Georg Cantor

Lore & Background

Cantor was born in Saint Petersburg, the eldest of six children, and moved with his family to Germany in 1856. He studied at the Swiss Federal Polytechnic in Zurich and later at the University of Berlin, where he attended lectures by Leopold Kronecker, Karl Weierstrass, and Ernst Kummer. After receiving his doctorate in 1867, he taught briefly at a Berlin girls' school before taking a position at the University of Halle, where he spent his entire career. He married Vally Guttmann in 1874 and had six children.

Cantor's theory of transfinite numbers was regarded as counter-intuitive and encountered resistance from contemporaries such as Leopold Kronecker, who publicly described him as a 'scientific charlatan' and 'corrupter of youth,' and from Henri Poincaré, Hermann Weyl, L. E. J. Brouwer, and Ludwig Wittgenstein. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians saw his work as a challenge to the uniqueness of absolute infinity, which Cantor vigorously rejected, though others like Konstantin Gutberlet and Cardinal Johann Baptist Franzelin accepted it.

Cantor suffered his first known bout of depression in May 1884, and recurring episodes continued to the end of his life, which some have attributed to the hostile attitude of contemporaries or to probable bipolar disorder. He curtailed his relationship with Gösta Mittag-Leffler after Mittag-Leffler asked him to withdraw a paper from Acta Mathematica, writing that it was 'about one hundred years too soon.' Cantor also studied Elizabethan literature, publishing pamphlets on the possibility that Francis Bacon wrote Shakespeare's plays.

Reader's Guide

Georg Cantor's work fundamentally reshaped mathematics by establishing set theory as a foundational discipline. His proof that the real numbers are more numerous than the natural numbers introduced the concept of different sizes of infinity, leading to the development of transfinite numbers and the arithmetic of cardinal and ordinal numbers. Despite fierce opposition from prominent mathematicians like Leopold Kronecker and later philosophers like Ludwig Wittgenstein, Cantor's ideas eventually became central to modern mathematics. The Royal Society awarded him the Sylvester Medal in 1904, and David Hilbert famously defended his work, declaring that no one should expel mathematicians from 'the paradise that Cantor has created.' Cantor's legacy includes not only his mathematical contributions but also the philosophical debates they sparked about the nature of infinity, with some theologians challenging and others accepting his theories. His personal struggles with depression, possibly exacerbated by professional opposition, add a human dimension to his story, but his ideas have endured and are now standard in mathematics curricula worldwide.

Did You Know?

• Cantor was an outstanding violinist as a child.
• He received a substantial inheritance upon his father's death in June 1863, which allowed him to transfer to the University of Berlin.
• Cantor's recurring bouts of depression from 1884 onward have been explained by some as probable manifestations of a bipolar disorder.
• He believed his theory of transfinite numbers had been communicated to him by God.

Notable Quotes (Wikiquote)

"In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi." — Georg Cantor "In mathematics the art of asking questions is more valuable than solving problems." — Georg Cantor "Doctoral thesis (De aequationibus secundi gradus indeterminatis, 1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it." — Georg Cantor "I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers." — Georg Cantor