Gödel's Proof

So I read this 1958 book which, according to a philosophy professor, is a must in any philosopher's library. It is concise and what's more it explains technical language, as it pertains to getting the gist of Gödel's paper, for a non-expert. The first few chapters explain the context in which Gödel, only 25 at the time, published his "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." In that paper Gödel proved that the axiomatic method had inherent limitations and, as the authors of Gödel's Proof Nagel and Newman indicate in the introduction to this book, Gödel proved that "it is impossible to establish the internal logical consistency of a very large class of deductive systems unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves." Indeed, this "internal logical consistency" refers to the fact that a proposition, G, and its formal negation, ~G, are formally demonstrable. If a formula and its formal negation are demonstrable it means that the axioms of that system are not consistent. Conversely, if the axioms of a system are consistent then neither the formula G nor its formal negation, ~G, are demonstrable. In short, Gödel undermined the axiomatic method. It proved that there are logical truths that cannot be demonstrated using an axiomatic method. Even if the axioms were extended so as to include the formal demonstration of a particular logical truth, there would still be other logical truths that the axioms did not cover.

What struck me as peculiarly interesting are the concluding comments by the authors in the last chapter. They give consideration to the question of whether there could ever be "calculating machines" powerful enough to match the human brain in mathematical intelligence. They acknowledge that there are some mathematical problems that are solvable by a computing machinery which are not solvable by a human beings. However, they go on, the human brain appears to embody a "structure of rules of operation" that far exceeds the structure of artificial machines. There doesn't appear to be a prospect for the replacement of human minds by robots. While these AI speculations, it must be said, only occupy a page or two in the entire 115-page book, they only serve to lead to the admonition that Gödel's paper is no excuse to start despairing. Instead, Gödel's finding "is an occasion...for a renewed appreciation of the powers of creative reason."