Georg Cantor

Georg Cantor (around 1910)

Georg Ferdinand Ludwig Philipp Cantor (* February 19 . Jul / In Saint Petersburg , † 6. January 1918 in Halle an der Saale ) was a German mathematician . Cantor made important contributions to modern mathematics . In particular, he is the founder of set theory and changed the concept of infinity . The revolutionary content of his work was only properly recognized in the 20th century.

Life

Georg Cantor (approx. 1870)

Cantor was the son of Georg Woldemar Cantor, a wealthy businessman and stockbroker , and Marie Cantor, nee. Böhm was born in St. Petersburg, the capital of Russia at the time . His father was born in Copenhagen and had come to St. Petersburg with his mother at a young age, where he had been raised in the German Lutheran mission there. Georg Cantor's statements that his father came from a Sephardic family and was only baptized Lutheran in Saint Petersburg can be supplemented as follows: The son, born on May 6, 1814 to the Jewish couple Lipman and Esther Cantor in Copenhagen, was given the name Hirsch and became Christened Georg Woldemar at a previously unknown date. The tea and china dealer Lipman Jacob Cantor married Esther, née Meyer, widowed Levy, in 1811. Lipman Cantor belonged to the Portuguese-Jewish community, but was very likely a descendant of Abraham Cantor from Hildesheim, who immigrated to Copenhagen around 1680. Georg Cantor's mother was born in St. Petersburg, a Roman Catholic , and came from a well-known Austrian family of musicians. The maternal grandparents, Franz Böhm and Marie Böhm, geb. Morawek, were both professional musicians (violinists), Franz Böhm was Kapellmeister of the Imperial Opera in Saint Petersburg and the brother of the violinist Joseph Böhm .

The children were raised in the Lutheran faith and in a German cultural environment. The father was very pious and instructed his son in religious matters. Georg Cantor remained a deeply religious person throughout his life. He attended elementary school in Saint Petersburg. When he was 11 years old, because of the poor health of his father, the family moved from St. Petersburg to the milder climate of the spa town of Wiesbaden in 1856 and a little later to Frankfurt am Main .

After graduating from secondary school in Darmstadt in 1860 ("with distinction"), he switched to the Darmstadt Higher Trade School, today's Technical University of Darmstadt . At the request of his father, he began vocational training for engineers there. In 1862 he managed to convince his father that his strengths lay more in mathematics, and he began to study mathematics at the Polytechnic in Zurich . In 1863 he moved to the university in Berlin. In 1866 he attended the University of Göttingen for a summer semester and in 1867 received his doctorate at the University of Berlin under Ernst Eduard Kummer . Karl Weierstrass , Ernst Eduard Kummer and Leopold Kronecker were among his teachers . Immediately afterwards he started working as a mathematics teacher at the Friedrich-Wilhelm-Gymnasium Berlin. At that time, he was already suffering from depression. After his habilitation in 1869 at the University of Halle on the subject of De transformatione formarum ternarium quadricarum , Cantor taught and worked in Halle until the end of his life , initially as a private lecturer, from 1872 as associate professor and from 1877 until his retirement in 1913 as a full professor. In Halle he was on friendly terms with Edmund Husserl , the founder of phenomenology .

In 1870 he succeeded in solving the mathematical problem of representing a function as the sum of trigonometric series . From 1872 further work on trigonometric series followed and in 1873 the proof that rational numbers are countable and that there is exactly one rational number for every natural number. In the following year he was able to draw the reverse conclusion that real numbers cannot be counted. In doing so, he also proved that almost all numbers are transcendent .

