A comparison of two cultural approaches to mathematics: France and Russia, 1890-1930

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Volumn 97, Issue 1, 2006, Pages 56-74

  Graham, Loren ^a   Kantor, Jean Michel ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

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[No Author keywords available]

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EID: 33646455898     PISSN: 00211753     EISSN: 15456994     Source Type: Journal    
DOI: 10.1086/501100     Document Type: Article

References (138)

1. Georg Cantor, "Über unendliche, lineare Punktmannichfaltigkeiten," Mathematische Annalen, 1883,21:545-591, on p. 545. Here and throughout this essay, translations are our own unless otherwise indicated.
2. 0. This is the way Cantor typically defined cardinal numbers (ordinal numbers are defined similarly for sets with a given order between their elements). Transfinite numbers are any of these cardinal or ordinal numbers constructed through various operations and compared, when possible, with each other. They were introduced by Cantor in 1882 in close connection with philosophical and even religious considerations about "the Absolute." This was the first introduction of infinity in mathematics (where it was called the "actual infinite" in opposition, following Aristotle, to the "potential infinite," thought of as a limit process). Set theory was born from the study of point sets of the real line. Later, as we will see, the efforts to describe and classify such sets would lead to descriptive set theory (see note 56, below).
3. See José Ferreirós, Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics (Basel/Boston: Birkhäuser, 1999);
4. Walter Purkert, "Georg Cantor und die Antinomien der Mengenlehre," Bulletin de la Société Mathématique de Belgique, 1986, 38:313-327;
5. Purkert, Georg Cantor, 1845-1918 (Basel/Boston: Birkhäuser, 1987).
6. There are hundreds of books on religion and science. A few that deserve notice are John Hedley Brooke, Science and Religion: Some Historical Perspectives (Cambridge: Cambridge Univ. Press, 1991);
7. Edward J. Larson, Trial and Error: The American Controversy over Creation and Evolution (Oxford: Oxford Univ. Press, 1989);
8. Michael Ruse, Can a Darwinian Be a Christian? The Relationship between Science and Religion (Cambridge: Cambridge Univ. Press, 2001);
9. Frederick Gregory, Nature Lost? Natural Science and the German Theological Traditions of the Nineteenth Century (Cambridge, Mass.: Harvard Univ. Press, 1992);
10. Max Jammer, Einstein and Religion: Physics and Theology (Princeton, N.J.: Princeton Univ. Press, 1999);
11. Louis Châtellier, Les espaces infinis et le silence de Dieu: Science et religion XVIe-XIXe siècle (Paris: Aubier, 2003);
12. C. Chevalley, Pascal, contingence et probabilités (Paris: Presses Univ. France, 1995);
13. T. Lævy, Figures de l'infini: Les mathématiques au regard des cultures (Paris: Seuil, 1987);
14. P. P. Gaidenko and V. N. Katanosov, eds., Nauka, filosofiia, religiia: V poiskakh obshchego znamenatelia (Moscow: IF RAN, 2003);
15. Stefan Blinderhöfer, Naturwissenschaftlicher Weltzugang und der Überschuss der Schöpfungsperspektive (Frankfurt: Lang, 2000).
16. An older book that is still of value is Charles Coulston Gillispie, Genesis and Geology: A Study in the Relations of Scientific Thought, Natural Theology, and Social Opinion in Great Britain, 1790-1850 (1951; Cambridge, Mass.: Harvard Univ. Press, 1969).
17. The classic statement of the conflict thesis is Andrew Dickson White, A History of the Warfare of Science with Theology in Christendom (New York: Appleton, 1896; Buffalo: Prometheus, 1993).
18. A book that revised the "warfare" model is David C. Lindberg and Ronald L. Numbers, eds., God and Nature: Historical Perspectives on the Encounter between Science and Christianity (Berkeley: Univ. California Press, 1986).
19. Leur tournure philosophique ne sera pas an obstacle pour le traducteur qui connaît Kant": Charles Hermite, "Lettres à G. Mittag-Leffler publiées et annot-acees par Pierre Dugac," Cahiers du Séminaire d'Histoire de Mathématiques, 1980, 5:49-285.
