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3. Biddle, Sheila, Internationalization: Rhetoric or Reality ? American Council of Learned Societies Occasional Paper, no. 56 (New York, 2002), 69.Google ScholarPubMed

4. Hacking, Ian, The Social Construction of What? (Cambridge, Mass., 1999), 44.Google Scholar

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6. Set theory is the mathematical science of the infinite; it starts from scratch with sets. A set is any well-defined collection of objects (the elements belonging to the set); the collection can be finite or infinite; correspondences between sets are functions; if there exists a one-to-one correspondence between two sets they are said to have the same cardinal number (this number can be infinite). As an example, any set in one-to-one correspondence with the set of integers is said to be denumerable; its cardinal is the Hebrew letter aleph with a subscript of 0 (N0), sometimes called “aleph-nought.” Ordinal numbers are similarly defined 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 Georg Cantor in 1882 in close connection with philosophical and even religious considerations about “the Absolute.” This was the first mathematical introduction of infinity in mathematics (called “actual infinite” in opposition, following Aristotle, to “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 led to descriptive set theory. See Ferreiros, Jose Dominquez, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Boston, 1999)CrossRefGoogle Scholar; Purkert, Walter, “Georg Cantor und die Antinomien der Mengenlehre,” Bulletin de la Société mathematique de Belgique 38 (1986): 313-27Google Scholar; and Purkert, W., Georg Cantor, 1845-1918 (Basel, 1987)Google Scholar.

7. Weyl, Hermann, God and the Universe: The Open World (New Haven, 1932), 8 Google Scholar.

8. Weyl, Hermann, “Über die neue Grundlagenkrise der Mathematik,” Mathematische Zeitschrift 10, nos. 1-2 (1921): 39–79 CrossRefGoogle Scholar. A particularly clear exposition of the crisis can be found in Segal, Sanford L., “The Crisis in Mathematics,” Mathematicians under the Nazis (Princeton, 2003), 14–41 CrossRefGoogle Scholar.

9. Dauben, J. W., Georg Cantor: His Mathematics and Philosophy of the Infinite (1979; reprint, Cambridge, Mass., 1990)Google Scholar. As mentioned above, the set containing all natural numbers is usually called א0.

10. See Loren Graham and Jean-Michel Kantor, “A Comparison of Two Cultural Approaches to Mathematics: France and Russia, 1890-1930,” Isis (March 2006): 56-74.

11. See, especially, Hadamard, J., “Cinq lettres sur la théorie des ensembles,” Bulletin de la Société Mathématiqued eFrance 33 (1905): 261-73CrossRefGoogle Scholar.

12. “Certains adeptes de la théorie des ensembles sont des scolastiques, qui auraient aimé â discuter les preuves de l'existence de Dieu, avec Saint-Anselme et son contradicteur, le moine de Noirmoutiers.” Picard, Emile, La science moderne et son état actuel (Paris, 1909), extract from chapter 2Google Scholar. See Cantorbery, Anselme de, Proslogion Allocution sur l'existence de Dieu suivi de sa réfutation par Gaunilon etde la réponse d'Anselme (Paris, 1993).Google Scholar

13. Polishchuk, E. S., ed., Imiaslavie: Antologiia (Moscow, 2002)Google Scholar; also see Solomina, O. L. and Khitrov, A. M., Zabytye stranitsy russkogo imiaslaviia: Sbornik dokumentov i publikatsii po afonskim sobytiiam 1910-1913gg. (Moscow, 2001)Google Scholar; Rasskazov, Iu., Sekrety imen: Ot imiaslavii do filosofii iazyka (Moscow, 2000)Google Scholar; Episkop Ilarion (Alfeev), Sviashchennaia taina tserkvi: Vvedenie v istoriiu i problematiku imiaslavskikh sporov, vols. 1-2 (St. Petersburg, 2002); Pravoslavnyi vzgliad na pochitanie imenii Bozhiia: Sobytiia na Afone 1913 g. (L'viv, 2003); Tom Dykstra, “Heresy on Mt. Athos: Conflict over the Name of God among Russian Monks and Hierarchs, 1912-1914” (MA thesis, St. Vladimir's Seminary, New York, 1988); Antoine Niviere, “Les moines onomatodoxes et l'intelligentsia Russe,” Cahiers du Monde russe et sovietique 29, no. 2 (1988): 181-94; Scott M. Kenworthy, “Church, State, and Society in Late- Imperial Russia: The Imiaslavie Controversy” (paper, American Association for the Advancement of Slavic Studies, Pittsburgh, November 2002); Eugene Clay, “Orthodox Missionaries and Orthodox Heretics in Russia, 1886-1917,” in Geraci, Robert P. and Khodarkovsky, Michael, eds., Of Religion and Empire: Missions, Conversion and Tolerance in Tsarist Russia (Ithaca, 2001), 38–69, especially 63-67.Google Scholar

