TITLE: Failures that lead eventually to new mathematics
QUESTION [23 upvotes]: Possible Duplicate:
Most interesting mathematics mistake? 

In the 25-centuries old history of Mathematics, there have been landmark points when a famous mathematician claimed to have proven a fundamental statement, then the proof turned out to be false or incomplete, and finally  the statement became a nice problem playing  an important role in the development of Mathematics. Two immediate examples come to me:

Fermat's last theorem,
the existence of a minimizer of the Dirichlet integral in calculus of variations, which led Weierstrass to introduce the notion of compactness.

This must have happened in almost all branches of Mathematics.

What are the best examples of such an evolution? How did they influence our every mathematical life?

REPLY [9 votes]: In the early 1960's Smale published a paper containing a conjecture whose consequence was that (in modern language) "chaos didn't exist". He soon received a letter from Norman Levinson informing him of an earlier work of Cartwright and Littlewood which effectively contained a counterexample to Smale's conjecture. Smale "worked day and night to resolve the challenges that the letter posed to my beliefs" (in his own words), trying to translate analytic arguments of Levinson and Cartwright-Littlewood into his own geometric way of thinking. This led him to his seminal discovery of the horseshoe map, followed by the foundation of the field of hyperbolic dynamical systems. For more details, see Smale's popular article "Finding a horseshoe on the beaches of Rio", Mathematical Intelligencer 20 (1998), 39-44.