20. On Bolzano's contribution to set theory see Bernard Bolzano, Les paradoxes de l'infini (Paris: Seuil, 1993);
21. J. Sebestik, Logique et mathématique chez Bernard Bolzano (Paris: Vrin, 1992).
22. Camille Jordan, Cours d'analyse de l'Ecole polytechnique, 2nd ed. (Paris: Gauthier-Villars, 1893-1896).
23. For the 1890 memoir see Henri Poincaré, "Sur le problème des trois corps et les q́uations de la dynamique," Acta Mathematica, 1890, 13:1-270;
24. see also A. Robadey, "Exploration d'un mode d'écriture de la généralité: L'article de Poincaré sur les lignes géodésiques des surfaces convexes (1905)," Revue d'Histoire des Mathématiques, 2004, 10.257-318.
25. The law of excluded middle (tertium non datur) states that for any proposition P, P or (not P) is true. It goes back to Aristotelian logic and would later be rejected by supporters of the intuitionist current in mathematics.
26. Le., they were represented by a continuous line with tangents everywhere except at a finite number of points. Discontinuous functions present jumps in their corresponding graphs. On the history of the theory of functions see (from one of our main characters) Nikolai Luzin, "Function," American Mathematical Monthly, 1988, 105:59-67 (Pt. 1), 263-270 (Pt. 2) (this text is translated by A. Schenitzer from the Great Soviet Encyclopaedia of 1934); of 1934);
27. more recently, Fyodor A Medvedev, Scenes from the History of Real Functions, trans. Roger Cooke (Boston: Birkhäuser, 1991).
28. "L'auteur nous paraît avoir une tournure d'esprit favorable à l'étude de ces questions qui sont à la frontière de la mathématique et de la philosophie": quoted in Hélène Gispert, ed., La France mathématique: La Société mathæmatique de France (1872-1914) (Paris: Société Frana̧ise d'Histoire des Sciences et des Techniques/ Société Mathématique de France, 1991), p. 375.
29. "On est parfois saisi de vertige en voyant à quels résultats on arrive quand on abandonne les hypothèses usuelles": ibid., p. 284.
30. "Natura non facit saltus, nous avons le sentiment, on pourrait dire la croyance, que daps la nature il n'y a pas de place pour la discontinuité": Emile Picard, La science moderne et son état actuel (Paris: Flammarion, 1905), Ch. 2: "Sciences mathématiques et astronomie," p. 54.
31. The thought that "nature does not make jumps" has been expressed by many writers, including Aristotle and Leibniz, but the Latin phrase "Natura non facit saltus" comes from Carl von Linné, Philosophia Botanica (Stockholm: Kiesewetter, 1751), aphorism 77.
32. See P. Guiraldenq, Emile Borel, 1871-1956: L'espace et le temps d'une vie sur deux siècles (Saint-Affrique: Progrès, 1999).
33. "J'ai été extrêmement séduit, dès l'âge de 20 ans, par la lecture des travaux de Cantor ... Georg Cantor a apporté dans l'études des mathématiques cet esprit romantique qui est I'un des côtés les plus séduisants de l'âme allemande": Emile Borel, Oeuvres complétes (Paris: Éditions du Centre National de la Recherche Scientifique, 1972), Vol. 4, p. 2101. Limit points are obtained as limits of a sequence of points: for example, 0 is the limit of the sequence of points (1/n); as n gets bigger the points representing 1/n come closer to the origin.
34. Normal numbers, e.g., would count as "down to earth"; see note 33, below.
35. "... à cause de la fatigue qu'elle lui imposait et qui lui faisait craindre des troubles sérieux s'il s'obstinait à ce travail": Paul Valèry, Cahiers, Vol. 1 (Paris: Gallimard, 1973), p. 797;
36. quoted in P Dugac, Histoire de l'analyse (Paris: Vuibert, 2003), p. 253.
37. On Baire's difficulties see the letter he sent to his brother in June 1932 and allusions to his very probable suicide: Dugac, "Notes et documents sur la vie et l'oeuvre de René Baire," Archive for History of Exact Sciences, 1976, 15:297-383, on p. 313.
