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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#231

Post by chrisjj »

rowanG077 wrote: post #101
sonicrocketman wrote: post #8

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I just don't see that to be true. If tommorow someone pulls a proof that n = np out of their ass but is not able to explain it, it will still have immense value.

With no explanation, how will you know whether it genuinely is proof?

Re: Mathematics in the age of AI

#232

Post by czgov »

mahdi7d1 wrote: post #221
czgov wrote: post #186

Earlier quoted context omitted.

If it is known that A is provably true then one can study the consequences of A being true. It changes things becuase the body of knowledge has expanded.

In this case won't this oracle also tell you what is the consequences as soon as it tells you RH is true and also much more? At this point what is the point of you knowing what is true and what is not?

There is no actual oracle. In this discussion oracle means Lean.

Re: Mathematics in the age of AI

#233

Post by chrisjj »

ipdashc wrote: post #204
highfrequency wrote: post #7

Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."

Someone just brought up this point to me a few days ago on here, I'm definitely increasingly convinced that it's one of the main reasons (maybe even the main reason?) AI prose is so annoying to read through, and so rarely seems able to convey true understanding. It assigns the same narrative importance and dramatic tone to everything (the load bearing whatever, the crucial insight, the smoking gun) even when it's tri…

What surprises me is anyone is surprised. Obviously a stocastic parrot has no understanding, so any conveyance of true understanding it delivers will be rare and accidental.

Re: Mathematics in the age of AI

#234

Post by amoss »

frozenseven wrote: post #224
amoss wrote: post #205

Earlier quoted context omitted.

If you place an LLM into a harness, alongside evaluation, feedback and problem decomposition / solution integration - it is still an LLM? There is an LLM acting as a component in a larger system. But that larger system is not an LLM. Calling it an "AI" is indeed an act of marketing as there is no learning / adjustment as we would expect from an intelligence, but calling it an LLM seems to be inaccurate. So what is it…

>Calling it an "AI" is indeed an act of marketing Again, this is literally the name of the field (and the tech). It's been around longer than you and probably your parents. Dartmouth workshop (1956): https://en.wikipedia.org/wiki/Dartmouth_workshop Random dusty undergrad textbook from the early 70s: https://m.media-amazon.com/images/I/816fxHVJkHL._SL1500_.jpg >there is no learning... If human-like continual learning…

Quoting directly from the dartmouth conference:

"every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it."

Re: Mathematics in the age of AI

#235

Post by gn_200826 »

a2ff6eeb0 wrote: post #25

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems. If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed. The human brain is being obsoleted, soon…

Most of mathematics is more akin to philosophy than physics or engineering. Sure, if AI can prove a result that leads to practical applications, who cares if humans can't understand it. If AI performs a series of convoluted arguments demonstrating the existence of souls (feel free to replace souls with abstract nonsense), but nobody understands why, then what's the point?

Re: Mathematics in the age of AI

#236

Post by czgov »

saithound wrote: post #222
czgov wrote: post #186

Earlier quoted context omitted.

If it is known that A is provably true then one can study the consequences of A being true. It changes things becuase the body of knowledge has expanded.

> If it is known that A is provably true then one can study the consequences of A being true But one can already study the consequences of P=NP right now. You don't need to know that it's provably true in order to do that. Knowing an actual proof would be useful, but an oracle revealing merely that it's true (or even provable) without telling you the proof does not let you do anything you couldn't do before.

Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. In the present discussion it’s not about letting me do something I can’t do now but about whether or not the endeavor is worthwhile.

A lot of people spent a lot of time and effort to prove or disprove the Jacobian Conjecture. AI solved it easily. It is increasingly becoming the case that humans are not as good at mathematics as computers. You are free to ignore computer generated proofs but I don’t think this position will win out in the long run.

Re: Mathematics in the age of AI

#237

Post by czgov »

customguy wrote: post #197
czgov wrote: post #187

Earlier quoted context omitted.

You have an unfortunate view of the value of knowledge. The frontiers of science would be static if everyone believed as you do.

Someone claimed it would be "useful", without saying what for. Hence the questions "what for?". To try to shame people for that question in the name of science of all things is wild.

Which product becomes better if it is correct?

Tying the worthiness of theoretical knowledge to the whether or not it improves a product is asinine in my opinion.

Re: Mathematics in the age of AI

#238

Post by unified101 »

butwaitTheres4 wrote: post #220
unified101 wrote: post #199

Earlier quoted context omitted.

You should definitely inform the wikipedia editors as well https://en.wikipedia.org/wiki/Terence_Tao#Recognition . (I don't know why you're so butthurt BTW - neigher of your ad-hominem comments actually outline your concern)

It's right there in my initial post; barely any of the work is his own. It's mostly memorization and recall and a single proof about primes he is well known for. It's akin to being well versed in Star Wars canon. If Tao can be replaced by a model he isn't that smart just hyper-optimized in a narrow scope. As a scientist such evidence has to be a part of the assessment; it's not hard; find gaps in a syntax system and…

I dunno about all that. For eg: I asked gemini who are the top 5 mathematicians by their contributions. It listed:

  Terence Tao
  Sir Andrew Wiles
  Grigori Perelman
  Peter Scholze
  Mikhail Gromov
Not sure you know more than the compiled knowledge of the world. I guess this conversation is becoming tedious..

Re: Mathematics in the age of AI

#239

Post by jbstack »

sonicrocketman wrote: post #8

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I don't agree with this rule of thumb at all. Let's say tomorrow someone comes up with a formally verified proof that a major encryption algorithm underpinning the security of the internet can be trivially broken, but they can't explain it. You're saying it should be kept under wraps and not published?

Tao is essentially saying that the only value in a proof is its ability to be understood, but that's wrong. A proof is also valuable because it establishes a new fact. The fact is useful in itself, even absent an explanation.

Re: Mathematics in the age of AI

#240

Post by ipdashc »

chrisjj wrote: post #233
ipdashc wrote: post #204

Earlier quoted context omitted.

Someone just brought up this point to me a few days ago on here, I'm definitely increasingly convinced that it's one of the main reasons (maybe even the main reason?) AI prose is so annoying to read through, and so rarely seems able to convey true understanding. It assigns the same narrative importance and dramatic tone to everything (the load bearing whatever, the crucial insight, the smoking gun) even when it's tri…

What surprises me is anyone is surprised. Obviously a stocastic parrot has no understanding, so any conveyance of true understanding it delivers will be rare and accidental.

I mean, having said what I said, we are literally in a thread about the future of mathematics being in question because LLMs are solving advanced problems. I feel like the "stochastic parrot" meme is a bit outdated by now. The bots' output may be annoying to read but they're clearly onto something, whether we call it "understanding" or not.
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