Should Wittgenstein be given partial credit for Gödel’s incompleteness theorem?

Question

1. Is there a connection between Wittgenstein's argument against the theory of types, and the proof of Gödel's incompleteness theorem? (It seems that Gödel's proof relies on referring to symbols as numbers, whereas Wittgenstein's argument is that you can do no such thing in Russel's theory of types.)

2. Was it really Wittgenstein's argument against type theory that changed logical positivism so drastically? If not, then what was it that happened before Gödel's proof which stopped analytic philosophy from thinking that it could axiomatize language? (It seems clear that Gödel's incompleteness theorem would have stopped this project in its tracks.)

3. Should Wittgenstein be given some credit for Gödel's incompleteness theorem?

Answer 1

1. Yes, there is a connection, as you point out. In the Tractatus, Wittgenstein writes:

   3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

   Gödel, as you know, proceeded to do precisely that.

2. Wittgenstein's argument against type theory is one of many factors that changed Logical Positivism. Russell's "barber paradox" was another. If the history of Logical Positivism interests you, I'd recommend a delightful graphic novel called Logicomix which covers the territory nicely.

3. Not really.

Answer 2

No.

Kurt Godel was interested in philosophy his whole life, his 2 favorite philosophers were Kant and Husserl. There is no historical evidence that his incompleteness theorems were influenced by Wittgenstein. I have read Godel's incompleteness academic papers, and Wittgenstein was not mentioned even once.

As a young man Godel studied a lot of elementary number theory, Russell's "Principia", and the theories of Peano and Frege. Later on in life he made several scathing remarks about Wittgenstein, saying that the latter "lost his mind". Wittgenstein once called incompleteness theorems liars paradox, and Godel got outraged by that remark because he thought that incompleteness theorems are just results in elementary number theory, having nothing to do with ancient greek sophistry.

Answer 3

In reality, Gödel fails to construct a sentence that can be interpreted as self-referential. When attempting self-reference, it enters a loop, because the substitution leaves another substitution pending, in a return to infinity. But since the trick of arithmetization is in the middle, it makes this failure unprovable in the formal system. But that does not make it a "true" statement in any scientifically important sense. If instead of arithmetization we use the English translation from Spanish, we would have:

La auto sustitución de { The self-substitution of x it is not demonstrable } no es demostrable

We have an unresolved self-substitution, so the statement is incomplete. But if the system does not have rules for interpreting English, it remains unproven, but it is still a mere malformation due to infinite regression.