In 1874 he married Vally Guttmann, with whom he had two sons and four daughters (the last child was born in 1886). The son Erich was a doctor, the daughter Else a concert singer and well-known music teacher. He spent his honeymoon in the Harz Mountains , where he was also able to discuss mathematics intensively with Richard Dedekind , a close friend he had met two years earlier while on vacation in Switzerland . In the same year he continued his publications on set theory with "About a property of the epitome of all real algebraic numbers". In 1877 he dealt with geometrical applications of set theory, for example whether a square with side length 1 contains as many elements as the line between 0 and 1. Although he originally assumed that it was not so, he was himself about his own Discovery and evidence surprised. “I see it, but I don't believe it,” he wrote himself. This had a major impact on previous geometrical views. The treatises he wrote on this and sent to Crelles Journal for publication were withheld by his former teacher Leopold Kronecker, who was a representative of finitistic mathematics, was skeptical of the concept of infinity and developed into an influential opponent of Cantor's set theory. Only the intervention of his friend Dedekind led to the publication. From 1879 he developed further revolutionary ideas for set theory. Until 1884 he published a series of articles entitled “On infinite linear point manifolds” . In it he established the foundations and main theorems of set theory. Part 5 of the series deals with the "basics of a general theory of manifolds" .

The resistance to his mathematical ideas put a strain on Cantor and led to the fact that he left his mathematical field for almost ten years and dealt with literary historical research, philosophical and theological topics. This happened almost at the same time as the onset of his illness, which dominated him more and more in the second half of his life. From 1884 onwards, Cantor repeatedly suffered from manic-depressive illness and had to seek psychiatric treatment for the first time. Cantor's preoccupation with the question of the “true” author of Shakespeare's works occurred in the early days of his mental illness. He advocated Francis Bacon as the author of several publications . Cantor made similar discussions with regard to the works of Jakob Böhme and John Dee . This very forced commitment to literary history is often seen as a result of his mental illness, but participation in the guesswork around Shakespeare was generally very widespread, and Cantor always showed great interest in questions outside his field of expertise, especially philosophy and (Catholic) theology, which were for him was closely related to the set-theoretical problems of infinity.

In these ten years he received numerous honors and also saw the increasing appreciation of his previous mathematical knowledge. He became a member of the German Academy of Sciences Leopoldina and took an active part in the founding of the German Mathematicians Association , which took place in 1890. Cantor was elected first chairman. It was not until 1895 that he consistently took up his work on set theory again. He publishes the " Contributions to transfinite set theory ", deals with the continuum hypothesis and in 1897 attended the first international mathematicians' congress in Zurich .

A second stay in a sanatorium followed in 1899. Shortly thereafter, Cantor's youngest son died suddenly (during a lecture by Cantor on the Bacon theory and Shakespeare). This tragedy aggravated his depression and impaired his mathematical work, which is why he was treated again in a sanatorium in 1903.

A year later, Julius König gave a lecture at the 3rd International Congress of Mathematicians in Heidelberg, in which he was supposedly able to prove that the thickness of the continuum among the Alephs does not occur at all. This contradicted Cantor's continuum hypothesis . As a reaction to this lecture, which was felt to be “sensational” in its effect, Cantor is said to have been upset and indignant that they had dared to refute his study (according to his statement transmitted by God), but also about the fact that his daughters and colleagues had to overhear the alleged refutation and the associated humiliation carried out on him. Although Ernst Zermelo demonstrated just a day later that Julius König's argumentation was wrong, Cantor remained shocked, angry and even began to doubt his belief. (Regarding Cantor's reaction to König's lecture, the participants in the congress also gave different descriptions.)

In 1911 Cantor was invited as one of the preferred foreign scholars on the 500th anniversary of the founding of the University of St. Andrews in Scotland . At this time, Bertrand Russell published the famous Principia Mathematica with Alfred North Whitehead , in which Russell frequently referred to Cantor's work. In the hope of meeting Bertrand Russell on the occasion, Cantor attended the founding ceremony of St. Andrews, but a meeting did not take place. A year later, the same university wanted to confer an honorary doctorate on Cantor, but Cantor was unable to attend personally because of his illness.

Cantor retired in 1913 and suffered from poverty and malnutrition during the First World War . The public celebration of his 70th birthday was canceled due to the war. On January 6, 1918, Georg Cantor died of heart failure in Halle in the sanatorium where he had spent the last year of his life. His grave is preserved in the Giebichenstein cemetery in Halle.

