This NYT Connections column has the most succinct description of Godel's most famous work that I recall reading:
Before Godel's incompleteness theorem was published in 1931, it was believed that not only was everything proven by mathematics true, but also that within its conceptual universe everything true could be proven. Mathematics is thus complete: nothing true is beyond its reach. Gdel shattered that dream. He showed that there were true statements in certain mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion that was both true and unprovable.But Godel's genius came with a price:
[jf: actually if memory serves a better statement might be: "He showed that there were true statements in certain nontrivial and interesting mathematical systems that could not be proven. And he did this with astonishing sleight of hand, producing a mathematical assertion that was both true and unprovable."]
... those leaps and connections could go awry. Godel was an intermittent paranoiac, whose twisted visions often left his colleagues in dismay. He spent his later years working on a proof of the existence of God. He even died in the grip of a perverse esotericism. He feared eating, imagined elaborate plots, and literally wasted away. At his death in 1978, he weighed 65 pounds.Genius, connections, intuition, courage, fascination with the infinite, then madness.
Thomas Nash, a nobel prize winner disabled for many years by paranoid schizophrenia comes to mind, but Godel was old for the onset of schizophrenia. What do we know, however, of the psychiatric disorders of genius? We are much more familiar with more conventional minds. I think also of Isaac Newton, who spent the latter half of his life wrapped up in Alchemy. Linus Pauling, who's powerful but misdirected intuition made him a peculiar pusher of vitamin C.
These extraordinary minds excelled at making connections and drawing inferences, at rethinking and radical leaps. Is the price of such excellence a predisposition to leaps beyond the bounds of reason?
"Whom the gods would destroy they first make mad".