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An OpenAI model has disproved a central conjecture in discrete geometry

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Re: An OpenAI model has disproved a central conjecture in discrete geometry

#631

Post by cwmoore »

From the meandering and self-loving article:

“ For decades, it was widely believed that this rate was essentially the best possible, and no construction could improve significantly over the square grid. In technical terms, Erdős conjectured an upper bound of n 1 + o ( 1 ) n 1+o(1) in which the additional o ( 1 ) o(1) indicates a term tending to 0 0 with n n.

Our new result disproves this conjecture. More precisely, for infinitely many values of n n, the proof constructs configurations of n n points with at least n 1 + δ n 1+δ unit-distance pairs, for some fixed exponent δ > 0 δ>0. (The original AI proof does not give an explicit δ δ, but a forthcoming refinement due to Princeton mathematics professor Will Sawin has shown one can take δ = 0.014 δ=0.014.)”

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#632

Post by make3 »

cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

To me, AI feels like the morbidity of Star Trek teleportation, where it's actually copying the person at to the other end and zapping the original one out of existence. The original human never benefits from the fast transportation.

Similarly, we're creating tools to improve knowledge, but we're progressively zapping the human out of the equation. Knowledge is created for something, but it's unclear if very soon humans will be able to understand it, or really benefit from it, except billionaires, etc.

It's too bad that we're not improving humans nearly as fast as we're replacing ourselves.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#634

Post by CamperBob2 »

School-Cotton wrote: post #415
CamperBob2 wrote: post #377

Earlier quoted context omitted.

What's happening is the verbal/linguistic equivalent of the invention of calculus. No intellectual field will ever be the same again. Who wouldn't find that exciting, and want to experience it?

I'm not sure I grasp the analogy to the invention of calculus. Calculus helped us solve new and interesting math/physics problems. Repeated for emphasis: helped *us* solve. This technology is solving interesting math/physics problems for us, which is completely different.

After Newton and Leibniz, math did things nobody thought it could do. After Vaswani et al., language does things nobody thought it could do.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#635

Post by doctorwho42 »

ted_dunning wrote: post #507
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

Does it terrify you to look at children? Not so many years from now, some of them will surpass you. A few years after that all (that survive to that point) will surpass you. Does that terrify you just as much?

AI is not a living or conscience entity, no matter what the hype men are selling society.

A child is a living, breathing, growing, and changing conscious entity. It is the natural order for the young to supplant the old, no matter what the politicians and billionaires desire.

"AI" - terrifies anyone who understands the pact our society rests upon: that labor is valued and can be exchanged for goods and services to survive. Thereby enabling a person to support their families without having to do everything themselves.

If AI replaced a noticeable fraction of society, destroying their capacity for work. That threatens and ultimately blows up this compact between working class and capital class... With it, the foundations of a modern technological society.... It may sound like hyperbole, or some fantastical prediction. But really it is basic economics, like econ 101... And personally the last few years have terrified me, not because of AI directly, but because how ignorantly blind many smart and tech savvy people are... You are marching us to collapse with a smile on your face...

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#636

Post by doctorwho42 »

trostaft wrote: post #518
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

I spent years grinding to learn mathematics because it was the language I needed to solve problems that excite me. If the tools I need to do so change, I can change too. Research training is not so rigid that it can only applied to the single set of skills I developed it in the context of. I can learn this too. Moreover, truth be told, I don't really see myself doing any less math and requiring less from my skills. A…

What about the future mathematician's yet to be born?

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#637

Post by doctorwho42 »

dekhn wrote: post #556
School-Cotton wrote: post #361

Earlier quoted context omitted.

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless. Along with all the rest of what humans find meaningful and fulfilling.

At least from my perspective, these sorts of tools could have the possibility of allowing us to reach post-scarcity (I guess a skynet future is another possible outcome, as is just grimdark industrial hell). If we reach that point, then anybody could (in principle- in reality utopias don't exist) pursue anything they wanted. This is just an application of the philosophy "automate yourself out of a job every 6 months"…

But that hasn't been done at scale... If everyone automated their job every 6 months, then millions would be out of work and starving.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#638

Post by 8n4vidtmkvmk »

doctorwho42 wrote: post #615
aroman wrote: post #410

Earlier quoted context omitted.

This is the beginning of thinking, not the end...

But when the bar to entry is beyond expertise in a field or subfield, how does an individual ever hope to attain an unexplored space to explore? It may be the beginning of thinking, but to many who view things on a longer timeline. It starts to look like it will breakdown the frameworks of which are required to get to that position. Otherwise, you just end up retreading explored ground. This removing the joy of disco…

Recently I've found my mind reawaken. It's about asking good questions now. The models can find the answers, but you have to know what to ask. Sometimes the model is wrong and you have to challenge it to find an alternative. Being able to explore problem spaces quickly is interesting.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#639

Post by dhosek »

cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

One of the challenges I had in graduate mathematics was just trying to keep all the concepts in my brain. It doesn’t help that you end up with things like homomorphism and homeomorphism tangling one’s brain thanks to their superficial similarities. Heck, just keeping track of basic theorems and definitions is a challenge.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#640

Post by justonepost2 »

libraryofbabel wrote: post #519
jryan49 wrote: post #457

Earlier quoted context omitted.

People are afraid for their livelihood. What do you expect?

Well yes, but there is a choice being made here and I would love to believe we can do better. The rational response to being afraid about your livelihood isn’t to spend time filling every HN thread on LLMs with embittered negativity. Not to mention all the flat denials that LLMs can do mathematics and write decent code, which is almost a self-contradictory position if you are worried they are going to replace you. Th…

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