His estate is kept by the Central Archives of German Mathematicians' bequests at the Lower Saxony State and University Library in Göttingen .

plant

Cantor initially dealt with number theory and turned to Fourier series in Halle under the influence of Eduard Heine . In 1869 he proved the uniqueness of the representation of functions by trigonometric series, published in the Journal for Pure and Applied Mathematics in 1870. More precisely, he proved that if

${\ displaystyle {\ frac {c_ {0}} {2}} + \ sum _ {k = 1} ^ {\ infty} (c_ {k} \ cdot \ cos (kx) + d_ {k} \ cdot \ sin (kx)) = 0.}$

for all that for all i. The theorem remains valid even with a finite number of exception points x (in which the Fourier series does not converge or is not equal to zero). ${\ displaystyle 0 \ leq x \ leq 2 \ cdot \ pi}$${\ displaystyle c_ {i} = d_ {i} = 0}$

When providing evidence, he built on the investigations of Bernhard Riemann and, in advance of the evidence, corresponded with his college friend Hermann Amandus Schwarz , who provided an important component of the evidence. The theory of the Fourier series was also the starting point of his preoccupation with set theory when he asked himself whether his uniqueness theorem would be preserved with an infinite number of exceptions.

Between 1874 and 1897 Cantor founded set theory , which he initially (1877) called the theory of manifolds. In 1895 he formulated the following often cited definition of the set :

"By a 'set' we mean any combination M of certain well-differentiated objects m of our perception or our thinking (which are called the 'elements' of M ) into a whole."

Cantor came to his set theory by considering unambiguous (today: "bijective" ) assignments of the elements of infinite sets. He described quantities for which such a relationship can be established as equivalent or “of equal power ”, also “equal power”. Accordingly, the set of natural numbers is equivalent to the set of rational numbers (fractions), which he showed by his diagonalization method. With his second diagonal argument he then proved that the set of real numbers is more powerful than that of natural numbers. A generalization was Cantor's theorem . The work was controversial among the mathematicians of his time because of the unanswered questions regarding the " actual infinite " and the introduction of transfinite numbers . In particular, Cantor came into profound scientific opposition to Leopold Kronecker . This is believed to be the reason for the delay in the publication of Cantor's article A Contribution to the Theory of Manifolds in Crelle's Journal . This controversy between Cantor and Kronecker is seen as a “prelude to the later dispute between intuitionists and formalists”. But Cantor also had early support from influential mathematicians, including David Hilbert , from whom the classic quote comes that Cantor had created a paradise from which no one could drive mathematicians (see also Cantor's Paradise and Henri Poincaré ). ${\ displaystyle \ {0, \, 1, \, 2, \, 3, \, 4, \, ... \}}$

Cantor himself was one of the first to discover the antinomies of naive set theory and proved with the two Cantor antinomies that certain classes are not sets. He can even be regarded as the creator of axiomatic set theory, because Cantor's set axioms from letters from 1889/99, which were only published posthumously, anticipate the axioms of the later Zermelo-Fraenkel set theory.

Cantor 's pairing function (also numbering function) goes back to Cantor .

Finally Cantor in 1870 created the so-called point set the foundations of the theory of the later of Benoît Mandelbrot so called fractals . The Cantor point set follows the principle of the infinite repetition of self-similar processes. The Cantor set is considered to be the oldest fractal ever.

Honors

Georg-Cantor-Gymnasium in Halle
Remembrance in Halle, Riebeckplatz

Fonts

• Georg Cantor: Collected treatises of mathematical and philosophical content. (With excerpts from the correspondence with Dedekind and Fraenkel's Cantor biography in the appendix.)

To number theory

• De aequationibus secundi gradus indeterminatis (dissertation).
• Two sentences from the theory of binary quadratic forms .
• About the simple number systems .
• Two propositions about a certain decomposition of numbers into infinite products .
• De transformatione formarum ternariarum quadraticarum (habilitation thesis).
• Algebraic note .
• On the theory of number theoretic functions .

To analysis

• About a theorem concerning the trigonometric series .
• Proof that a function f (x) given by a trigonometric series for every real value of x can only be represented in this form in one single way .
• On the expansion of a theorem from the theory of trigonometric series, 1872 .
• Note on trigonometric series .
• Further remark about trigonometric series .
• About a new and general condensation principle of the singularities of functions .
• Comment with reference to the article: On the Weierstrass-Cantor theory of irrational numbers .

