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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#161

Post by ziiinq »

chongli wrote: post #133
noosphr wrote: post #127

Earlier quoted context omitted.

Math is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.

A proof by contradiction that p = np is of no use to anyone, given that it wouldn't help you find any polynomial time algorithms for np-hard problems.

I don’t recall anything specific off the top of my head but I am confident that such a proof would have immediate actionable implications.

Furthermore, careful analysis of the latter would as likely as not yield further understanding and, actually /would/ help finding such algorithms.

Finally, it has been observed time and time again that often (again, nothing comes up and i don’t want to ask AI) the certainty that something is possible and has been done is motivation and inspiration enough for people to independently solve a problem. Sometimes it is even enough for someone new to simply not know that something is “hard” to solve.

It even “motivates” llms, it seems (eg https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...)

Of course this is all pure speculation concerning a hypothetical proof that most likely doesn’t exist, or indeed might be so complicated as to not be approachable even after hundreds of lifetimes of study.

Nevertheless your conclusion does not follow from the premise

Re: Mathematics in the age of AI

#162

Post by ziiinq »

Scarblac wrote: post #154
noosphr wrote: post #127

Earlier quoted context omitted.

Math is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.

How would the mere knowledge that it holds, without any understanding why, be useful?

[dead]

Re: Mathematics in the age of AI

#163

Post by qsera »

GPerson wrote: post #139
qsera wrote: post #108

Earlier quoted context omitted.

> is trying to help humanity. That is what all marketing wants you to think...

It’s not marketing. This guy could have signed up for one of those hundred million dollar salaries with a phone call and did not. I know several people who have met him and everyone says he’s the genuine article. He’s actually just devoted to human mathematics. One tragic thing about all of this is that unlike almost every profession, mathematics actually has a kind of honesty. You honestly solve the problem or you d…

Legit people are used by marketing all the time.

Re: Mathematics in the age of AI

#164

Post by HWR_14 »

noosphr wrote: post #127
zahlman wrote: post #83

Earlier quoted context omitted.

I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point : the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.

Math is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.

If a magic oracle tells you p=np, that's useless. How would that change anything?

Re: Mathematics in the age of AI

#165

Post by unified101 »

butwaitTheres4 wrote: post #160
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.

Tao has yet to produce work that outshines those whose work he studied and memorized. Not worth the reverence merely being a VHS copy of history. He's a typical person otherwise, politically aware of how he barters for food; until proven otherwise this can be seen as little more than social moat defense. To paraphrase a quote attributed to Upton Sinclair; hard to get a worker to understand something when their payche…

wtf?! Tao is the only mathematician I can name, and widely considered the foremost living one.

Re: Mathematics in the age of AI

#166

Post by glimshe »

ves wrote: post #153
glimshe wrote: post #112

Earlier quoted context omitted.

Math isn't "cooperative". Math is about truths. The length of circumference. The area of a triangle. The formulas for these are true in an objective sense irrespective of whether you understand them. That said, without understanding, Math can't evolve. Comprehension of a proof is very important, but not what Math is fundamentally about. Computer programs are Math. You can use them without understanding how they work.

This hasn't been a remotely reasonable characterization of math since at least Hilbert's time.

I think Hardy would be very much on the same page with me, as well as Godel and many others. Mathematical truths exist independently from our feelings and processes to discover them. People do and should argue about which truths are interesting to pursue and refine, but all of them are out there to be discovered... or not.

Whatever philosophy you prefer, math is about establishing objectively valid logical results, completely independent of the human process used to arrive at them.

Re: Mathematics in the age of AI

#167

Post by chongli »

nilkn wrote: post #158
chongli wrote: post #155

Earlier quoted context omitted.

And that's an issue why? The issue is SNR: signal to noise ratio. Generating exponentially more mathematics, particularly if the process is indiscriminate or optimized for something other than usefulness or mathematical relevance (such as optimizing for machine-provability), does not imply that we get exponentially more applications. We may end up halting the progress of applications altogether as the entire capacity…

It could just as easily be the opposite: it could end up being far easier to reasonably direct and evaluate the research direction and output of AI systems than human mathematicians, who are forced to specialize over decades and essentially cannot pivot and often can't even meaningfully evaluate each other's work. Moreover, the disdain you have for low-value output in mathematics is not unique to you. Talented mathem…

Moreover, the disdain you have for low-value output in mathematics is not unique to you

I didn't say anything about low-value output. No one actually knows the value of any particular piece of mathematics within that deluge. Mathematicians don't have a magical ability to differentiate high-value mathematics from low-value merely by reading paper titles and abstracts.

The dirty secret in the mathematical world -- that has been going on for a long time already -- is that papers get attention based on the reputation of the authors, not on the rigour or validity of the proof. The big headline-grabbing papers are getting read by mathematicians because AI researchers have leveraged media exposure to bypass the reputation network, but media exposure doesn't scale.

When everyone is using LLMs to generate proofs, only reputable mathematicians will be able to get their work read. And herein lies the crux of the problem: an exponential takeoff in the volume of output from respected mathematicians will leave a critical shortage of readers.

it could end up being far easier to reasonably direct and evaluate the research direction and output of AI systems than human mathematicians

That's baseless speculation. All indications so far are that LLMs produce proofs far longer and far more complicated than humans are capable of, such that only machines can check the proofs for validity. Digesting them into a human-readable interpretation of the results is an open problem.

Re: Mathematics in the age of AI

#168

Post by demibabs »

BeetleB wrote: post #51
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…

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading: https://arxiv.org/abs/math/9404236 He wrote it in 1994. He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field. When he discovered this, he realized his error was that he was focusing on producing res…

> He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field.

This is hilarious lmao

Re: Mathematics in the age of AI

#169

Post by ziiinq »

thechao wrote: post #95
zahlman wrote: post #83

Earlier quoted context omitted.

I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point : the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.

I hope I'm remembering this right: a mathematician claims to have a proof for the ABC conjecture, but can't conceive any other mathematician it's right — it's "too weird", so the proof is rejected?

[dead]

Re: Mathematics in the age of AI

#170

Post by a2ff6eeb0 »

Scarblac wrote: post #154
noosphr wrote: post #127

Earlier quoted context omitted.

Math is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.

How would the mere knowledge that it holds, without any understanding why, be useful?

Can you explain to your cat how Amazon uses graph theory to deliver their packages of cat food?

Is Amazon still delivering food to your cat?

Humans don't need to understand what AI generates. We still can get the rewards.

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