A Conversation for Hilbert's Infinite Hotel
Hilbert, Godel, & Wittgenstein
Steve K. Started conversation Jun 13, 2005
Good article. I had to take a lot of math in college, but it was of the "applied" variety for my engineering major. My wife, OTOH, was a math major. Her textbooks might as well have been in Klingon as far as I was concerned. 
I happen to be reading "Incompleteness: The Proof and Paradox of Kurt Godel" by R. Goldstein. Both Hilbert & Wittgenstein are key figures. An excerpt:
QUOTE
Hilbert certainly wanted to avoid the extreme limitation that the intuitionists would place on mathematical practice ... he spoke of "completely clarifying the nature of the infinite" ...
... if it could be shown that such a formal system was both complete, allowing us to prove all arithmetical truths, as well as consistent, the linchpin of the Hilbert program would have been secured ...
Enter Godel.
END QUOTE
Godel's Incompleteness Theorem (there are actually two of them) sank Hilbert's ship. Hilbert eventually reconciled himself to this 
, but Wittgenstein never did.
QUOTE
[Wittgenstein] acknowledged the incompatibility [between his and Godel's views on the foundations of mathematics] and countered that Godel therefore could not have proved what he thought he had proved.
...
[Wittgenstein in the Tractatus] had spoken of necessary silence. Godel, one suspects, would have liked that silence to envelope the philosopher himself.
END QUOTE
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