14. See Georges Florovsky, Ways of Russian Theology, pt. 2, vol. 6, in Haugh, Richard S., ed., The Collected Works of Georges Florovsky, trans. Nichols, Robert L. (Belmont, Mass., 1987)Google Scholar.

15. For a modern translation of Memphite Theology, see Clagett, Marshall, Ancient Egyptian Science (Philadelphia, 1989), 1:305-12, 595-602Google Scholar. We are grateful to John Murdoch for this suggestion.

16. Scholem, Gershom, Major Trends infewish Mysticism (New York, 1995).Google Scholar

17. Eberle, R. A., Nominalistic Systems (Dordrecht, 1970)CrossRefGoogle Scholar; Kaluza, Zenon, Les querelles doctrinales a Paris: Nominalistes et realistes aux confins du XTVe et du XVe siécles (Bergamo, 1988)Google Scholar.

18. The full title of the book was Na gorakh Kavkaza, beseda dvukh startsev podvishnikov o vnutrennem edinenii s Gospodom nashikh serdets chrez molitvu Iisus Khristova Hi dukhovnaia deiatel'nost’ sovremennykh pustinnikov, sostavil pustynnozhitel’ Kavkavskikh gor skhimonakh Ilarion (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) (1st ed., Batalpashinsk, 1907; 2d corrected and expanded ed., 1910; 3d ed., Kievskaia Pecherskaia Lavra, 1912). The roots of the Jesus Prayer go back centuries; the start of the tradition is often cited as Apostle Paul's instructions to the faidiful 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 (Kazan, 1884). The book popularized the prayer and was translated into many languages.

19. Salinger has Franny observing to her incredulous friend Lane, “Well, the starets tells him about the Jesus Prayer first of a l l … . 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, 1961), 36-37, emphasis in the original.

20. See description of Ilarion and his followers in N. K. Bonetskaia, “Bor'ba za Logos v Rossii v XX veke,” Voprosy filosqfii 7 (1998): 148-69, especially 150-51; Polishchuk, ed., Imiaslavie, 490, emphasis in the original.

21. Polishchuk, ed., Imiaslavie, 479-518. These events were also reported in the foreign press; see “Heresy at Mount Athos: A Soldier Monk and the Holy Synod,” London Times, 19 June 1913; “Heresy at Mount Athos, London Times, 23 August 1913; “The Mount Athos Heresy Case: Voluntary Exile in Siberia,” Times, 23 August 1913.

22. According to Vladimir Gubanov, the Holy Synod urged the tsar to squelch the heresy before it split the faith and the nation, but the monk Grigorii Rasputin, who was close to the court, defended the Name Worshippers. The tsar evidently hesitated but in the end gave in to the synod. Gubanov, Vladimir, Tsar'Nikolai II i novye mucheniki: Prorochestva, chudesa, otkrytiia i molitvy: Dokumenty (St. Petersburg, 2000)Google Scholar. Tom Dykstra has also written that Rasputin may have supported the Name Worshippers. Dykstra, “Heresy on Mt. Athos.“ On Rasputin's role, see also Ilarion (Alfeev), Sviashchennaia taina tserkvi, 2:15 and 64.

