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

Infinities, impossibilities, and the man in the white linen suit

iain.so

Post reply on HN
Print view View original

21–30 of 65 posts

Re: Infinities, impossibilities, and the man in the white linen suit

#21

Post by hardbass »

skippyfish wrote: post #5

Aside from seemingly being 100% AI-generated, this article makes a really spurious use of Godel's argument, which I think is best exemplified by the following passage near the end: "These four threads share a common ancestor in what Gödel proved in 1931, and Turing sharpened in 1936. Rule-based systems cannot fully account for themselves. A system cannot certify its own trustworthiness. A learning framework cannot de…

The worse confusion they seem to have is they think mathematical theorems are only applicable to "computers". Mathematical theorems have no escape, they apply just as much to human brains as to computers. If these theorems were a blocker for developing general intelligence then how do humans exist.

Re: Infinities, impossibilities, and the man in the white linen suit

#22

Post by hardbass »

iainharper wrote: post #2

I wrote this partly because nobody seems to know about Gödel who is arguably a more important figure than Turing. The piece follows four threads where formal limits show up in AI practice today: learnable problems that mathematics cannot sort, neural networks that exist but cannot be trained, self-improving systems that dropped their safety proofs for benchmark scores, and the fact that “will this AI cause harm” is,…

If the theorem is truly general ie any algorithm not just certain classes of neural net training methods, then is a moot point. Computers, humans or any other physical system can't solve it.

Re: Infinities, impossibilities, and the man in the white linen suit

#23

Post by pdonis »

c2c2c wrote: post #11
skippyfish wrote: post #5

Aside from seemingly being 100% AI-generated, this article makes a really spurious use of Godel's argument, which I think is best exemplified by the following passage near the end: "These four threads share a common ancestor in what Gödel proved in 1931, and Turing sharpened in 1936. Rule-based systems cannot fully account for themselves. A system cannot certify its own trustworthiness. A learning framework cannot de…

I think this is selling Gödel a little bit short. He proved for the first time that in any system of mathematics with enough complexity to be interesting, there are statements that cannot be proven true or false. The Gödel sentence might seem like a trivial example, but other examples of such things have been discovered since, such as the continuum hypothesis which the article mentions.

> n any system of mathematics with enough complexity to be interesting, there are statements that cannot be proven true or false.

Note that the article itself is not as careful as you are here about correctly stating what Godel's Incompleteness Theorem actually says. The article states it this way: "any set of rules powerful enough to handle basic arithmetic will contain true statements it cannot prove". But that's wrong, because, when you take into account Godel's Completeness Theorem, you realize that, for any formal system complex enough for the proof of Godel's Incompleteness Theorem to go through, there must be a semantic model of that system in which the Godel sentence for that system is false! The article's misstatement is unfortunately very common, but it's still a misstatement.

Re: Infinities, impossibilities, and the man in the white linen suit

#24

Post by skybrian »

skippyfish wrote: post #5

Aside from seemingly being 100% AI-generated, this article makes a really spurious use of Godel's argument, which I think is best exemplified by the following passage near the end: "These four threads share a common ancestor in what Gödel proved in 1931, and Turing sharpened in 1936. Rule-based systems cannot fully account for themselves. A system cannot certify its own trustworthiness. A learning framework cannot de…

I think you’re right about practicality. Seems like nonstandard models of arithmetic suggest that there are things we cannot know about all natural numbers, because some of them (nearly all of them) are “too high to work with.” They are too high when all you can do with them is write proofs about them using induction.

There’s a sense in which the set of natural numbers is too big to be practically useful, but no smaller set makes sense when writing proofs. The set of numbers you can reach without using induction (say, by using a computer to check) isn’t well-defined.

And that has nothing to do with AI’s limitations.

Re: Infinities, impossibilities, and the man in the white linen suit

#25

Post by skippyfish »

hardbass wrote: post #20
azakai wrote: post #9

The overall point feels strained to me. Yes, formal mathematics has such limits. We can't expect machines to be perfect and provably perfect. But the industry isn't assuming that. Why would it? Natural intelligence is not perfect or provably perfect, either. Rather than certainty, measurement is often enough. We can't prove a program will always halt, but we can check it halts in a specific execution. Approximation i…

I don't understand the point at all. Mathematical theorems like godel's incompleteness theorem or the halting problem are applicable to everything including the human brain. If they were a limitation to developing general intelligence then human beings wouldn't have existed.