38. David Hilbert, "Sur les problèmes futurs des mathématiques," trans. L. Laugel, in Compte-rendu du deuxième congrès international des mathematiciens (Paris: Congrès International des Mathematiciens, 1900), pp. 58-114; shortly before the mathematics congress there was an international congress of philosophy in which the mathematicians Borel, Poincaré, and Jacques Hadamard participated.
39. Cantor showed in 1874 that the set of all real numbers (represented by the continuum of the straight line) is not denumerable. In other words, there cannot be a one-to-one correspondence between all integers and all real numbers. This conclusion led him to the Continuum Hypothesis, in which he asked whether any part of the continuum could be put in one-to-one correspondence either with the set of integers or with the whole line.
40. For one arithmetical approach see R. Dedekind, Stetigkeit und irrationale Zahlen (Braunschweig: Vieweg, 1872).
41. Il y a entre la théorie générale des fonctions de variables réelles et la géomætrie pure des liens que je sens étroits, encore qu'ils restent un peu mystérieux pour moi": introduction à la "Notice sur les travaux scientifiques de M. Henri Lebesgue de 1922" (unpublished), in Lebesgue, Oeuvres complètes (Genève: L'Enseignement Mathématique, 1972-1973), Vol. 1, p. 90.
42. For a detailed analysis of Borel's and Lebesgue's work on measure theory (which studies how to extend the notion of length of intervals to highly complex parts of the continuum) see G. H. Moore, "Lebesgue's Measure Problem and Zermelo's Axiom of Choice: The Mathematical Effects of a Philosophical Dispute," in History and Philosophy of Science: Selected Papers (Annals of the New York Academy of Sciences, 412) (New York: New York Academy of Sciences, 1983), pp. 129-154.
43. On the classification of functions see R Baire, Oeuvres scientifiques, ed. P. Lelong with P. Dugac (Paris: Gauthiers-Villars, 1990).
44. An ordered set is a set where any two elements can be compared with respect to an "order relation"; a "well-ordered set" is an ordered set with similar properties as the set of positive integers. E. Zermelo, "Beweis, dass jede Menge wohlgeordnet warden kann (Aus emem an Herm Hilbert gerichteten Briefe)," Math. Ann., 1904, 59:514-516.
45. See also G. H. Moore, Zermelo's Axiom of Choice: Its Origin, Development, and Influence (Berlin: Springer, 1990).
46. "Zermelo arriva, et ce fut la bagarre": H. Lebesgue, Oeuvres scientifiques (Genève: L'Enseignement Mathématique, 1972-1973), p. 294.
47. The letters were published as J. Hadamard, R. Baire, H. Lebesgue, and E. Borel, "Cinq lettres sur la théorie des ensembles," Bulletin de la Société Mathématique de France, 1905, 23:261-273.
48. "La notion de correspondance qui peut être décrite est en dehors des mathm̧atiques, rel-greve du domaine de la psychologie et est relative à une propriété de notre esprit": Hadamard, et al., "Cinq lettres sur la théorie des ensembles," letter 1.
49. "Raisonner sur un ensemble c'est raisonner sur des objets pris dans un sac C sur lesquels on sait seulement qu'ils ont une propriété B en common n'appartenant pas aux autres objets de C. On ne sait par conséquent rien qui permette d'aborder la d-acefinition de l'ordre des éléments; on ne sait même pas les distinguer": Henri Lebesgue to Emile Borel, Feb. 1905, in Lebesgue, Les lendemains de l'intégrale: Lettres à Emile Borel (Paris: Vuibert, 2004), p. 101.
50. C. Burali-Forti, "Una questione sui numeri transfiniti," Rendiconti del Circolo Matematico di Palermo, 1897, 11:154-164;
51. B. Russell, "On Some Difficulties in the Theory of Transfinite Numbers and Order Types," London Mathematical Society, 1906, 4:29-53;
52. J. Richard, "Lettre à Monsieur le édacteur," Revue Générale des Sciences, June 1905, 12:12.
53. For all three references see also Ferreirós, Labyrinth of Thought (cit. n. 2).
54. See, e.g., A. Heyting, Les fondements des mathématiques: Intuitionnisme, th-aceorie de la démonstration (Paris: Gauthier-V llars, 1955).