To set theory

• About a property of the epitome of all real algebraic numbers .
• A contribution to the theory of manifolds, 1878 .
• About a theorem from the theory of continuous manifolds .
• About infinite linear point manifolds .
• Sur various théorèmes de la théorie des ensembles de point situés dans un espace continu an dimensions .
• De la puissance of the ensemble parfait de points .
• About various theorems from the theory of point sets in an n-fold extended continuous space Gn. Second communication .
• On an elementary question of the theory of manifolds, 1890/91 .
• Contributions to the foundation of the transfinite set theory 1895/1897 .

Others

• On the various points of view regarding the actual infinite, 1886 .
• Herbert Meschkowski (Ed.): Letters. Springer, Berlin 1991.

Movie

• Georg Cantor - The discoverer of infinity. Documentary, Germany, 2018, 44:13 min., Script and director: Ekaterina Eremenko , production: Saxonia Entertainment, MDR , first broadcast: March 4, 2018 in MDR, table of contents and online video by MDR. Among others with the mathematicians Felix Günther, Walter Purkert , Karin Richter, Galina Sinkevich, Eberhard Knobloch, Alexander Bobenko .

Commons : Georg Cantor  - collection of images, videos and audio files

Works

Biographies

Sources and Notes

1. ^ Letter from Cantor to Paul Tannery dated January 6, 1896, in which the question of the relationship with Moritz Cantor was concerned. (In: Anne-Marie Décaillot, Cantor und die Franzosen , Springer, 2011, p. 173.) Cantor was of the opinion that he was not related to Moritz Cantor. But the latter himself wrote in a letter to Tannery that he was also from the same Sephardic family, only from a branch that had gone to Amsterdam instead of Copenhagen. On the other hand, Purkert and Ilgauds agree (in: Georg Cantor 1845–1918 , Birkhäuser, Basel 1987, p. 15) that his paternal grandfather was not Jewish, despite research from various quarters.
2. Georg Singer: New Findings on the Descent of Georg Cantors. In: MAAJAN - The source: Yearbook of the Swiss Association for Jewish Genealogy. Volume 4 (= year 33; issue 119). Zurich, 2019. pp. 170–201.
3. heise online: 100th anniversary of the death of Georg Cantor: The master of quantities. Retrieved September 13, 2019 .
5. ^ Mathematics Genealogy Project
6. ^ Fröba, Wassermann, The most important mathematicians, marix Verlag 2012, section Georg Cantor
7. ^ Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 79 ff .
8. ^ A b Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 160 .
9. ^ A b Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 161 .
10. Georg Cantor: Proof that a function f (x) given by a trigonometric series for every real value of x can only be represented in this form in one single way . In: Journal for pure and applied mathematics . tape 72 , 1870, p. 139–142 ( uni-goettingen.de [accessed on July 5, 2013] digitized at the University of Göttingen).
11. ^ Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 34 .
12. David Foster Wallace: The Discovery of Infinity. 4th edition, p. 295 ff.
13. Contributions to the foundation of transfinite set theory. In: Mathematical Annals . Volume 46, p. 481.
14. ^ Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 51 ff .
15. ^ Walter Purkert, Hans Joachim Ilgauds: Georg Cantor 1845–1918 . 1987, p. 53 .
16. Nobody should be able to drive us out of the paradise that Cantor created for us , Hilbert, Über das Unendliche, Mathematische Annalen, Volume 95, 1926, p. 170, digitized version (from p. 161)
17. ^ Gazetteer of Planetary Nomenclature
18. Minor Planet Circ. 41573
19. ^ E. Zermelo (ed.): Collected treatises of mathematical and philosophical content. Springer, Berlin 1932. (Reprint: Springer, 1980.)
20. p. 443f.
21. p. 452f.
22. Contributions to the foundation of transfinite set theory. 1. Article. In: Mathematical Annals. 46, 1895.
23. Contributions to the foundation of transfinite set theory. 2. Article. In: Mathematical Annals. 49, 1897.