Georg Cantor (1845-1918)

This is intended to be a complete list of the publications of Georg Cantor as well as translations of those works. It is followed by works which contain his correspondence and by a list of some of the most valuable English language historical books and papers dealing with Cantor and his work. Corrections and suggestions would be most welcome. Send email to fred-rickey@usma.edu .  Many of these are available in digital form:  http://dz-srv1.sub.uni-goettingen.de/cache/browse/AuthorMathematicaArticleC8.html

1. De aequationibus secundi gradus indeterminatis.
   Diss. Phil. Berlin 1867.
   = Gesammelte Abhandlungen,
      
2. Über einfache Zahlensysteme.
   Zeitschrift für Math. und Physik 14(1869), 121-128.
   = Gesammelte Abhandlungen,
   
3. De transformatione formarum ternarium quadricarum.
   Habilitationsschrift Halle 1869.
   = Gesammelte Abhandlungen,
      
4. Über einen die trigonometrischen Reihen betreffenden Lehrsatz.
   Journal f. reine und angew. Math. 72(1870), 130-138.
   = Gesammelte Abhandlungen,
      
5. Beweis, daß eine für jeden reellen Wert von x durch eine trigonometrische Reihe gegebene Funktion f(x) sich nur auf eine einzige Weise in dieser Form darstellen läßt.
   Journal f. reine und angew. Math. 72(1870), 139-142.
   = Gesammelte Abhandlungen,
      
6. Über trigonometrische Reihen.
   Math. Annalen 4(1871), 139-143.
   = Gesammelte Abhandlungen,
      
7. Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen.
   Math. Annalen 5(1872), 123-132.
   = Gesammelte Abhandlungen,
   David R. Wilkins has made this available, in German, on the web at  http://www.maths.tcd.ie/pub/HistMath/People/Cantor/Ausdehnung/
      
8. Historische Notizen über die Wahrscheinlichkeitsrechnung.
   Sitzungsberichte der Naturforschenden Gesellschaft zu Halle 1873, 34-42.
   = Gesammelte Abhandlungen,
      
9. Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen.
   Journal f. reine und angew. Math. 77(1874), 258-262.
   = Gesammelte Abhandlungen, 115-118.
   French translation entitled "Sur une propriété du systeéme de tous les nombres algébriques réels," Acta Mathematica, 2 (1883), 305-310. 
   English translation entitled "On a property of the set of real algebraic numbers," in volume 2,  pp. 839-843 of From Kant to Hilbert. A Source Book in the Foundations of Mathematics edited by William Ewald, Oxford: Clarendon Press, 1999.
      
10. Ein Beitrag zur Mannigfaltigkeitslehre.
    Journal f. reine und angew. Math. 84(1878), 119-133.
    = Gesammelte Abhandlungen,
       
11. Über einen Satz aus der Theorie der stetigen Mannigfaltigkeiten.
    Göttinger Nachr. 1879, 127-135.
    = Gesammelte Abhandlungen,
       
12. Über ein neues und allgemeines Condensationsprinzip der Singularitäten von Funktionen.
    Math. Annalen 19(1882), 588-594.
    = Gesammelte Abhandlungen,
       
13. Über unendliche, lineare Punktmannigfaltigkeiten.
    Mathematische Annalen 15(1879), 1-7; 17(1880), 355-358; 20(1882), 113-121; 21(1883), 51-58 und 545-586; 23(1884), 453-488.
    = Gesammelte Abhandlungen, 139-145, 145-148, 149-157, [?] 157-164, 165-208, 210-244.
        French translation of the first four parts as "Sur les ensembles infinis et linéares de points," Acta Mathematica, 2(1883), 349-356, 357-360, 361-171, 372-380.
        Reprinted in  Über unendliche, lineare Punktmannigfaltigkeiten. Arbeiten zur Mengenlehre aus den Jahren 1872--1884. Edited and with a foreword and commentary by G. Asser. Teubner-Archiv zur Mathematik, 2. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1984. 180 pp. MR 87g:01061 by Dauben. 
    
14. Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: B. G. Teubner.
    = Gesammelte Abhandlungen, 165-208.
        French translation of portions of this as "Fondaments d'une théorie générale des ensembles," Acta Mathematica, 2 (1883), 381-408.
        English translation by U. Parpart as "Foundation of the theory of manifolds," The Campaigner (The Theoretical Journal of the National Caucus of Labor Committees), 9 (January and February 1976), 69-96. Dauben (1979, p. 328) points out that this translation does not respect Cantor's distinction between reellen and realeen Zahlen (real numbers as opposed to complex numbers vs. real numbers in the concrete, ontological sense) as well as between Zahlen and Anzahlen, translating both as 'number'.
        
       
15. Sur divers théorèmes de la théorie des ensembles de points situés dans un espace continu à n dimensions. (Première communication. Extrait d´une lettre adressée à l´éditeur)
    Acta Mathematica, 2(1883), 409-414.
    = Gesammelte Abhandlungen, 247-251.
       
