Georg Ferdinand Ludwig Philipp Cantor (
/ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.[2]
Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God - [4] on one occasion equating the theory of transfinite numbers with pantheism[5] - a proposition which Cantor vigorously refuted. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. 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".[8] 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,[9] though some have explained these episodes as probable manifestations of a bipolar disorder.[10]
The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."[13]
Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist, having inherited his parents' considerable musical and artistic talents. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm’s brother) was the well-known musician and the soloist in the Russian empire in an imperial orchestra.[14] Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the University of Zürich. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies 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, then and later a very important center for mathematical research. In 1867, Berlin granted him the PhD for a thesis on number theory, De aequationibus secundi gradus indeterminatis.
In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin. After teaching briefly in a Berlin girls' school, Cantor 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.[15]
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite 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 Swiss holiday.
Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.[16] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[17] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.[18] Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor ever to leave Halle.
In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.[19] Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.[20] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party:
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"Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica."[21] |
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Cantor suffered his first known bout of depression in 1884.[22] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
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"... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness."[23] |
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This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.[24]
Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–1884. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time.
After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.[22] Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics.[25] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. (Konig is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.) Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.[26] Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God.[11] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years.[27] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.
In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.
Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[28] The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.
Cantor's work between 1874 and 1884 is the origin of set theory.[29] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle.[30] No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.
In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets).[31]
Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable.
Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter Failed to parse (Missing texvc executable; please see math/README to configure.): \aleph
(aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.
The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.[32] The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.
Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set p(n) of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S' as its set of zeros, where S' is the set of limit points of S. If p(1) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p(1). Similarly for p(2), the set of limit points of p(1), and so on. By taking the intersection of p(1), p(2), p(3),... he formed p(ω), and then he noticed that p(ω) had a set of limit points p(ω + 1), and so on.[33] Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics".[34] Cantor also published an erroneous "proof" of the inconsistency of infinitesimals[35]
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article,[29] "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").[37] This article was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).[38] Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof is more complex than the more elegant diagonal argument that he gave in 1891.[39] Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.[40]
Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers[41] as a sequence a1, a2, a3, …. In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence — that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.[42] Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.[43]
Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.[44] Cantor also introduced the Cantor set during this period.
The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.
In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem.Cantor wrote on the Goldbach conjecture in 1894.
In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory.[45] The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schroeder theorem.
Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[46] The result that he found so astonishing has implications for geometry and the notion of dimension.
In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.
This paper displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass supported its publication.[47] Nevertheless, Cantor never again submitted anything to Crelle.
Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[9]
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").[48]
Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program.[49] In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.[11]
In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called limitation of size,[50] according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.
One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.[51]
The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[52] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[53] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God,[54] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[12]
Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.[55] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[6] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[56] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[6] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[8]
Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[4] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[57] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[58]
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... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.[59] |
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Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism—and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.[60]
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[61] as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[5] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[58]
Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom."[62] These ideas parallel those of Edmund Husserl.[63]
Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".
Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering:
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... 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.[64] |
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Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.
The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor",
Vasilievsky Island, Saint-Petersburg.
"Very little is known for sure about the origin and education of George Woldemar Cantor."[65] Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on his grandparents.[66] Cantor was sometimes called Jewish in his lifetime,[67] but has also variously been called Russian, German, and Danish as well.
Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they converted to Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:
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"Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ..."[65] |
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("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.[68]
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
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"More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew .."[65] |
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It is also later said in the same document:
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"Also efforts for a long time by the librarian Josef Fischer, one of the best experts on Jewish genealogy in Denmark, charged with identifying Jewish professors, that Georg Cantor was of Jewish descent, finished without result. [Something seems to be wrong with this sentence, but the meaning seems clear enough.] In Cantor's published works and also in his Nachlass there are no statements by himself which relate to a Jewish origin of his ancestors. There is to be sure in the Nachlass a copy of a letter of his brother Ludwig from 18 November 1869 to their mother with some unpleasant antisemitic statements, in which it is said among other things: ..."[65] |
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(the rest of the quote is finished by the very first quote above). In Men of Mathematics, Eric Temple Bell described Cantor as being "of pure Jewish descent on both sides," although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor," the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).
A letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306) mentions Cantor's belief that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde..." ("He was born in Copenhagen of Jewish (lit: "Israelite") parents from the local Portuguese-Jewish community.")[69] In addition, Cantor's maternal great uncle,[70] a Hungarian violinist Josef Böhm, has been described as Jewish,[71] which would imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.[72]
In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows:
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Neither my father nor my mother were of german blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.[73] |
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Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927)—largely the correspondence with Mittag-Leffler—and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst".[74] Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell—including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.[75] A critique of Bell's book is contained in Joseph Dauben's biography.[76]
- ^ Grattan-Guinness 2000, p. 351
- ^ The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
- ^ Dauben 2004, p. 1.
- ^ a b Dauben, 1977, p. 86; Dauben, 1979, pp. 120, 143.
