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

Mathematics in the age of AI

arxiv.org

Post reply on HN
Print view View original

131–140 of 292 posts

Re: Mathematics in the age of AI

#131

Post by energy123 »

fultonn wrote: post #62
nilkn wrote: post #48

Earlier quoted context omitted.

I believe this rule of thumb will come to fail. The combination of superhuman mathematical reasoning and synthesis in upcoming AI models plus the rapid build-out of scalable formal verification infrastructure means this exponential in math is going to take off quite explosively, and we've barely seen anything yet. Mathematics is going to decisively move beyond human ability fairly soon (within our lifetimes, if not m…

People say similar things about automation of software engineering. Different, but similar. I'm deeply suspicious. I do not yet have a concise statement for why, but a lot of literature on the sociology of knowledge work sort of points at my thoughts. Section 5 of the Thurston article cited by Tao touches the elephant. Raduchel's article on the economics of software [2] also touches it. I've tried to put words to thi…

Many eminent mathematicians did their best work while not talking about it with anyone, sometimes in isolation. Newton's calculus, Perelman's poincare, much of Grothendiek's work, Wiles's fermat, Ramanujan's earlier days. That's not all top mathematicians as you can look at Von Neumann as a sociable counter-example. But it shows that discussion of your current ideas is not a requirement. Grothendiek goes so far as to say it is a net negative for mathematical creativity because it is difficult to resist thinking like the herd without some level of seclusion.

Re: Mathematics in the age of AI

#133

Post by chongli »

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.

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.

Re: Mathematics in the age of AI

#134

Post by chongli »

d4v3 wrote: post #119
gsinclair wrote: post #115

Earlier quoted context omitted.

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?

No, they're saying that what is true in mathematics is contingent, not absolute. It all depends on which set of axioms use, what assumptions you make.

The area of a triangle doesn't have 1 unique formula, it has many, depending on the system you use. A triangle in plane geometry has a different area than a triangle in spherical geometry, and different again in hyperbolic geometry.

When you get to studying the geometry of manifolds, you realize the area of a triangle can be any damn thing you want, depending on how you construct the manifold you embed it in.

Re: Mathematics in the age of AI

#135

Post by [deleted] »

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.

[deleted]

Re: Mathematics in the age of AI

#136

Post by GPerson »

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?

Some of the best mathematicians in the world tried to study his work, found flaws he did not address, and somehow there’s someone every week suggesting there’s a conspiracy against this guy. It’s really baffling. AI will probably help him move on by lean verifying his proof is wrong…

Re: Mathematics in the age of AI

#137

Post by GPerson »

raincole wrote: post #118
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.

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 on…

The tastes and interests, and cultural pretext, of human beings are not arbitrary just because they’re not easily described.

Re: Mathematics in the age of AI

#138

Post by xdavidliu »

jltsiren wrote: post #80
TMWNN wrote: post #54

Earlier quoted context omitted.

>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." I noticed a long time ago, that the more people focus on trivialities like typos when arguing against someone online, the more compelling the original argument is. B…

I noticed another thing a long time ago. Some academic cultures have a tradition of formal debates. They are based on the premise that an educated person should be able to argue convincingly for or against any idea, regardless of whether they believe in it. A natural corollary is that you should not let convincing arguments convince you, as the merits of the argument have little to do with the merits of the idea itse…

> They are based on the premise that an educated person should be able to argue convincingly for or against any idea, regardless of whether they believe in it.

In many situations, people doing this, skillfully even, has had quite pernicious consequences.

Re: Mathematics in the age of AI

#139

Post by GPerson »

qsera wrote: post #108
GPerson wrote: post #23

Earlier quoted context omitted.

You should give it a try. Tao is a wonderful person and is trying to help humanity.

> 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 don’t. The pecking order in mathematics at least used to have a grounding in actual abilities. People respect this guy because he’s actually legit.

Re: Mathematics in the age of AI

#140

Post by a2ff6eeb0 »

Jblx2 wrote: post #93
a2ff6eeb0 wrote: post #41

Earlier quoted context omitted.

What's wrong with sitting on the beach with a bottle of wine for eternity, with no need to do anything, knowing that all your needs and desires will be automatically taken care of? I don't think you can be coherently pro-AI without thinking that the human brain will be obsolete, unless you believe in some inherent magic that the brain is imbued with. The only other option is that you haven't thought through the long…

Are dolphins obsolete?

They're certainly endangered.
Post reply on HN