55. A thorough philosophical discussion has been offered by J. Bouveresse, "Sur les sens du mot platonisme dans l'expression 'platonisme mathématique,'" paper presented at a conference of the Société romande de philosophie, groupe genevois, University of Genève, 19 Nov. 1998; see http://un2sg4.unige.ch/ athena/bouveresse/bou_plat.html.
56. The connection drawn by Hadamard is reported by E. Borel in "L'infini mathématique et la réalité," Revue du Mois, 1914, 18:71-84.
57. See Emile Borel: Philosophe et homme d'action, pages choisies et présentées par M. Fréchet (Paris: Gauthier-Villars, 1967), p. 183.
58. "Peut-on s'assurer de l'existence d'un être mathématique sans le définir? Définir veut toujours dire nommer une propriété du défini": Hadamard et al., "Cinq lettres sur la théorie des ensembles" (cit. n. 17).
59. "Ces spéculations sur l'infini forment un chapitre tout nouveau dans la science mathématique de ces dernières années, mais il faut reconnaiître que ce chapitre n'est pas exempt de paradoxes. C'est ainsi que l'on a pu définir certains nombres appartenant et n'appartenant pas tout à la fois à des ensembles déterminés. Toutes les difficultés de ce genre résultent de ce qu'on ne s'entend pas sur le mot existence. Certains adeptes de la théorie des ensembles sont des scolastiques, qui auraient aimé à discuter les preuves de l'existence de Dieu, avec Saint-Anselme et son contradicteur, le moine de Noirmoutiers." E. Picard, La science moderne et son état (Paris: Flammarion, 1909), extract from Ch. 2.
60. See Anselme de Cantorbery, Proslogion Allocution sur l'existence de Dieu suivi de sa réfutation par Gaunilon et de la r-aceponse d'Anselme (Paris: Flammarion, 1993).
61. Borel's rejection of the Axiom of Choice was spelled out in Atti del IV Congresso Internazionale dei Matematici: Roma, 6-11 Apr. 1908 (Rome: Tip. Della R. Accademia dei Lincei, 1909), Vol. 2, pp. 15-17.
62. His opposition to nondenumerable infinities is maintained throughout the prefaces of all four later editions of his famous Legons sur la théorie des fonctions up to 1950;
63. see, e.g., Leçons sur la théorie des fonctions (Élæments et principes de la theéorie des ensembles: Applications à la théorie des fonctions), 3rd ed. (Paris: Gauthier-Villars, 1928).
64. See Moore, Zermelo's Axiom of Choice (cit. n. 16).
65. See Moore, Zermelo's Axiom of Choice, for more positive reactions in England, Germany, Italy, and the United States.
66. The tradition of democracy and an emphasis on the importance of education were revived during the Dreyfus controversy, in which many friends of Borel participated.
67. "... diviser chacune des difficultés que j'examinerais, en autant de parcelles qu'il se pourrait, et qu'il serait requis pour les mieux résoudre. Š conduire par ordre mes pensées, en commençant par les objets les plus simples et les plus aisés à connaître, pour monter peu à peu, comme par degrés, jusque la connaissance des plus composés; et supposant même de l'ordre entre ceux qui ne se précèdent point naturellement les uns et les autres to": René Descartes, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Paris: Michel Bobin et Nicolas Le gras, 1668).
68. "Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisémenf": Nicolas Boileau-Despréaux, Les quatre poétiques d'Aristote, d'Horace, de Vida, de Despréaux (Paris: Saillant et Nyon-Desaint, 1771), Vol. 1, p. 153.
69. "J'ai toujours eu, comme il convient, un respect infini pour Descartes ... Il faut juger Descartes savant sur l'orientation toute nouvelle donnée à la science par ses intuitions géniales et par une méthode": E. Picard, Une édition nouvelle du discourse de la méthode de Descartes (Paris: Gauthier-Villars, 1934), p. 54.
70. "Dès lors qu'il s'agit de mathématiques et non de philosophie, le désaccord ne peut provenir que d'un malentendu": Borel, "L'infini mathématique et la réalité" (cit. n. 22), p. 73.
