This was the man that set out to give us a complete/consistent mathematics. We know of this man who shattered his dreams by his incompleteness theorem. We know that this man demands a new kind of science. We also know of his friend who is an authority on the limits of mathematics.
What has all of this led us to? Are proofs of theorems absolute? Is Math certain of what it is saying? How should Math be done? Read this article from the Seed Magazine.
To me, a layman interested in the historical development of mathematics and philosophy, all this sends two signals.
1. Math is not absolute. (We sort of knew this by intuition, don’t we? ;))
2. Math is getting more and more difficult to do. (I am not a mathematician. So it is just an impression from whatever I have read)