Indeed. Godel's theorem is very technical, and any use outside the very technical realm of its immediate application should be viewed with great suspicion.
For example, if you take the statement you quoted from Wikipedia and replace "natural number" with "real number", it doesn't work anymore: it's been proven that the arithmetic of real numbers is decidable[1]. That means that the sentence you quoted from OP's comment is not true.
Anyone inclined to use Godel's theorem in these philosophical contexts should maybe read the great little book "Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse" by Torkel Franzén. I'll leave here a quote from a review[2]:
In addition to obvious nonsense, there are among the nonmathematical ideas inspired by Gödel’s theorem many that by no means represent postmodernist excesses, but rather come to mind naturally to many people with very different backgrounds when they think about the theorem. It is especially such naturally occurring misunderstandings that Franzén intends to correct.
It was Brin, not me, who makes the connection, and says that Godel refutes Hegel. The scholar I mentioned, JN Findlay, has a rigorous understanding of both authors, but I couldn't quickly find an article where he makes the argument. Nothing to do with "postmodernist excesses" or whatever.
Also read my comment, see this article here[0] about how Godel adopted phenomenology, which is the philosophical backbone of much of "postmodernism," so it would be entirely fair to make a connection between Godel and, say, Derrida, for instance, since they both claim to be in the same philosophical tradition. But that's just what the scholarly evidence suggests.
In any case, Godel's proof has little to do with "math" in the sense of calculation but rather is a refutation of Russel & Whiteheads attempts at a logical foundation of mathematics, which is a philosophical endeavour. The mathematical aspect is secondary and merely follows from the philosophical argument which it entails. It is the simply the case that, Russel & Whitehead were themselves engaging with "Hegel" in Principia Mathematica, who of course had his own system of logic (cf. the Science of Logic), but they failed insofar as Godel's critique is accepted, and insofar as you accept Godel's critique you could make the inference (though by no means on an entirely solid basis) that Godel's work constitutes, in a certain sense, a re-interpretation of Hegel, though not directly.
> Nothing to do with "postmodernist excesses" or whatever.
To clarify, I wasn't implying you or Brin were commiting "postmodernist excesses". The part of the review I quoted was explicitly saying that the book aims to correct misunderstandings that "by no means represent postmodernist excesses, but rather come to mind naturally to many people with very different backgrounds".
It's easy to see Derrida as a bullshitter who doesn't understand the texts that he abuses but hard to make the criticism stick because Derrida himself is unclear and hard to read.
Badiou clearly does understand the math that he's abusing and you can't find anything really wrong with it except for the idea that anyone would care about Marxism when we know so much more about the science of civilization now. Many people come to the conclusion that Badiou is a bullshitter, but if he is a bullshitter he's much more rigorous in terms of working within the systems he works with and also much more clear in his exposition in that you really can follow what he says.
If Badiou derives atheism from the topless nature of Nature (no universal set/set of all sets), perhaps I should try to interest him in Algebraic Theology? https://news.ycombinator.com/item?id=41258636
— Herschel, they say you don't believe in G*d?
— What? Who says that?
— You know, people; lots of people are saying it.
— People? People say all kinds of things; you know better than that. Why didn't you just ask G*d directly if I believe in H*m or not?
I’m no Badiouian myself, however Derrida and Badiou are no more difficult to read than Kant and Hegel. Just because its not easy to read doesn’t mean its not worth your time to read.
For example, if you take the statement you quoted from Wikipedia and replace "natural number" with "real number", it doesn't work anymore: it's been proven that the arithmetic of real numbers is decidable[1]. That means that the sentence you quoted from OP's comment is not true.
Anyone inclined to use Godel's theorem in these philosophical contexts should maybe read the great little book "Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse" by Torkel Franzén. I'll leave here a quote from a review[2]:
[1] https://en.wikipedia.org/wiki/Decidability_of_first-order_th...[2] https://www.ams.org/notices/200703/rev-raatikainen.pdf