alt Hacker NewsAside 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 determine its own boundaries. A safety strategy cannot verify its own completeness.
None of this is softened by the fact that a neural network feels organic rather than rule-like. A model’s weights are numbers, and its training is arithmetic, all of it running on von Neumann’s realisation of Turing’s imaginary device. AI is not adjacent to this mathematics. AI is made of it.
The AI industry, understandably, would rather not dwell on this. "
This is a complete non-sequitur. Godel's argument isn't even that profound: it just says that a system of axioms good enough to embed standard natural number arithmetic must be independent of a certain constructed sentence, i.e., it will run the same whether that sentence is true or false and you can't tell from its axioms which of these two options is in effect. While that sucks for philosophers, it doesn't make mathematics less useful, less dependable, or less correct in a practical sense. It has no practical implications to anything that the AI industry is promising today.
Replies
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!"
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.
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.
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.