23. Demidov, S. S., “Professor Moskovskogo universiteta Dmitrii Fedorovich Egorov i imeslavie v Rossii v pervoi treti XX stoletiia,“ Istoriko-matematicheskieissledovaniia, 2d ser., 39 (1999): 129-30.Google Scholar

24. Polishchuk, ed., Imiaslavie, 513.

25. Ilarion (Alfeev), Sviashchennaia taina tserkvi, vol. 2.

26. 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: Parshin, “Svet i slovo (k filosofii imeni),” in Polishchuk, ed., Imiaslavie, 529-44.

27. Egorov's deep religiosity is described in Demidov, “Professor Moskovskogo universiteta Dmitrii Fedorovich Egorov,” 137. Luzin's conversion by Florenskii to a religious viewpoint is described in various sources, including Ford, Charles, “The Influence of P. A. Florensky on N. N. Luzin,” Historia Mathematica 25, no. 3 (August 1998): 332-39CrossRefGoogle Scholar. See also Ford, , “Dmitrii Egorov: Mathematics and Religion in Moscow,” Mathematical Intelligencer, no. 2 (1991): 24–30 Google Scholar. Florenskii's best-known published statement of faith is probably his Stolp i utverzhdenie istiny (Moscow, 1914), published in English as The Pillar and Ground of the Truth, trans. Boris Jakim (Princeton, 1997).

28. Polishchuk, ed., Imiaslavie, 513.

29. Ilarion (Alfeev), Sviashchennaia taina tserkvi, 2:114.

30. Cantor, Georg, “Über unendliche, lineare Punktmannichfaltigkeiten,” MathematischeAnnalen 21 (1883): 545 Google Scholar.

31. Lebesque as cited in Hadamard, “Cinq lettres sur la dieorie des ensembles.“

32. At one point Luzin wrote in his notes “Everything seems to be a daydream, playing widi symbols, which however, yield great things.” At another moment Luzin scribbled in infelicitous but understandable French: “nommer, c'est avoir individu.” Archive of the Russian Academy of Sciences, Moscow, f. 606, op. 1, ed. khr. 34. Courtesy of Roger Cooke, “N. N. Luzin on the Problems of Set Theory” (unpublished paper, January 1990), 1-2, 7.

33. Lavrent'ev, M. A., “Nikolai Nikolaevich Luzin,” Russian Mathematical Surveys 29, no. 5 (1974): 173-78CrossRefGoogle Scholar, and Uspekhi mathematicheskikh nauk 29, no. 5 (1974): 177-82.

34. Phillips, Esther R., “Nicolai Nicolaevich Luzin and the Moscow School of the Theory of Functions,” Historia Mathematica 5, no. 3 (August 1978): 293 CrossRefGoogle Scholar.

35. Charles Coulston Gillispie, ed., Dictionary of Scientific Biography (New York, 1970- 1990). Luzin is in 8:557-59; Egorov in 4:287-88; Aleksandrov in 17.2:11-15; Uryson in 13:548-49; Stepanov in 13:35-36.

36. While we know that Egorov was a leader of this circle, we have no concrete evidence that Luzin was a member or even ever attended meetings. We do know that Luzin was a friend of Father Florenskii, that he was familiar with the Name Worshipping movement, and that in his mathematical research he put great emphasis on “naming.” Luzin was more cautious than Egorov and probably made more of an attempt to conceal his religious views from the Soviet authorities.

37. 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 December 1937. Rehabilitated in 1956, he has slowly gained attention since then as a philosopher of language and culture, a theologian, and, most recently, as an influence on Russian mathematics. See Richard Gustafson, “Introduction,” in Florenskii, The Pillar and Ground of the Truth, ix-xxiii. 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 September 1931. We are told that he died in die arms of the wife of the mathematician N. G. Chebotarev, who was a doctor in the hospital. Chebotarev's son G. N. Chebotarev wrote, “On umer na maminykh rukakh” (He died in my mother's arms): G. N. Chebotarev, “Iz vospominanii ob ottse,” in Iu. B. Ermolaev, ed., Nikolai Grigor'evich Chebotarev (Kazan, 1994), 56. Luzin was submitted to an ideological trial in which many of his former colleagues turned against him. See Demidov, S. S. and Esakov, V. D., “'Delo akademika N. N. Luzina’ v svete stalinskoi reformy sovetskoi nauki,“ Istoriko-matematicheskieissledovania, 2d ser., 39, no. 4 (1999): 156-70Google Scholar. Demidov, S. S. and Levshin, B. V., eds., Delo akademika Nikolaia Nikolaevicha Luzina (Petersburg, St., 1999)Google Scholar; Iushkevich, A. P., “Delo akademika N. N. Luzina,” Vestnik ANSSSR, no. 4 (1989): 102-13Google Scholar; Levin, Alexey E., “Anatomy of a Public Campaign: ‘Academician Luzin's Case’ in Soviet Political History,” Slavic Review 49, no. 1 (Spring 1990): 90–108 CrossRefGoogle Scholar; Bogoliubov, A. N. and Rozhenko, N. M., “Opyt’ ‘vnedreniia’ dialektiki v matematiku v kontse 20-kh nachale 30-kh godov,” Voprosy filosofii, no. 9 (1991): 32–43.Google Scholar