> Godel's incompleteness theorem or the halting problem are applicable to everything including the human brain.

Maybe. That's if you assume that the universe is a conventional computer. Far from settled and the alternative isn't necessarily just "God does it" - it might be that the universe is a "computer-plus", with some processes that produce results not computable by Turing means. It would be strange, but so is quantum mechanics, so...

Re: Infinities, impossibilities, and the man in the white linen suit

#26

Post by moi2388 »

hardbass wrote: post #20
azakai wrote: post #9

The overall point feels strained to me. Yes, formal mathematics has such limits. We can't expect machines to be perfect and provably perfect. But the industry isn't assuming that. Why would it? Natural intelligence is not perfect or provably perfect, either. Rather than certainty, measurement is often enough. We can't prove a program will always halt, but we can check it halts in a specific execution. Approximation i…

I don't understand the point at all. Mathematical theorems like godel's incompleteness theorem or the halting problem are applicable to everything including the human brain. If they were a limitation to developing general intelligence then human beings wouldn't have existed.

Are they? Are there any actual Turing machines? Doesn’t that require infinite memory? I don’t know of a single thing which actually has that, nor one that comes close.

Re: Infinities, impossibilities, and the man in the white linen suit

#27

Post by inigyou »

skippyfish wrote: post #5

Aside from seemingly being 100% AI-generated, this article makes a really spurious use of Godel's argument, which I think is best exemplified by the following passage near the end: "These four threads share a common ancestor in what Gödel proved in 1931, and Turing sharpened in 1936. Rule-based systems cannot fully account for themselves. A system cannot certify its own trustworthiness. A learning framework cannot de…

Axiom systems feel like the ordinal numbers: you can keep building upwards as far as the eye can see, and even farther and you run out of useful reasons to do it long before you run out of axiom systems or numbers.

"One!" "Two!" "Three!" "Infinity!" "Infinity plus one!" "Infinity plus two!" "Infinity plus infinity!" "Infinity times infinity!" "Infinity to the power of infinity!" "Infinity to the power of infinity to the power of infinity!" "The supremum of all ordinals that can be expressed in a finite sequence of symbols!" "The supremum of all ordinals that can be expressed in a finite sequence of symbols plus one!"

Re: Infinities, impossibilities, and the man in the white linen suit

#28

Post by pegasus »

skippyfish wrote: post #25
hardbass wrote: post #20

Earlier quoted context omitted.

I don't understand the point at all. Mathematical theorems like godel's incompleteness theorem or the halting problem are applicable to everything including the human brain. If they were a limitation to developing general intelligence then human beings wouldn't have existed.

> Godel's incompleteness theorem or the halting problem are applicable to everything including the human brain. Maybe. That's if you assume that the universe is a conventional computer. Far from settled and the alternative isn't necessarily just "God does it" - it might be that the universe is a "computer-plus", with some processes that produce results not computable by Turing means. It would be strange, but so is qu…

Yep, a possibility that has found support with Roger Penrose, who wrote two books about it.

Re: Infinities, impossibilities, and the man in the white linen suit

#29

Post by jibal »

Turing proved that no TM can determine whether all TMs halt ... not that no TM can determine whether some specific TM halts ... the difference a common misunderstanding of the proof. Analogously, while we can't prove that no AI is or isn't harmful, we can prove that certain AIs are or aren't harmful.

Re: Infinities, impossibilities, and the man in the white linen suit

#30

Post by jibal »

moi2388 wrote: post #26
hardbass wrote: post #20

Earlier quoted context omitted.

I don't understand the point at all. Mathematical theorems like godel's incompleteness theorem or the halting problem are applicable to everything including the human brain. If they were a limitation to developing general intelligence then human beings wouldn't have existed.

Are they? Are there any actual Turing machines? Doesn’t that require infinite memory? I don’t know of a single thing which actually has that, nor one that comes close.

Finite machines such as FSMs (finite state machines) are a subset of TMs, so yes they are actual Turing Machines. What they aren't is UTMs ... universal Turing Machines that can emulate every TM.
Post reply on HN