First, a short quote that makes me feel better about my past failures at REALLY understanding the proof of GIT:

"The details of Gödel's proof in his epoch-making paper are too difficult to follow without considerable mathematical training." (page 7)Nevertheless, the book aims to convey "the basic structure of his demonstrations" to "readers with very limited mathematical and logical preparation." Check!

The bulk of the introduction discusses, in very general terms, deductive reasoning, axiomatic systems, and logical proofs. Since we'll have many occasions to go into these topics in great depth over the next few months, I won't go into it now. In fact, I'll devote my next post to formal systems.

Going back to the introduction, it makes several strong claims that I want to investigate after I master the proof, namely:

- GIT are "revolutionary in their broad philosophical import." I definitely want to read about the philosophy of mathematics. But aren't there big claims made about the impact of GIT on the philosophy of mind, artificial intelligence, etc.? Controversial stuff, no doubt!
- GIT "undermined deeply rooted preconceptions and demolished ancient hopes that were being freshly nourished by research on the foundations of mathematics." I assume this is a reference to formalism and David Hilbert's program.
- GIT "introduced into the study of foundation questions a new technique of analysis comparable in its nature and fertility with the algebraic method that René Descartes introduced into geometry." I wonder what "new technique" the authors are referring to here. Gödel numbering or some other aspect of the proof?

As far as #1 goes I know that Roger Penrose was convinced at one time that GIT implied that human minds were not "algorithmic" in that they could not be modeled as turing machines. He wrote two books on the subject - Shadows of the Mind (https://en.wikipedia.org/wiki/Shadows_of_the_Mind) was one of them.

ReplyDeleteAt some point he gave up on this position, I'm not sure why.

Hey Nick,

DeleteAs it turns out, I own a copy of Penrose's earlier book "The Emperor's New Mind" but I never finished it. I'll give it another try after I am "done" with the proof of GIT.

Would you happen to have a reference for where he talks about his change of mind?