undefined

points

[-]

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.

by skippyfish1 days ago|

parent|

[-]

The article is massively overstating the implications of Godel to the AI industry, so to be clear - are you defending the article, or are we bikeshedding about my phrasing? Because my "not that profound" should be interpreted as relative to the article.

by srean1 days ago|

parent|

[-]

Ah that "with me or against me" dichotomy again.

by Animats1 days ago|

parent|

prev|

[-]

> enough complexity to be interesting

That's the loophole. There is mathematics without infinity.[1] Until recently, it was considered somewhat fringe. But there's been enough development that it's now clear you can do mathematics without infinity. In that view, infinities are a labor-saving device to get rid of special cases. They're not essential. Mathematicians used to hate special cases, but now that the active ones all grew up with computers, they're not so scared of IF statements.

You can have "arbitrarily large" but not "infinite". Boyer-Moore theory is like that. It starts out like Peano arithmetic. There is (ZERO) and (ADD1 (ZERO)) and (EQUALP A B) and (LESSP A B) as predicates. That's the axiomatic basis. Numbers are (ADD1 (ADD1 (ADD1 (ZERO)))), etc.

You can have recursive functions, but they must terminate. They must have a value which, for each recursion, is LESSP than the previous iteration. That's how you prove termination.

You can do most of mathematics that way. But not infinitesimals. You never can get to

    1 + 1/2 + 1/4 + 1/8 ... = 2

in that kind of constructive mathematics. That requires an infinite recursion, which is not allowed.

[1] https://engtechnica.com/rethinking-mathematics-without-infin...

by pdonis1 days ago|

parent|

prev|

[-]

> 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.

by inigyou1 days ago|

prev|

[-]

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!"

by skybrian1 days ago|

prev|

[-]

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.

by hardbass1 days ago|

prev|

[-]

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.

by Diogenesian1 days ago|

parent|

[-]

Gödel believed the human brain used non-mathematical reasoning (i.e. inexpressible with a Turing machine) to derive the axioms and thus could "see outside" of any particular axiom schema.

The Church-Turing thesis throws cold water on this: since the human body (including brain) is describable by a finite system of Schrödinger equations, and these equations can be solved numerically by a Turing machine, the human process of creating an axiom scheme should be Turing-computable. But some recent results on very large finite numbers (busy beaver) suggest there may be a subtlety here, e.g. complexity blows up to the point that it takes far more energy than the sun to simulate one human.

The more interesting subtlety: for a physically meaningful result you would need to define configuration space very carefully, e.g. not screwing up the boundary consitions or causal order of subsystems. Perhaps defining this is actually not computable, and after every delta(t) in a computer simulation, a human has to check the physics and redefine certain parameters of the system. Solving the Schrodinger equation numerically is certainly Turing-computable, but the process of ensuring that solution is physically meaningful isn't even slightly formalized. It may be unformalizable.

by hardbass11 hours ago|

parent|

[-]

Some problem being too chaotic or computationally expensive to perform is a completely different thing that claiming something supernatural. And there is hardly any reason for computers to be confined to silicon digital logic, using coprocessors is a standard procedure, so whatever element if any which exhibits hypercomputation (which is highly doubtful as the universe barely even reaches even a small fraction of the possibilities of turing machines let alone beyond them) or otherwise more efficient classical computation, we can isolate the element and use it as a chip.

by Diogenesian9 hours ago|

parent|

[-]

To be clear the limitation here isn't silicon logic, it's theoretical logic (specifically general recursive functions). It sure seems like any possible computation can be expressed as a general recursive function, but that's a scientific thesis, not a mathematical theorem. As we have yet to formalize "define a physical system" it is possible that this task isn't actually expressible in 21st century mathematical logic. I suppose some custom hardware which doesn't use logic at all might help, but then Gödel's theorem wouldn't apply at all. (Likewise with modal logic.)