71. See B. Belhoste, "L'enseignement secondaire français et les sciences an début du XXe siècle: La réforme de 1902 des plans d'études et des programmes," Revue d'Histoire des Sciences, 1990, 53(4):371-400.
72. For a thorough analysis of Pascal's differences with Descartes on the nature of mathematics see Catherine Chevalley, Contingence et probabilités (Paris: Presses Univ. France, 1995).
73. Normal numbers are numbers-say, between 0 and 1-that, if written in decimal expansion, present each sequence of n letters with the frequency that would be expected for "random" numbers. They were introduced by Borel (Leçons sur la théorie des fonctions [Paris: Gauthier-Villars,1914], p. 197).
74. Absolutely normal numbers are numbers that have the analogous property on any basis of expansion (not only decimal expansion). Although the absolutely normal numbers have measure 1, it is difficult to give explicit ("effective" was the word used then) constructions. For references and recent progress see V. Becher and S. Figueira, "An Example of a Computable Absolutely Normal Number," Theoretical Computer Science, 2002, 270.948-958.
75. See Jean-Michel Kantor, "Dans le gouffre du continu," in preparation.
76. Allen Shields, "Years Ago: Luzin and Egorov," Mathematical Intelligencer 1987, 9:24-27;
77. Smilka Zdravkovska and Peter L. Duren, eds., Golden Years of Moscow Mathematics (Providence, R.I.: American Mathematical Society, 1993);
78. Alexander Vucinich, "Mathematics and Dialectics in the Soviet Union: The Pre-Stalin Period," Historia Mathematica, 1999, 26:107-124;
79. P. S. Aleksandrov, Matematika v SSSR za 15 let (Moscow: Gostekhizdat, 1932).
80. It is hardly necessary to prove the importance of religion to Florenskii, a priest; his best-known published statement of faith is probably to be found in his Stolp i utverzhdenie istiny (Moscow: Put', 1914), published in English as The Pillar and Ground of the Truth, trans. Boris Jakim (Princeton, N.J.: Princeton Univ. Press, 1997).
81. The deep religiosity of Egorov is described in S. S. Demidov, "Professor Moskovskogo universiteta Dmitrii Fedorovich Egorov i imiaslavie v Rossii v pervoi treti XX stoletiia," Istoriko-Matematicheskie Issledovaniia, 2nd Ser., 1999, 39.123-156, esp. p. 137.
82. Luzin's conversion by Florenskii to a religious viewpoint is described in various sources, including Charles Ford, "The Influence of P. A. Florensky on N. N. Luzin," Hist. Math., 1998, 25:332-339.
83. See also Ford, "Dmitrii Egorov: Mathematics and Religion in Moscow," Math. Intelligencer, 1991, 13:24-30.
84. Ford has done very valuable work on this topic, and we are grateful for his assistance, but we do not agree with him that Egorov's deep religious views and mathematical conceptions were unconnected.
85. Florenskii was first arrested in 1928, then released, then arrested again in 1933 and sentenced to ten years in labor camps in Siberia. He was executed on 8 Aug. 1937. In 1956 he was rehabilitated, and since then he has slowly gained attention as a philosopher of language and culture, a theologian, and, most recently, an influence on Russian mathematics.
86. See Richard Gustafson, "Introduction," in Florenskii, Pillar and Ground of the Truth, trans. Jakim, pp. ix-xxiii.
87. Egorov was rebuked by the Communist Party in 1929, arrested in 1930, and sent to prison. There he went on a hunger strike. Just before his death he was taken under guard to a hospital in Kazan; he died on 10 Sept. 1931. We are told that he died in the arms of the wife of the mathematician N. G. Chebotaryov, who was a doctor in the hospital. Chebotaryov's son G. N. Chebotaryov wrote, "On umer na maminykh rukakh" ("He died in my mother's arms"): G. N. Chebotaryov, "Iz vospominanii ob ottse," in Nikolai Grigor'evich Chebotarev, ed. Iu. B. Ermolaev (Kazan: Izdatel'stvo Kazanskogo Universiteta, 1994), pp. 54-68, on p. 56.
