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

An OpenAI model has disproved a central conjecture in discrete geometry

openai.com

Post reply on HN
Print view View original

501–510 of 1001 posts

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

#501

Post by dogwalker5000 »

doginasuit wrote: post #471
libraryofbabel wrote: post #448

This HN thread depressed me. I’m still thinking about why. Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers…

I find it understandable, it is common to evaluate human intelligence vs AI as a zero-sum competition, because that is how employers typically understand it and LM providers market it. AI proving itself moves the needle in an uncomfortable direction for all of us without very robust job security. > I wonder if we’ll still be doing this two years hence. It is going to take some time for people to recognize that AI has…

> because that is how employers typically understand it and LM providers market it.

Every few months you get an article of some executive bragging that he fire an entire department of people because of AI.

It was adversarial from the start. The idle rich who don’t have to work for a living and their sycophants who somehow believe they won’t be replaced vs … everyone else.

I used to think that the common tale of AI rebelling in Hollywood movies was unlikely. Turns out we don’t even need rogue AI, our fellow men are quite willing to wipe the rest of us out.

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

#502

Post by virgildotcodes »

davebren wrote: post #398
mooreat wrote: post #366

Earlier quoted context omitted.

I currently operate under the assumption that humans are at most as powerful as Turing Machines. And from what I understand these models internally are modeling increasingly harder and larger DFAs, so they're at least as powerful as regular languages. Assuming humans are more powerful than regular languages I could maybe agree that these methods may not eventually yield entirely human like intelligence, but just bett…

Well yeah there is likely an equivalence between computability and epistemology, but I'm not sure it matters when comparing LLM intelligence to human intelligence. There is clearly a missing link that prevents the LLM from reaching beyond its training data the way humans do.

If you look at the life efforts and accomplishments of the ~100 billion humans who have ever lived, how many lifetimes would you discount as having "non-human intelligence" based on the lack of "novel" contributions to frontier of our species' scientific understanding according to the same high bar you apply to LLMs?

Do you pass that bar yourself?

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

#503

Post by ex-aws-dude »

davebren wrote: post #323
mooreat wrote: post #164

I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by…

> I think that's just a matter of having them able to work on longer and longer time horizons. No this will never do the kind of math that humans did when coming up with complex numbers, or hell just regular numbers ex nihilo. No matter how long it's given to combine things in its training data.

You're just stating the opposite of the commenter with no additional discussion

Its like just commenting "I disagree" its totally pointless for discussion.

That's why you're getting downvoted if you're wondering.

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

#504

Post by drdeca »

slashdave wrote: post #473
paulddraper wrote: post #150

Earlier quoted context omitted.

"LLMs just interpolate their training data" Cracks me up. What exactly do we think that human brains do?

You have to define what you mean by "interpolate". The mechanisms that LLM use are not mysterious, and they are not the same as used by humans.

If you interpret “interpolate” in the literal sense, and apply it to the mechanisms behind LLMs, then the claim that they only interpolate, is straightforwardly false.

Taking it instead as a metaphorical claim may be more valid, but in that case it doesn’t depend on our understanding of how LLMs work.

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

#505

Post by ted_dunning »

isotypic wrote: post #463
trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a…

In Erdös idiosyncratic nomenclature, all the best proofs are "in the book" and it was always a joyful thing to not only find a proof, but to find the proof that is in the book.

Who cares if it is God's book or the machine's Xeroxed copy?

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

#506

Post by k_roy »

isotypic wrote: post #463
trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a…

And you just expressed the thoughts of every engineer that writes code for a living who is either left behind, or embracing the technology to hit KPIs and QVRs.

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

#507

Post by ted_dunning »

School-Cotton wrote: post #361
trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

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?

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

#508

Post by ted_dunning »

mlcrypto wrote: post #491
rogerrogerr wrote: post #389

Earlier quoted context omitted.

People who enjoy thinking. Ya know, the "intellectual" part.

The so called "progressives" prove that they were the same ones crying after the printing press, automobile, calculator, washing machine, etc

You made up a group in the past and you made up things they say and then draw the inference that a different group in the present is somehow morally disadvantaged by obvious inference.

Perhaps your name-calling is not actually as logically grounded as you think. It definitely seems to depend on unfounded leaps.

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

#509

Post by awdfeswavcra »

__0x01 wrote: post #423

From the companion paper: > The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. Can someone please elaborate on this?

The last two are straightforward. The proof relies on a result called the Golod-Shafarevich theorem that gives a criterion for a group to be infinite. Golod and Shafarevich proved this a long time ago (1964). Moreover, if you look at how Golod and Shafarevich used this criterion, it's the same way it's used in the proof: They apply it to some Galois groups that appear in number theory, prove these are infinite in certain cases, and deduce that there exists an infinite tower of number fields with some surprising properties.

Much more recently (2021), Hajir, Maire, and Ramakrishna figured out how to apply the Golod-Shafarevich theorem to a slightly different Galois group to produce an infinite tower of number fields with some even more surprising properties. This is used in the new proof. It requires very slightly modifying the construction of Hajir, Maire, and Ramakrishna to produce the fields needed in this proof, but the explanation of how to do this takes only a paragraph in the human-written summary. (The explanation is more laborious in the original AI writeup).

The relation to Ellenberg-Venkatesh is more indirect. This is where "in retrospect" comes in because this work was not cited in the original AI proof. This has to do with the next step of the proof, after you construct the number field, you need to find many elements of this field with the same norm to produce many vectors of the same length. To do this, the proof uses a pigeonhole argument which uses small split primes of the field (constructed via Hajir, Maire, and Ramakrishna's argument) to construct many ideals. By the pigeonhole principle, you can guarantee two ideals lie in the same class. When two ideals lie in the same class, you get an element of the field. You can rig things so these elements all have the same norm. Ellenberg and Venkatesh had an argument which also used the pigeonhole prnciple to guarantee two ideals lie in the same class to produce elements of the field. They were working on a different problem so their argument was slightly different, but similar.

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

#510

Post by ex-aws-dude »

libraryofbabel wrote: post #448

This HN thread depressed me. I’m still thinking about why. Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers…

Lets just be real its because a lot of programmer's ego is built on intelligence/being a coding wizard and this threatens that ego

If suddenly anyone can code we're not that special anymore.

Post reply on HN