38. Shields, Allen, “Years Ago: Luzin and Egorov,” Mathematical Intelligencer 9 (1987): 24–27 CrossRefGoogle Scholar; Zdravkovska, Smilka and Duren, Peter L., eds., Golden Years of Moscow Mathematics (Providence, 1993)Google Scholar; Vucinich, Alexander, “Mathematics and Dialectics in the Soviet Union: The Pre-Stalin Period,” Historia Mathematica 26, no. 2 (May 1999): 107-24CrossRefGoogle Scholar; Aleksandrov, P. S., Matematika v SSSR la 15 let (Moscow, 1932).Google Scholar

39. “Exigences mathématiques et exigences philosophiques sont constamment associées, on peut meme dire fondues.” Lebesgue, Henri, “Préface,” in Luzin, Nicolas, Lecons sur les ensembles analytiques et leurs applications (Paris, 1930), xi.Google Scholar

40. “M. Lusin examine les questions d'un point de vue philosophique et aboutit ainsi â des résultats mathématiques: originalite sans précédent!” Lebesgue, “Préface,” ix.

41. In the early development of set theory, the works of Bernard Bolzano and Georg Cantor, both of whom had strong philosophical and religious beliefs, await deeper contextual analysis. The work of the Russian mathematician and geometer A. D. Alexandrov, who had strong philosophical commitments, also beckons. The Bourbaki mathematics group in France awaits further contextual examination. The great mathematician Alexander Grothendieck, who was interested in mysticism at many points in his life, should be similarly examined.

42. Kaiser, David, “A ip is just ai)j? Pedagogy, Practice, and the Reconstitution of General Relativity, 1942-1975,” Studies in the History and Philosophy of Modern Physics, 29 no. 3 (1998): 336.CrossRefGoogle Scholar

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44. Graham, Loren R., “Do Mathematical Equations Display Social Attributes?” Mathematical Intelligencer 22, no. 3 (Summer 2000): 31–36.CrossRefGoogle Scholar

45. Fock asserted his Marxism in dozens of publications and conversations, both with one of the authors (Loren Graham) and with others. Graham met with Fock in Leningrad in the spring of 1961 and engaged in a published discussion with him in 1966: Loren R. Graham, “Quantum Mechanics and Dialectical Materialism,” Slavic Review 25, no. 3 (September 1966): 381-410; see Fock's reply in his “Comments,” Slavic Review 25, no. 3 (September 1966): 411-13. After the fall of the Soviet Union, when there was no longer any political reason to affirm Fock's Marxism, people who knew him and his work continued to describe him as an intellectual Marxist. Two years after the end of the USSR, in 1993, the historian of Russian physics Gennadii Gorelik wrote, “There is no doubt that in the 1930s Fock was already sincerely devoted to dialectical materialism.” Gorelik, G. E., “V. A. Fok: Filosofiia tiagoteniia i tiazhest'filosofii,” Priroda, no. 10 (1993): 92 Google Scholar.

46. As reflected in the English translation of the title of one of his best-known works: V. A. Fock, The Theory of Space, Time, and Gravitation, trans. N. Kemmer (New York, 1959).

