Kurt Godel published the "incompleteness theorem" in 1931.

Douglas Hofstadter, author of 'Godel, Escher and Bach' wrote about him in Time magazine's March 29, 1999 issue,  'The Century's Greatest Minds'.
He describes the time before Godel as a striving for formalism,  where "the beauty of mathematics was that it eliminated all need for thought or judgment."   Godel changed that.  He explains that "the familiar methods of mathematical analysis could be brought to bear on the very pattern-sprouting processes that formed the essence of formal systems - of which mathematics itself was supposed to be the primary example.  Thus mathematics twists back on itself, like a self-eating snake."
And summarizes the theorem:  "All formal systems turn out to be incomplete because they are able to express statements that say of themselves that they are unprovable.  It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules. 

To you that might not come as a shock, but to mathematicians in the 1930's, it upended their entire world view, and math has never been the same since."
Another Ritalin candidate, if he were alive today, Godel "was every bit as eccentric as his theories.  He and his wife Adele, a dancer, fled the Nazis in 1939 and settled at the Institute for Advanced Study ain Princeton, where he worked with Einstein.  In his later years, Godel grew paranoid about the spread of germs, and he became notorious for compulsively cleaning his eating utensils and wearing ski masks with eye holes wherever he went.  He died at age 72 in a Princeton hospital, essentially because he refused to eat.  Much as formal systems, thanks to their very power, are doomed to incompleteness, so living beings, thanks to their complexity, are doomed to perish, each in its own unique manner."

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