd guess that verdict (or its opposite) is to come within the next 24 hours.

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

#158

Post by aureianimus »

emil-lp wrote: post #27
azaras wrote: post #19

It did not use Lean or other proof assistant?

There's really no good proof system mature enough to do advanced graph theory. The leading library in Lean is Graphlib, and it's really not ready for research level theorems.

Graphlib? Do you have a link to this for me?

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

#159

Post by nilkn »

varjag wrote: post #155
nilkn wrote: post #150

Earlier quoted context omitted.

I disagree. It's the only way to scale AI mathematics far beyond human mathematics. Any interesting verified result would, obviously, be rewritten back into natural language for human understanding and consumption (as well as potentially for the benefit of AI conjecturers too). You are falsely assuming that advances in formal mathematics would not feed back into similar (potentially massive) advances into informal ma…

I didn't say they have no value. Just limited value. A novel readable proof that expands the horizons of human insight is certainly more valuable than a megabyte sized trychnobezoar of machine generated predicates.

You are assuming that the latter, once autonomously discovered and verified at scale, could not simply be translated into the former, also perhaps autonomously at scale (or otherwise selectively as determined by human interest, taste, and relevance).

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

#160

Post by plorkyeran »

hks0 wrote: post #151
charcircuit wrote: post #141

Earlier quoted context omitted.

>At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change In my experience you get taught the definition of a derivative of a function at a point is equal to the instantaneous rate of change and that integrals are defined as a Reimann Sum, the sum of the area under the curve. Everything in the class comes from building on top of those definitions.

That you think this way (and if like me, it makes you excited!) I think it's because it has clicked for you. For many that light bulb above their head doesn't flash on, hence they get to dislike the subject or forget it after they are done with their studies. I was lucky enough to appreciate math that much to redo it in my free time after high-school and make it click for me.

No, my calculus class in HS very literally started with finding the area under a curve “manually” and introduced integration as a generalization of that. I’m not surprised to hear that calculus is sometimes taught very poorly, but it’s not universal.
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