16. De la puissance des ensembles parfaits des points. (Extrait d´une lettre adressée à l´éditeur)
    Acta Math. 4 (1884), 381-392.
    = Gesammelte Abhandlungen,
       
17. Über verschiedene Theoreme der Punktmengen in einem n-fach ausgedehnten stetigen Raume G_n. Zweite Mitteilung
    Acta Math. 7(1885), 105-124.
    = Gesammelte Abhandlungen,
       
18. Ludwig Scheeffer (Nekrolog).
    Bibliotheka mathematica 1(1885), 197-199.
    = Gesammelte Abhandlungen,
       
19. Rezension der Schrift von G. Frege "Die Grundlagen der Arithmetik".
    Dtsch. Lit. Ztg. 6(1885), 728-729.
    = Gesammelte Abhandlungen,
       
20. Über die verschiedenen Standpunkte in bezug auf das aktuale Unendliche.
    Zeitschr. für Philos. und philos. Kritik 88(1886), 224-233.
    = Gesammelte Abhandlungen,
       
21. Mitteilungen zur Lehre vom Transfiniten.
    Zeitschr. für Philos. und philos. Kritik 91(1887), 81-125; 92(1888), 240-265.
    = Gesammelte Abhandlungen,
       
22. Über eine elementare Frage der Mannigfaltigkeitslehre.
    Jahresber. der Dt. Math.-Verein. 1(1890/91), 75-78.
    = Gesammelte Abhandlungen, 278-280.
        Spanish translation in Mathesis, 3 (1987), no. 2, 95--98.
        English translation entitled "On an elementary question in the theory of manifolds," in volume 2,  pp. 923-940 of From Kant to Hilbert. A Source Book in the Foundations of Mathematics edited by William Ewald, Oxford: Clarendon Press, 1999.
        This paper contains Cantor's well known diagonal method for showing that there are more real numbers than natural numbers. 
       
23. Beiträge zur Begründung der transfiniten Mengenlehre.
    Mathematische Annalen 46(1895), 481-512; 49(1897), 207-246.
    = Gesammelte Abhandlungen, 282-311.
        French translation by F. Marotte in Sur les fondements de la théorie des ensembles transfinis, Paris: Hermann, 1999.
        English translation by Philip E. B. Jourdain in Contributions to the Founding of the Theory of Transfinite Numbers, 1952, Dover publications (originally Open Court, 1915), pp. 85-136 and 137-201, with original pagination indicated. These translations are preceded by a long and interesting introduction to Jourdain.
       
24. Ressurrectio Divi Quirini Francisci Baconi Baronis de Verulam Vicecomitis Sancti Albani CCLXX annis post obitum eius IX die aprilis anni MDCXXVI.
    (Pro manuscripto.) Cura et impensis G. C. Halis Saxonum MDCCCXCVI (ed. G. C.)
       
25. Confessio fidei Francisci Baconi Baronis de Verulam ... cum versione latina a G. Rawley nunc denuo typis excusa cura et impensis G. C. Halis Saxonum
    MDCCCXCVI (ed. G. C.)
       
26. Die Rawleysche Sammlung von zweiunddreißig Trauergedichten auf Francis Bacon.
    Ein Zeugnis zugunsten der Bacon-Shakespeare-Theorie mit einem Vorwort herausgegeben von Georg Cantor, Halle 1897
       
27. Shakespearologie und Baconianismus ...
    Magazin für Literatur 69(1900), 196-203
       
28. Bemerkungen zur Mengenlehre.
    Jahresber. der Dt. Math.-Verein. 12(1903), 519.
    = Gesammelte Abhandlungen,
       
29. Ex oriente lux. Gespräche eines Meisters mit seinem Schüler über wesentliche Punkte des urkundlichen Christentums.
    Berichtet vom Schüler selbst Georg Jacob Aaron, cand. sacr. theol. Erstes Gespräch.
    Hrsg. von Georg Cantor, Halle 1905.
    
30. Ein Brief von Carl Weierstrass über das Dreikörperproblem.
    Rendiconti del Circolo Mathematico di Palermo 19(1905), 305-308.
    
31. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernst Zermelo. Berlin: J. Springer, 1932. Reprinted Hildesheim: Georg Olms, 1962 and by Springer in 1980: MR 82i:01069.
        Very favorably reviewed by A. A. Bennett, American Mathematical Monthly, 40(1933), 103-104.
    
32. Principien einer Theorie der Ordnungstypen, (Erste Mittheilung), edited by Ivor Grattan-Guinness. Paper originally dated November 6, 1884, but this is its first publication.
    Acta Mathematica 124, 65-107. MR 41 #5171 by Kenneth May. 