- ^ a b Dauben, 1977, p. 102.
- ^ a b c Dauben 1979, p. 266.
- ^ Dauben 2004, p. 1. See also Dauben 1977, p. 89 15n.
- ^ a b Rodych 2007
- ^ a b Dauben 1979, p. 280:"...the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
- ^ Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression".
- ^ a b c Dauben 1979, p. 248
- ^ a b Dauben 2004, pp. 8, 11, 12-13.
- ^ Hilbert 1926, p. 170; see Reid 1996, p. 177
- ^ ru: The musical encyclopedia (Музыкальная энциклопедия)
- ^ O'Connor, John J, and Robertson, Edmund F (1998). "Georg Ferdinand Ludwig Philipp Cantor". MacTutor History of Mathematics. http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html.
- ^ Dauben 1979, p. 163.
- ^ Dauben 1979, p. 34.
- ^ Dauben 1977, p. 89 15n.
- ^ Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
- ^ Dauben 1979, p. 138.
- ^ Dauben 1979, p. 139.
- ^ a b Dauben 1979, p. 282.
- ^ Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
- ^ Dauben 1979, pp. 281–283.
- ^ Dauben 1979, p. 283.
- ^ For a discussion of König's paper see Dauben 1979, 248–250. For Cantor's reaction, see Dauben 1979, p. 248; 283.
- ^ Dauben 1979, p. 283–284.
- ^ Dauben 1979, p. 284.
- ^ a b Johnson 1972, p. 55.
- ^ This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later.
- ^ A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
- ^ Reid 1996, p. 177.
- ^ Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences 45 (4): 281, DOI:10.1007/BF01886630.
- ^ Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. doi:10.1007/s10699-011-9223-1 See link
- ^ Ehrlich, P. (2006) The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60, no. 1, 1–121.
- ^ This follows closely the first part of Cantor's 1891 paper.
- ^ Cantor 1874. English translation: Ewald 1996, p. 840–843.
- ^ For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous — see Moore 1995, p. 114.
- ^ For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., Suppes 1972.
- ^ Education.fr
- ^ The real algebraic numbers are the real roots of polynomial equations with integer coefficients.
- ^ For more details on Cantor's article, see Cantor's first uncountability proof and Gray 1995. Gray 1995 (p. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
- ^ Cantor's construction starts with the set of transcendentals T and removes a countable subset {tn} (for example, tn = e / n). Call this set T’. Then T = T’ ∪ {tn} = T’ ∪ {t2n-1} ∪ {t2n}. The set of reals R = T ∪ {an} = T’ ∪ {tn} ∪ {an} where an is the sequence of real algebraic numbers. So both T and R are the union of three disjoint sets: T’ and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T’, f(t2n-1) = tn, and f(t2n) = an. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1932, p. 142).
- ^ Dauben 1977, p. 89.
- ^ The English translation is Cantor 1955.
- ^ Wallace 2003, p. 259.
- ^ Dauben 1979, p. 69; 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
- ^ Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
- ^ Dauben 1979, pp. 240–270; see especially pp. 241, 259.
- ^ Hallett 1986.
- ^ Weir 1998, p. 766: "...it may well be seriously mistaken to think of Cantor's Mengenlehre [set theory] as naive..."
- ^ Dauben 1979, p. 295.
- ^ Dauben, 1979, p. 120.
- ^ Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
- ^ Dauben 1979, p. 225
- ^ Snapper 1979, p. 3
- ^ Davenport 1997, p.3
- ^ a b Dauben, 1977, p. 85.
- ^ Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
- ^ Dauben 1979, p. 296.
- ^ Dauben, 1979, p. 144.
- ^ Dauben 1977 pp. 91–93.
- ^ On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
- ^ Dauben 1979, p. 96.
- ^ a b c d Purkert and Ilgauds 1987, p. 15.
- ^ E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
- ^ For example, Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–1897, "List of Jewish Celebrities of the Nineteenth Century", p.119; this list has a star against people with one Jewish parent, but Cantor is not starred.
- ^ For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert and Ilgauds 1985; the letter is from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
- ^ Paul Tannery, Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, 1934, p. 306
- ^ Georg Cantor: his mathematics and philosophy of the infinite, Princeton University Press, 1990, By Joseph Warren Dauben, page 274
- ^ Modern Jews and their musical agendas, Ezra Mendelsohn, Oxford University Press, 1993, page 9
- ^ Ismerjükoket?: zsidó származású nevezetes magyarok arcképcsarnoka, István Reményi Gyenes Ex Libris, (Budapest 1997), pages 132-133
- ^ Bertrand Russell, Autobiography, vol. I, p. 229. In English in the original; italics also as in the original.
- ^ Grattan-Guinness 1971, p. 350.
- ^ Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p.1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
- ^ Georg Cantor, Joseph Dauben, Harvard University Press (1979). Reprinted as a paperback by Princeton University (1990).
