Georg Cantor (1845-1918)
Of all the great mathematicians, Cantor most perfectly fulfils the (Hollywood) stereotype that a genius for maths and mental illness are somehow inextricable. Cantor's most brilliant insight was to develop a way to talk about mathematical infinity.
His set theory lead to the counter-intuitive discovery that some infinities were larger than others.
The result was mind-blowing. Unfortunately he suffered mental breakdowns and was frequently hospitalized. He also became fixated on proving that the works of Shakespeare were in fact written by Francis Bacon.
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