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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#111

Post by fukaiall »

kriro wrote: post #35
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give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published.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.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

Mate you managed to provoke with this comment. But you know what you’re saying right?

Re: Mathematics in the age of AI

#112

Post by glimshe »

zahlman wrote: post #83
kriro wrote: post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

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 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.

Re: Mathematics in the age of AI

#113

Post by skybrian »

nilkn wrote: post #107
skybrian wrote: post #49

Earlier quoted context omitted.

Maybe that will be true when it's math with practical applications, but most theoretical math isn't like that. If it's not practical and it's not for mathematians to understand, what good is it?

We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains. Why does this tend to be the case, even when some of the smartest people in the world have historically predicted incorrectly that certain branches of math would forever be useless (e.g., number theory)? I can only offer my own theory on that, but my guess…

It sounds like the idea is to turn on a math generator and keep running it until it generates something interesting. And it might be fun to try it. But if it’s too much output to read and we don’t understand the output either, how does anyone recognize when it’s done something that’s practically interesting?

The output might make a cool screen saver as-is, but we probably need a way to evaluate it somehow.

Re: Mathematics in the age of AI

#114

Post by sanxiyn »

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?

The consensus is that proof is in fact incorrect. People tried really hard (like putting in a year of effort) and most converged to the same place, that proof of 3.12 is incorrect or has a gap. Peter Scholze (who won Fields Medal) and Jakob Stix did a writeup. People seem to think Shinichi Mochizuki correctly reduced ABC conjecture to 3.12, but didn't prove 3.12, and also are doubtful about the whole program because 3.12 doesn't seem any easier than ABC conjecture while complicating everything.

https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture...

Re: Mathematics in the age of AI

#115

Post by gsinclair »

glimshe wrote: post #112
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 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.

First of all, mathematics is about so much more than the area of a triangle etc. that any analogy based on such simple things is overwhelmingly likely to be too simple to be of value.

Secondly, there is no truly objective truth to the area of a triangle. At bottom, this “truth” is simply “everyone is convinced, and for good reason”.

Without persuading other people of the “truths” that you discover, there is no real mathematics.

Re: Mathematics in the age of AI

#116

Post by nilkn »

skybrian wrote: post #113
nilkn wrote: post #107

Earlier quoted context omitted.

We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains. Why does this tend to be the case, even when some of the smartest people in the world have historically predicted incorrectly that certain branches of math would forever be useless (e.g., number theory)? I can only offer my own theory on that, but my guess…

It sounds like the idea is to turn on a math generator and keep running it until it generates something interesting. And it might be fun to try it. But if it’s too much output to read and we don’t understand the output either, how does anyone recognize when it’s done something that’s practically interesting? The output might make a cool screen saver as-is, but we probably need a way to evaluate it somehow.

Yes, of course we'd need a way to evaluate it. I don't right now have a fully conceived answer to what that will look like. But I'm confident at least in saying we would not evaluate it, like Tao is suggesting, by only accepting something once a human can easily teach it unassisted to another human. That sets the bar dramatically too low and would quickly become an extraordinary impediment to progress. You'd have to think of yourself less like a researcher and more like the director of the world's largest research institute. It's highly unlikely you'll understand or even care about every single paper every one of your researchers is producing, but you'll care about the overall research direction and whether the intermediate results are accumulating into outcomes you consider meaningful. How to do this where the institute is based on superhuman AI mathematicians is an unsolved problem, but I see no reason to imagine it's unsolvable.

Let me make up an example of where I could imagine this going. Something we essentially cannot do right now is predict coarse-grained phenomena from systems that involve millions or trillions or more of interacting components. Over hundreds/thousands of years of experiment and theory we've derived laws that essentially do this in a few special cases, but we have no systematic theoretical way of doing it in general, and frankly I think it's beyond human ability. Whatever deep patterns or structures exist for doing this in a general way I think are simply out of reach for us.

Re: Mathematics in the age of AI

#117

Post by raincole »

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.

The problem is that there will be far more formally verified proofs than that human mathematicians around the world can read, much less explain. What then? Would the role of mathematicians just become explainers of AI generated proofs?

Re: Mathematics in the age of AI

#118

Post by raincole »

glimshe wrote: post #112
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 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.

Math is not about truths, at least not by the meaning of "truth" as a word in daily use.

Math has been almost purely arbitrary since ~ late 19th/early 20th century. There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols. Even if you have a tape with infinite length (which is already longer than the whole physical universe!) filled with theorems, they are still only 0% of all correct theorems. That's how arbitrary math is.

> The area of a triangle

Yes, even this is arbitrary. The rigorous definition of triangle is arbitrary. People just subconsciously choose something that vaguely approximates their physical intuition.

Re: Mathematics in the age of AI

#119

Post by d4v3 »

gsinclair wrote: post #115
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.

First of all, mathematics is about so much more than the area of a triangle etc. that any analogy based on such simple things is overwhelmingly likely to be too simple to be of value. Secondly, there is no truly objective truth to the area of a triangle. At bottom, this “truth” is simply “everyone is convinced, and for good reason”. Without persuading other people of the “truths” that you discover, there is no real m…

> Without persuading other people of the “truths” that you discover, there is no real mathematics.

Why? If I sat around and studied math by myself and discovered something true yet not yet known but didn't share it, it's still true. Are you saying I didn't "do math" because I didn't share the result? Math exists on another plane and it has 'truths' that we haven't discovered, yet are still 'true', no?

Re: Mathematics in the age of AI

#120

Post by odyssey7 »

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.

This is a statement about what Tao values in the proofs that he consumes, as a world-class, human mathematician.

For many of the rest of us, mere consumers of mathematical results, it’s sufficient to know that a^2 + b^2 = c^2 was proven by somebody or some machine at some point.

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