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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#251

Post by frozenseven »

amoss wrote: post #234
frozenseven wrote: post #224

Earlier quoted context omitted.

>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."

Yes, that's correct. Intelligence is not magic.

Re: Mathematics in the age of AI

#252

Post by GPerson »

ziiinq wrote: post #192
GPerson wrote: post #137

Earlier quoted context omitted.

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

Yes and therefore? You seem to be vaguely waving in the general direction of a point, without making any concrete claims or bothering to engage with the GP’s argument. The tastes and interests of humans are absolutely not arbitrary. They are dictated by fate, the sun and the moon gods. Or maybe by the unitary evolution of the universe’s quantum state. Or by the probability distribution of finding ourselves in a parti…

You seem rather aggressive so replying to you feels pointless and uncomfortable.

Nonetheless, the person writes, “ Math has been almost purely arbitrary”.

This is simply a misusage of the word arbitrary, which is a word with a specific meaning you can look up if you are unaware, since mathematics is (obviously) not arbitrary in the sense this person wants to convey, in part for the reasons I state. Humans are not choosing arbitrary logical statements to prove true or false.

Re: Mathematics in the age of AI

#253

Post by aleph_minus_one »

js8 wrote: post #200
HWR_14 wrote: post #164

Earlier quoted context omitted.

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

Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm. In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems. Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is no…

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

> Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.

OK, I tell you that P=NP, and that I am a magic oracle. So, you have you psychological boost for finding a practical algorithm for free. :-)

Re: Mathematics in the age of AI

#254

Post by GPerson »

qsera wrote: post #163
GPerson wrote: post #139

Earlier quoted context omitted.

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.

I think the villains who run these companies benefit from Tao’s actions because he’s a de facto thought leader of mathematics not taking a very hard anti-AI stance. However, I don’t think a very hard anti-AI stance is correct for someone in his position. Honestly, do you want a bunch of AI labs to 100% dictate the future of whatever your field is or do you want the best leaders of your field to at least try to shape how AI will change it? And if all of the legitimate people in math ignore AI entirely, then historically important parts of the field will get entirely taken over by dilettantes and AI labs. I’m all for math institutions being a safe haven for AI free math work. But I don’t want to see some space X intern solve all the most important problems on a whim and no leaders in math be aware of it.

(I’m not sure the extent to which you think this is marketing. I’m operating under the assumption that you agree with his Hypothesis 4.1. If you don’t, then I’d assume you haven’t seen the long list of prominent open math problems that these systems are providing answers to. And if this doesn’t sway you about Hypothesis 4.1 I’d just halt and ask why.)

Re: Mathematics in the age of AI

#255

Post by auggierose »

esafak wrote: post #75
auggierose wrote: post #67

Earlier quoted context omitted.

Whose requirements?

The employer's, whose salary affords your subsistence.

I don't think anyone wants to be employed. They want the money that comes with employment. If you can make get money without employment, like rich kids, for example, I think most people would prefer that.

Re: Mathematics in the age of AI

#256

Post by [deleted] »

auggierose wrote: post #255
esafak wrote: post #75

Earlier quoted context omitted.

The employer's, whose salary affords your subsistence.

I don't think anyone wants to be employed. They want the money that comes with employment. If you can make get money without employment, like rich kids, for example, I think most people would prefer that.

[deleted]

Re: Mathematics in the age of AI

#257

Post by skinner_ »

czgov wrote: post #236
saithound wrote: post #222

Earlier quoted context omitted.

> 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…

> Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement.

No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.

Re: Mathematics in the age of AI

#258

Post by applecoffeecake »

jbstack wrote: post #239
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…

Tao is clearly referring to mathematics journals. Any kind of practically useful result can easily be published in an engineering or applied scientific journal.

Re: Mathematics in the age of AI

#259

Post by loicd »

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…

> There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols.

What are you talking about? A theorem is a statement that has been proved from some axioms. A statement itself is a finite sequence of symbols satisfying some syntactical rules. The set of symbols for set theory, arithmetic, etc is finite, so the set of statements, and a fortiori the set of theorems, is at most countable. Where do you get the uncountability?

I suspect you are confusing theorems and theories. Assuming the set theory ZF (for instance) is consistent, then a consequence of Gödel's incompleteness theorem is that there are indeed uncountably many inequivalent extensions of ZF that are complete and consistent. Also, not a single one of these extensions can be described in symbols in the sense that there does not exist a computer program that enumerates a possible set of axioms for the extension.

As for your general point about arbitrariness, the late 19th century was a period when mathematicians started being concerned with the rigorous formalization of mathematics. Sure, there are some arbitrariness in the particular choice of formalization in the same way that the particular form of a programming language like C is arbitrary. However, the Gödel stuff has nothing to do with that arbitrariness, it is about the limit of formalization itself. The programming analogue is the undecidability of the halting problem. Saying that mathematics are arbitrary sounds to me a bit like saying that an algorithm like Quicksort is arbitrary because you saw an implementation in C and the particular form of the C language is arbitrary. Obviously, if you don't like the C language, you can implement Quicksort in another language. The same is true for mathematics. If some day, somebody finds a contradiction in ZF, or simply a new formalization that people find more convenient, then most mathematics will simply get translated and very little will change.

Re: Mathematics in the age of AI

#260

Post by czgov »

skinner_ wrote: post #257
czgov wrote: post #236

Earlier quoted context omitted.

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…

> Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.

If a counterexample to a conjecture is found then all work toward proving consequences of the conjecture will cease. No one is trying to discover consequences of the Jacobian Conjecture now.

At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.

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