88. S. S. Demidov and V. D. Esakov, "'Delo akademika N. N. Luzina' v svete stalinskoi reformy sovetskoi nauki," Istor Matemat. Issled., 2nd Ser., 1999, 39:156-170;
89. Demidov and B. V. Levshin, eds., Delo akademika Nikolaia Nikolaevicha Luzina (St. Peterburg: Russkii Khristianskii Gumanitarnyi Institute, 1999);
90. A. P Iushkevich, "Delo akademika N. N. Luzina," Vestnik AN SSSR, 1989, 4:102-113;
91. Alexey Levin, "Anatomy of a Public Campaign: 'Academician Luzin's Case' in Soviet Political History," Slavic Review, 1990,49:211-252;
92. A. N. Bogoliubov and N. M. Rozhenko, "Opyt' 'vnedreniia' dialektiki v matematiku v kontse 20-kh-nachale 30-kh godov," Voprosy Filosofii, 1991, 9:32-43.
93. On 15 July 1908 Luzin wrote Florenskii: "Two times I was very close to suicide-then I came here... looking to talk with you, and both times I felt as if I had leaned on a 'pillar.'... I owe my interest in life to you." Ford, "Influence of Florensky on Luzin" (cit. n. 36), p. 338.
94. "Perepiska N. N. Luzina s P. A. Florenskim," Istor. Matemat. Issled., 1989, 31:136.
95. Florenskii, Pillar and Ground of the Truth, trans. Jakim (cit. n. 36), p. 160.
96. For a philosophical discussion of discontinuity, especially in the Russian context, see M. D. Akhundov, Problema preryvnosti i neprerynosti prostranstva i vremeni (Moscow: Nauka, 1974).
97. See also J. Bouveresse, "Weyl, Wittgenstein, et le problème du continu," in Le labyrinthe du continu, ed. J.-M. Salanskis and H. Sinaceur (Paris/Berlin: Springer, 1992).
98. P. A. Florenskii, "Vvedenie k dissertatsii 'Ideia preryvnosti kak element mirosozertsaniia,'" Istor Matemat. Issled, 1986, 30:159-176, on p. 170.
99. Ibid, p. 164. On the introduction of terms from set theory at Moscow University see F. A. Medvedev, "O kurse lektsii B. K. Mlodzeevskogo po teorii funktsii deistvitel'nogo peremennogo, prochitannykh osen'iu 1902 g. v Moskovskom universitete," Istor Matemat. Issled, 1986, 30:130-147; deistvitel'nogo peremennogo, prochitannykh osen'iu 1902 g. v Moskovskom universitete"
100. on Cantor see Joseph W. Daubers, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, Mass.: Harvard Univ. Press, 1979).
101. E. S. Polshchuk, ed., Imiaslavie: Antologiia (Moscow: Faktorial, 2002).
102. See also O. L. Solomina and A. M. Khitrov, Zabytye stranitsy russkogo imiaslaviia: Sbornik dokumentov i publikatsii po afonskim sobytiiam 1910-1913 gg. (Moscow: Palomnik, 2001);
103. Iu. Rasskazov, Sekrety imen: Or imiaslavii do filosofii iazyka (Moscow: Labirint, 2000);
104. Ilarion, Sviashchennaia taina tserkvi: Vvedenie v istoriiu i problematiku imiaslavskikh sporov (St. Petersburg: Aleteiia, 2002);
105. Georges Florovsky, Ways of Russian Theology, Pt. 2, in The Collected Works of Georges Florovsky, ed. Richard S. Haugh, trans. Robert L. Nichols (Belmont, Mass.: Buchervertriebsanstalt, 1987), Vol. 6, p. 376;
106. Tom Dykstra, "Heresy on Mt. Athos: Conflict over the Name of God among Russian Monks and Hierarchs, 1912-1914"
107. (M.A. essay, St. Vladimir's Seminary, New York, 1988)
108. Antoine Niviere, "Les moines onomatodoxes et l'intelligentsia Russe, Cahiers du Monde Russe et Soviétique, 1988, 29(2):181-194
109. Scott M. Kenworthy, Church, State, and Society in Late-Imperial Russia: The Imiaslavie Controversy," paper presented at the American Association for the Advancement of Slavic Studies National Convention, Pittsburgh, Nov. 2002. Although the debates over Name Worshipping became particularly intense in the first third of the twentieth century, the roots of the controversy go far back in the history of Eastern Orthodoxy and can be found in some of the writings and sayings of Basil the Great, John Chrysostom, and other Church figures. See Florovsky, Ways of Russian Theology.