More papers by Cantor (there are more to include from the DSB):

1895
"V\'erification jusqu'\\^a 1000 du th\'eor\`me empirique de Goldbach," Association Fran\c caise pour l'Advancement des Sciences, {\sl Comptes rendus de la 23me [superscript] session (Caen 1894), pt. 2 (1895), 117-134.
 

Correspondence

1937
Noether, Emmy (1882-1935) and Cavaillès, Jean (1903-1944), editors.
Briefwechsel Cantor-Dedekind, Paris: Hermann, 1937, 60 p.

1965
Meschkowski, Herbert, "Aus den Briefbüchern Georg Cantors," Arch. History Exact Sci. 2 (1965), 503--519. MR 33 #3868.

1971
Grattan-Guinness, Ivor, "The correspondence between Georg Cantor and Philip Jourdain," Jahresbericht der Deutschen MathematikerVereiningung, 73 (1971-1972), 111-130.

1986
Lipschitz, Rudolf (1832-1903).
Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass und anderen, [Freiburg i. Br.] : Deutsche Mathematiker Vereinigung ; Braunschweig : F. Vieweg, 1986, xviii, 253 p. Edited by Winfried Scharlau. MR 87i:01051.

Secondary Literature

1937
Bell, Eric Temple (1883-1960), Men of Mathematics, New York: Simon and Schuster, 1937.
    The last chapter of this classic is a biography of Cantor. Today this book is scorned by historians for its historical inaccuracies and numerous prejudices, but the book has encouraged many to take up the study of mathematics. An interesting project would be to study this chapter (or the whole book) as a historical document. How good a use of the available historical documents did Bell make? Precisely where did he fabricate stories or facts? What prejudices, of Bell or his times, are discernable today?

1964
Meschkowski, Herbert, Ways of Thought of Great Mathematicians, San Francisco: Holden-Day, 1964.
    The chapter "Weierstrass and his school," pp. 85-89, contains the first publication of a  letter from H. A. Schwarz to Cantor of February 25, 1870, which gives the first rigorous proof  that a differentiable function whose derivative vanishes is constant.
    The chapter on "Georg Cantor," pp. 91-104, contains a letter of June 18, 1886 from Cantor to F. Goldscheider, a Berlin Gymnasium teacher and admirer of Cantor's work. This example laden letter is a little textbook of set theory.

1970
Grattan-Guinness, Ivor, "Towards a biography of Georg Cantor," Annals of Science, 27 (1970), 345-391.

1971
Dauben, Joseph W., "The trigonometric background to Georg Cantor's theory of sets," Archive for History of Exact Sciences, 7 (1970-71), 181-216.

Grattan-Guinness, Ivor, "Towards a biography of Georg Cantor," Annals of Science, 27 (1971), 345-391 + 3 plates.

Meschkowski, Herbert, "Cantor, Georg," pp. 52-58 in volume 3 of the Dictionary of Scientific Biography, edited by C. C. Gillispie, New York: Charles Scribner's Sons.

1974
Dauben, Joseph W., "Denumerability and dimension: the origins of Georg Cantor's theory of sets," Rete, 2, no. 2, 105-134. MR 57#2838.

Grattan-Guinness, Ivor, "The rediscovery of the Cantor-Dedekind correspondence," Jahresbericht der Deutschen Mathematiker-Vereiningung, 76 (1974), 104-139. MR 58 #4878.

1975
Dauben, Joseph W., "The invariance of dimension: problems in the early development of set theory and topology," Historia Mathematica 2 (1975), 273-288. 

1979
Dauben, Joseph W., Georg Cantor. His Mathematics and the Philosophy of the Infinite, Cambridge MA: Harvard University Press, 1979. Reprinted by Princeton University Press, 1990.Reviewed by Gregory H. Moore, History and Philosophy of Logic, 1 (1980), 238-240 and by H. C. Kennedy, MR 80g:01021..

1989
Purkert, Walter, "Cantor's views on the foundations of mathematics," pp. 49-65 in The History of Modern Mathematics, Volume I, edited by David E. Rowe and John McCleary, Boston: Academic Press, 1989. MR 91a:01028.

1993
Cooke, Roger, "Uniqueness of trigonometric series and descriptive set theory, 1870-1985," Archive for History of Exact Sciences, 45 (1993) 281-334.

Ferreirós, José, "On the relations between Georg Cantor and Richard Dedekind," Historia Mathematica, 20 (1993), 343-363.

1999
Laubenbacher, Reinhard and Pengelley, David, Mathematical Expeditions: Chronicles by the Explorers, New York: Springer, 1999.
    Chapter 3, Set Theory: Taming the Infinite, pp 54-94, provides a very nice introduction to the work of Cantor and contains repritns of translations by Jourdaine.
 
2000
Grattan-Guinness, Ivor, The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor Through Russell and Gödel, Princeton University Press, 2000.
    Chapter 3, pp. 75-125, is entitled "Mathematics as Mengenlehre."
 
 
 
 
 

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