Three crises in mathematics

Brian Davies' makes the argument in his paper Whither Mathematics that


... developments of the classical Greek view of mathematics do not adequately represent current trends in the subject. It proved remarkably successful for many centuries, but three crises in the twentieth century force us to reconsider the status of an increasing amount of current mathematical research


The three crises stem from Godel's Incompleteness Theorem and proofs which are too long and complex for mathematicians to be confident that they are actually correct. Davies also discusses a bit about the use of computers in mathematics and the formal verfication of proofs. The paper is short and very readable.

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