Georg Cantor
Founder of set theory and transfinite numbers
Georg Cantor (1845–1918) was a mathematician who helped create set theory, now a core part of mathematics. He showed that matching items from two sets one-to-one was key, defined what infinite and well-ordered sets are, and demonstrated that there are more real numbers than natural numbers. His proof also implies there are infinitely many different sizes of infinity. He introduced cardinal and ordinal numbers and how to do arithmetic with them. Cantor knew his work had deep philosophical meaning.
Originally, his ideas about transfinite numbers seemed strange and shocking. Many mathematicians resisted them, including Leopold Kronecker, Henri Poincaré, and later Hermann Weyl and L. E. J. Brouwer. The philosopher Ludwig Wittgenstein also raised objections. Cantor, a devout Lutheran, believed God had revealed the theory to him. Some Christian theologians, especially neo-Scholastics, saw it as challenging God’s unique absolute infinity, even calling it pantheism—which Cantor strongly denied. But not all theologians opposed him; a prominent neo-Scholastic philosopher supported it, and Cardinal Johann Baptist Franzelin accepted it after Cantor clarified certain points.
The opposition could be fierce. Kronecker publicly attacked Cantor, calling him a “scientific charlatan,” a “renegade,” and a “corrupter of youth.” He objected to Cantor’s proofs that algebraic numbers are countable and transcendental numbers are uncountable—results now standard in math. Decades after Cantor’s death, Wittgenstein complained that mathematics was “ridden through and through with the pernicious idioms of set theory,” dismissing it as “utter nonsense,” “laughable,” and “wrong.” Cantor suffered from depression from 1884 onward, which some blame on this hostility, though others suggest bipolar disorder.
Later, Cantor received high honors. In 1904, the Royal Society gave him its Sylvester Medal, its top math award. David Hilbert defended him, saying, “No one shall expel us from the paradise that Cantor has created.”
Cantor was born on March 3, 1845, in Saint Petersburg, Russia, and lived there until age eleven. The oldest of six children, he was an excellent violinist. His father, Georg Waldemar Cantor, was Danish and had fled to Russia during the Napoleonic Wars; his mother, Maria Anna Böhm, was Austro-Hungarian and born in Saint Petersburg. The family was Lutheran, though his mother was raised Catholic. He said his paternal grandparents belonged to Copenhagen’s Portuguese Jewish community. His maternal grandfather, Franz Böhm, was a well-known musician and soloist in a Russian imperial orchestra.
His father worked as a stockbroker in Saint Petersburg until illness in 1856 forced the family to move to a milder climate. They went to Germany, first to Wiesbaden, then Frankfurt. In 1860, Cantor graduated with distinction from a Realschule in Darmstadt, noted for his math skills, especially trigonometry. In August 1862, he graduated from the Höhere Gewerbeschule Darmstadt (now Technische Universität Darmstadt). He then entered the Swiss Federal Polytechnic in Zurich. After his father died in June 1863, leaving a large inheritance, Cantor transferred to the University of Berlin, where he attended lectures by Leopold Kronecker, Karl Weierstrass, and Ernst Kummer. He spent summer 1866 at the University of Göttingen, then a major math center. He was a good student and earned his doctorate in 1867.
Cantor submitted his dissertation on number theory at Berlin in 1867. After briefly teaching at a girls’ school in Berlin, he took a position at the University of Halle, where he stayed his whole career. He earned his habilitation with a thesis on number theory in 1869.
In 1874, he married Vally Guttmann. They had six children, the last born in 1886. His father’s inheritance let him support his family despite modest academic pay. On his honeymoon in the Harz mountains, he spent much time discussing math with Richard Dedekind, whom he had met two years earlier in Switzerland and with whom he had a lively correspondence.
Cantor became an extraordinary professor in 1872 and a full professor in 1879. Reaching that rank at 34 was notable, but he wanted a chair at a more prestigious university, especially Berlin, then Germany’s leading school. His work faced too much opposition for that. Kronecker, who led math at Berlin until his death in 1891, grew uncomfortable with Cantor as a colleague, seeing him as a “corrupter of youth” for teaching his ideas to younger mathematicians.
- born
- 3 March 1845 (O.S. 19 February 1845) in Saint Petersburg, Russian Empire
- died
- 6 January 1918
- field
- Mathematics
- nationality
- German (born in Russian Empire to Danish and Austro-Hungarian parents)
- known_for
- Creation of set theory, transfinite numbers, cardinal and ordinal numbers, proof that real numbers outnumber natural numbers
Lore & Background
Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Konstantin Gutberlet was in favor of it, and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a 'scientific charlatan', a 'renegade' and a 'corrupter of youth'. Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is 'ridden through and through with the pernicious idioms of set theory', which he dismissed as 'utter nonsense' that is 'laughable' and 'wrong'. Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder. The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal. David Hilbert defended it from its critics by declaring, 'No one shall expel us from the paradise that Cantor has created.'
Reader's Guide
Cantor was born on March 3, 1845, in Saint Petersburg, Russian Empire, and was brought up in the city until the age of eleven. The eldest of six children, he was regarded as an outstanding violinist. His father, Georg Waldemar Cantor, was Danish, but fled with his family to Russia during the Napoleonic Wars; his mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg. Cantor's family was Lutheran, though his mother was raised Catholic. He described his paternal grandparents as having belonged to the Portuguese Jewish community of Copenhagen. His maternal grandfather Franz Böhm was a well-known musician and a soloist in a Russian imperial orchestra. Cantor's father worked as a broker in the Saint Petersburg stock exchange until an illness in 1856, which forced the family to seek out a more temperate climate. They moved to Germany, first to Wiesbaden, then to Frankfurt. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the 'Höhere Gewerbeschule Darmstadt', now the Technische Universität Darmstadt. In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen. Cantor was a good student, and he received his doctoral degree in 1867. Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle. In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on holiday in Gersau in Switzerland, and with whom a lively mathematical correspondence had develop
Did You Know?
- Cantor was an outstanding violinist.
- He believed his theory of transfinite numbers was communicated to him by God.
- The Royal Society awarded Cantor the Sylvester Medal in 1904.
- Cantor's father died in June 1863, leaving him a substantial inheritance.
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
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