- Older sources on Cantor's life should be treated with caution. See Historiography section above.
- Primary literature in English
- Cantor, Georg (1955) [1915], Philip Jourdain, ed., Contributions to the Founding of the Theory of Transfinite Numbers, New York: Dover, ISBN 978-0-486-60045-1, http://www.archive.org/details/contributionstot003626mbp .
- Ewald, William B., ed. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, New York: Oxford University Press, ISBN 978-0-19-853271-2 .
- Primary literature in German
- Cantor, Georg (1874), "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik 77: 258–262, http://bolyai.cs.elte.hu/~badam/matbsc/11o/cantor1874.pdf .
- Cantor, Georg (1932), Ernst Zermelo, ed. (PDF), Gesammelte Abhandlungen mathematischen und philosophischen inhalts, http://philosophons.free.fr/philosophes/cantor1932.pdf . Almost everything that Cantor wrote.
- Hilbert, David (1926), "Über das Unendliche", Mathematische Annalen 95: 161–190, DOI:10.1007/BF01206605, http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002270641 .
- Secondary literature
- Aczel, Amir D. (2000), The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind, New York: Four Walls Eight Windows Publishing . ISBN 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned.
- Dauben, Joseph W. (1977), "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite", Journal of the History of Ideas 38 (1): 85–108 .
- Dauben, Joseph W. (1979), Georg Cantor: his mathematics and philosophy of the infinite, Boston: Harvard University Press . The definitive biography to date. ISBN 978-0-691-02447-9
- Dauben, Joseph W. (June 1983), "Georg Cantor and the Origins of Transfinite Set Theory", Scientific American 248 (6): 122–131
- Dauben, Joseph (1993, 2004), "Georg Cantor and the Battle for Transfinite Set Theory", Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA), pp. 1–22 . Internet version published in Journal of the ACMS 2004.
- Davenport, Anne A. (1997), "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century", Isis 88 (2): 263–295 .
- Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought, Basel, Switzerland: Birkhäuser . ISBN 3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
- Grattan-Guinness, Ivor (1971), "Towards a Biography of Georg Cantor", Annals of Science 27: 345–391 .
- Grattan-Guinness, Ivor (2000), The Search for Mathematical Roots: 1870–1940, Princeton University Press . ISBN 978-0-691-05858-0
- Gray, Robert (1994), "Georg Cantor and Transcendental Numbers", American Mathematical Monthly 101: 819–832 .
- Hallett, Michael (1986), Cantorian Set Theory and Limitation of Size, New York: Oxford University Press . ISBN 0-19-853283-0
- Halmos, Paul (1998, 1960), Naive Set Theory, New York & Berlin: Springer . ISBN 3-540-90092-6
- Hill, C. O.; Rosado Haddock, G. E. (2000), Husserl or Frege? Meaning, Objectivity, and Mathematics, Chicago: Open Court . ISBN 0-8126-9538-0 Three chapters and 18 index entries on Cantor.
- Johnson, Phillip E. (1972), "The Genesis and Development of Set Theory", The Two-Year College Mathematics Journal 3 (1): 55–62 .
- Meschkowski, Herbert (1983), Georg Cantor, Leben, Werk und Wirkung (George Cantor, Life, Work and Influence, in German), Wieveg, Braunschweig
- Moore, A.W. (April 1995), "A brief history of infinity", Scientific American 272 (4): 112–116 .
- Penrose, Roger (2004), The Road to Reality, Alfred A. Knopf . ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
- Purkert, Walter; Ilgauds, Hans Joachim (1985), Georg Cantor: 1845–1918, Birkhäuser . ISBN 0-8176-1770-1
- Reid, Constance (1996), Hilbert, New York: Springer-Verlag . ISBN 0-387-04999-1
- Rucker, Rudy (2005, 1982), Infinity and the Mind, Princeton University Press . ISBN 0-553-25531-2 Deals with similar topics to Aczel, but in more depth.
- Rodych, Victor (2007), "Wittgenstein's Philosophy of Mathematics", in Edward N. Zalta, The Stanford Encyclopedia of Philosophy .
- Snapper, Ernst (1979), "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism", Mathematics Magazine 524: 207–216 .
- Suppes, Patrick (1972, 1960), Axiomatic Set Theory, New York: Dover . ISBN 0-486-61630-4 Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
- Wallace, David Foster (2003), Everything and More: A Compact History of Infinity, New York: W.W. Norton and Company . ISBN 0-393-00338-8
- Weir, Alan (1998), "Naive Set Theory is Innocent!", Mind 107 (428): 763–798 .
- O'Connor, John J.; Robertson, Edmund F., "Georg Cantor", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html .
- O'Connor, John J.; Robertson, Edmund F., "A history of set theory", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Beginnings_of_set_theory.html . Mainly devoted to Cantor's accomplishment.
- Georg Cantor at the Mathematics Genealogy Project
- Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech.
- Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gynmasium Halle
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