This blog entry will be the start of trying to do that, beginning with a quick review of a book on Godel's theory by Margerat Goldstein.
"Incompleteness: The Proof and Paradox of Kurt Gödel"
Kurt Godel is the kind of guy you think of when you think of a mad mathematician. He's reclusive, uncomfortable in public and not at all patient when dealing with people with less intelligence, which is almost everybody. One of his best friends in life was Einstein, who he often walked with while at Princeton. Indeed, Einstein was one of his few intellectual equals, and he once surprised the physicist on his birthday with a reformulation of his relativity theory that allowed for time travel. Einstein also drove Godel to his citizenship test in the US, where Godel worried him by pointing out he had found a flaw in the constitution of that country that could make the whole government illegal. Despite Einstein's warning not to, Godel illucidated this theory of his to the examiner in his test. Luckily it did not count against him and he became a US citizen.
This book explores the beginning of his intellectual life in an elitist intellectual club known as the Vienna Circle. The story then discuess his move to Princeton before the second world war. Although not Jewish, he was eligible for conscription to the German army, and through being stubborn and difficult he very nearly did not make it to the US to work there.
An amusing part of the story is when Godel presented his paper at a European conference and it was almost completely ignored. Only a single person there understood the great significance of the theory; John von Neumann.
Although a book like this can't hope to teach all the mathematics you need to understand a complex paper like the two proofs of incompleteness, from my layman view it certainly seemed to present a tangible feel for how Godel developed the proof, and what it consisted of.
People looking for a deeper understanding may want to check out "Godel, Escher Bach: An Eternal Golden Braid" which I will talk about in another post.
The Vienna Circle represented the philosophy that experience is the only valid source of knowledge, and that only formal logic can be used to manipulate this knowledge to study philisophy. Although a member of the group, Godel did not share their views. Although his proof is a magnificent piece of work in formal logic, what he achieved was to show that not everything we can know can be produced in such a system. His later life was spent studying Liebniz, looking for clues in the work of that great genius, for how human knowledge differed from knowledge that was entirely empirical or generated mechanically from a formal system.
Sadly he became very paranoid in later life and shut out everybody, eventually starving himself to avoid being poisoned by his imagined enemies. This paranoia was no doubt caused by some mental illness, but seemed to reflect the the intellectual loneliness he suffered both in Vienna and in Princeton.
To summarise Goldstein takes us on a tour that explores Godel's life in a way that is very accessible and highly interesting, yet still apparently with plenty of depth.