110. For a scriptural reference to human inability to comprehend God see, e.g., Philippians 4:7: and the peace of God, which surpasses all understanding, will guard your hearts and minds through Christ Jesus." Both mathematics and religion make heavy, though very different, use of symbols. And in both the question of the autonomy of the symbols arises. Although Florian Cajori does not directly confront this issue, see his useful A History of Mathematical Notations (Chicago: Open Court, 1929)
111. see also Paul D. L. Avis, God and the Creative Imagination: Metaphor, Symbol, and Myth in Religion and Theology (New York: Routledge, 1999).
112. The full name of the book was In the Mountains of the Caucasus: A Conversation between Two Elder Ascetics Concerning the Inner Union of Our Hearts with the Lord through the Prayer of Jesus Christ, or, The Spiritual Activity of Contemporary Hermits, Composed by the Hermit of the Caucasus Mountains, the Monk Ilarion (Na gorakh Kavkaza, beseda dvukh startsev podvishnikov o vnutrennem edinenii s Gospodom nashikh serdets chrez molitvu Iisus Khristova ili dukhovnaia deiatel'nost' soveremennykh pustinnikov, sostavil pustynnozhitel' Kavkavskikh gor skhimonakh Ilarion] (1907), 3rd ed. (Kiev: Kievskaia Perchersskaia Lavra, 1912). The roots of the Jesus Prayer go back centuries; the start of the tradition is often cited as the Apostle Paul's instruction to the faithful to "pray without ceasing" (1 Thess. 5:17). The tradition exists in both Catholic and Orthodox liturgies, and there is an extensive literature on it. In Russia the Jesus Prayer acquired a special prominence, especially after the publication of the folklore classic The Way of the Pilgrim in Kazan in 1884. The book popularized the prayer and was translated into many languages.
113. A. F. Losev, "Imiaslavie," http://www.ccel.org/contrib/ru/ Other/Losev/ONOMATOD.HTM, p. 3; and Polshchuk, ed., Imiaslavie: Antologiia (cit. n. 44), p. 490.
114. Salinger has Franny observe to her incredulous friend Lane, "Well, the starets tells him about the Jesus Prayer first of all... If you keep saying that prayer over and over again-you only have to just do it with your lips at first-then eventually what happens, the prayer becomes self-active. Something happens after a while. I don't know what, but something happens, and the words get synchronized with the person's heartbeats, and then you're actually praying without ceasing. Which has a really tremendous, mystical effect on your whole outlook. I mean that's the whole point of it, more or less. I mean you do it to purify your whole outlook and get an absolutely new conception of what everything's about." J. D. Salinger, Franny and Zooey (Boston: Little, Brown, 1961), pp. 36-37.
115. Polshchuk, ed., Imiaslavie: Antologiia, pp. 479-518.
116. According to Vladimir Gubanov, the Holy Synod was urging the tsar to squelch the heresy before it split the faith and the nation, but the monk Grigory Rasputin, who was close to the court, defended the Name Worshippers. The tsar evidently hesitated but in the end gave in to the synod. See Vladimir Gubanov, Tsar' Nikolai II i novye mucheniki: Prorochestva, chudesa, otkrytiia i molitvy: Dokumenty (St. Petersburg: Obshchestvo svt. Vasilliia Velikogo, 2000). urging the tsar to squelch the heresy before it split the faith and the nation, but the monk Grigory Rasputin, who was close to the court, defended the Name Worshippers. The tsar evidently hesitated but in the end gave in to the synod. See Vladimir Gubanov, Tsar' Nikolai II i novye mucheniki: Prorochestva, chudesa, otkrytiia i molitvy: Dokumenty (St. Petersburg: Obshchestvo svt. Vasilliia Velikogo, 2000).
117. Tom Dykstra also has written that Rasputin may have supported the Name Worshippers in "Heresy on Mt. Athos" (cit. n. 44).
