I don't consider Godel's theorem to be philosophy. Its status is similar to Cantor's theorem stating that reals are uncountable. Surely is inspired by philosophy and has philosophical consequences, but it is a part of mathematics. Trivia: Godel used the Chinese remainder theorem in his proof http://mathoverflow.net/questions/19857/has-decidability-got.... I acknowledge that philosophy can be inspiration for mathematics, but this is rather unsatiating, as very many things can be.

In the meantime, I found a very good defense of philosophy here: http://www.ditext.com/russell/rus15.html