I think that it has a lot to do with embodied cognition [0]. But every mathematician I've explained that too find the idea repugnant.

[0] http://www.amazon.com/Where-Mathematics-Come-From-Embodied/d...

Anyone interested in this might also like Lakoff's "Where Mathematics Comes From," which I read after seeing a Bret Victor recommendation of it:

http://www.amazon.com/Where-Mathematics-Come-From-Embodied/d...

Absolutely fascinating book.

GEB was a considerable waste of time and contributed nothing to my understanding of intelligence or AI. The time would have been be better spent elsewhere.

If you want to understand Godel's proofs then I recommend the book "Godel's Proof" by Ernest Nagel and James R. Newman:

http://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/081475837...

Instead of Hofstadter's GEB, read some of his papers, e.g., "Analogy as the Core of Cognition" http://prelectur.stanford.edu/lecturers/hofstadter/analogy.h...

But there are others who have focused longer on analogy, e.g., George Lakoff:

"Metaphors we Live by"

http://www.amazon.com/Metaphors-We-Live-George-Lakoff/dp/022...

"Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being":

http://www.amazon.com/Where-Mathematics-Come-Embodied-Brings...

"Women, Fire, and Dangerous Things"

http://www.amazon.com/Women-Fire-Dangerous-Things-Lakoff/dp/...

Everything. But we clean up the incorrectness, sweep any inconsistencies under the rug and then publish the corrected proof as if it sprang, fully-formed, from our mind. This, much to the bewilderment and bafflement of students thereafter! mwahahaha! http://upload.wikimedia.org/wikipedia/commons/8/84/Evillaugh...

If your viewpoint is the history of mathematical proof, then the answer might be "Everything up to the early Greeks." Here's a nice link: "The History and Concept of. Mathematical Proof" by Steven G. Krantz http://www.math.wustl.edu/~sk/eolss.pdf

But if you want to really understand then take a look at the book

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being by G. Lakoff & R. Núñez. http://www.amazon.com/Where-Mathematics-Comes-Embodied-Bring...

The introduction and first four chapters [PDF] are available at

http://www.cogsci.ucsd.edu/~nunez/web/INTR-04.PDF

Before digging into the various books others have suggested, you would do well to read "Where Mathematics Comes From" by George Lakoff and Rafael Nunez: http://www.amazon.com/Where-Mathematics-Comes-Embodied-Bring...

That book explains the origins and understanding of the basic items of mathematical analysis: infinity, sets, classes, limits, the epsilon-delta of calculus and alternatives, infinitesimals, etc. The explanation is from the viewpoint of psychological understanding. It details how we build up a scaffolding of tools (starting with basic counting) sufficient to slay the dragons of modern physics and mathematics.