118. Demidov, "Professor Moskovskogo universiteta Dmitrii Fedorovich Egorov i imiaslavie v Rossii v pervoi treti XX stoletiia" (cit. n. 36), pp. 129-130.
119. Regarding the survival of Name Worshipping see Polshchuk, ed., Imiaslavie: Antologiia (cit. n. 44), p. 513;
120. Ep. Marion (Alfeev), Sviashchennaia taina tserkvi, Vol. 2, http://st-jhouse.narod.ru/biblio/texts.htm.
121. On its present interest to intellectuals see the recent essay by A. N. Parshin, well-known mathematician, pupil of Igor Shafarevich, and corresponding member in the Department of Mathematics of the Russian Academy of Sciences: "Suet i Slovo (k filosofii imeni)," in Imiaslavie: Antologiia, ed. Polschuk, pp. 529-544.
122. Polshchuk, ed., Imiaslavie: Antologiia, p. 513; and http://st-jhouse. narod.ru/biblio/texts.htm.
123. For a modern translation of Memphite theology see Marshall Clagett, Ancient Egyptian Science, Vol. 1 (Philadelphia: American Philosophical Society, 1989), pp. 305-312,595-602.
124. We are grateful to John Murdoch for this suggestion. On the Jewish mystical tradition see Gershom Scholem, Major Trends in Jewish Mysticism (New York: Schocken, 1995).
125. Henri Lebesgue, "Contribution a l'étude des correspondances de M. Zermelo," Bull. Soc. Math. France, 1907, 35:227-237, esp. pp. 228, 236.
126. Later, after the intervention of Zermelo and also Richard's letter, the adjective "effective" or "denumerable" would be used, e.g., by Borel: see Hadamard et al., "Cinq lettres sur la théorie des ensembles" (cit. n. 17).
127. Roger Cooke, "N. N. Luzin on the Problems of Set Theory," unpublished draft, Jan. 1990, pp. 1-2, 7.
128. Luzin's notes are held in the Archive of the Academy of Sciences of the USSR, Moscow, fond 606, op. 1, ed. khr. 34 (the "scribble" is in infelicitous but understandable French).
129. M. A. Lavrent'ev, "Nikolai Nikolaevich Luzin," Russian Mathematical Surveys, 1974,29(5):173-178 [Uspekhi Mathematicheskikh Nauk, 1974, 29(5):177-182];
130. Esther R. Phillips, "Nicolai Nicolaevich Luzin and the Moscow School of the Theory of Functions," Hist. Math., 1978, 5:275-305, on p. 293.
131. Descriptive set theory was a means to analyze the "complexity" of subsets of real numbers using the complexity of their description. A major aim of descriptive set theory is to describe all of the "naturally occurring" sets of real numbers by using various constructions to build a strict hierarchy, beginning with simplest sets made of open intervals of real numbers.
132. Waclaw Sierpinski, Les ensembles projectifs et analytiques (Paris: Gauthier-Villars, 1950), pp. 44-47.
133. For more information on Suslin, who died tragically young, see V. I. Igoshin, "A Short Biography of Mikhail Yakovlevich Suslin," Russian Math. Surv., 1996, 51(3):371-383.
134. Sierpinski, Les ensembles projectifs et analytiques.
135. Hermann Weyl, God and the Universe: The Open World (New Haven, Conn.: Yale Univ. Press, 1932), p. 8.
136. While we know that Egorov was a leader of this circle, we have no concrete evidence that Luzin was or that he ever attended its meetings. We do know that Luzin was a friend of Florenskii, that he was familiar with the Name Worshipping movement, and that he put great emphasis on "naming" in his mathematical research. Luzin was more cautious than Egorov and probably made more of an effort to conceal his religious views from the Soviet authorities.
137. "... exigences mathématiques et exigences philosophiques sont constamment associées, on pent meme dire fondues": Henri Lebesgue, "Préface," in Nicolas Lusin, Leçons sur les ensembles analytiques et leurs applications (Paris: Gauthier-Villars, 1930), p. xi.
138. "M. Lusin examine les questions d'un point de vue philosophique et aboutit ainsi à des résultats mathématiques: originalité sans précédent!" ibid., p. ix.