47. See Holton, Gerald, “Introduction: Einstein and the Shaping of Our Imagination,“ in Holton, Gerald and Elkana, Yehuda, eds., Albert Einstein: Historical and Cultural Perspectives (Princeton, 1982), xv.Google Scholar

48. As Max Planck commented, “The concept of relativity is based on a more fundamental absolute than the erroneously assumed absolute which it has supplanted.” Max Planck, TheNew Science (Greenwich, Conn., 1959), 146. Planck expressed the same idea in earlier publications, for example, Das Weltbild derNeuen Physik (Leipzig, 1929), 18.

49. Fok [Fock], “Protiv nevezhestvennoi kritiki sovremennykh fizicheskikh teorii,“ 172.

50. “Es ist nicht überflüssig zu unterstreichen, dass das Verhältnis von Körpern oder Prozessen zum Bezugssystem ebenso objektiv ist (d.h. unabhängig von unserem Bewusstsein) wie uberhaupt alle physikalischen und anderen Eigenschaften der Korpern” (It is not superfluous to emphasize that the relationship of bodies or processes to the reference system is just as objective [i.e., independent of our consciousness] as are physical and other properties in general). Fock, V. A., “Über philosophische Fragen der modernen Physik,“ Deutsche Zeitschrift für Philosophie, no. 6 (1955): 742 Google Scholar.

51. Fock, Theory of Space, Time, and Gravitation, xv, xvi, 351. See also V. A. Fok [Fock], “Poniatiia odnorodnosti, kovariantnosti i otnositel'nosti,” Voprosy filosofii, no. 4 (1955): 133.

52. Fock, Theory of Space, Time, and Gravitation, 193; also see 366-75.

53. Siegfried Miiller-Markus, a German historian and philosopher of science, noting leading physicists’ praise of Fock's work, ended up writing a book positively interpreting Fock's views of general relativity, even though his original intention had been to criticize them. See Siegfried Miiller-Markus, Einstein und die Sotujetphilosophie, vol. 2 (Dordrecht, 1970).

54. See John Wheeler's response to Loren R. Graham, “The Reception of Einstein's Ideas: Two Examples from Contrasting Political Cultures,” in Gerald Holton and Yehuda Elkana, eds., Albert Einstein: Historical and Cultural Perspectives (Princeton, 1982), 107-36 (Wheeler's response appears on p. 135).

55. See “Naravne s Einshteinom,” Poisk, 29 January 1999, 8.

56. Logunov, A. A., Relativistic Theory of Gravity (Commack, N.Y., 1998), 65 Google Scholar; see also Logunov, , The Theory of Gravity (Moscow, 2001)Google Scholar.

57. Stigler, Stephen M., The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, Mass., 1986)Google Scholar; Stigler, , Statistics on the Table: The History of Statistical Concepts and Methods (Cambridge, Mass., 1999)Google Scholar; Daston, Lorraine, Classical Probability in the Enlightenment (Princeton, 1988)Google Scholar; Porter, Theodore, The Rise of Statistical Thinking: 1820-1900 (Princeton, 1986)Google Scholar; Desrosieres, Alain, The Politics of Large Numbers: A History of Statistical Reasoning, trans. Camille Naish (Cambridge, Mass., 2002)Google Scholar.

58. Desrosieres, Politics of Large Numbers, 11.

59. Mosteller, Frederick and Moynihan, Daniel P., eds., On Equality of Educational Opportunity (New York, 1972)Google Scholar; Mosteller, Frederick and Wallace, David L., Inference and Disputed Authorship: The Federalist Papers (Reading, Mass., 1964)Google Scholar.

60. Gardner, Howard, Multiple Intelligences: New Horizons (New York, 2006), 5.Google Scholar

61. See also Mayr, Ernst, This Is Biology: The Science of the Living World (Cambridge, Mass., 1997).Google Scholar

62. See Hacking, Social Construction of What? The initial and best-known episode in the controversy was Alan D. Sokal's spoof of social constructivists in his “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity,” Social Text, no. 46/47 (Spring/Summer 1996): 217-52. Sokal revealed that this article was a hoax designed to parody diose who “socially construct” science in his “A Physicist Experiments with Cultural Studies,” Lingua Franca (May/June 1996): 62-64. Sokal's original article was very clever and he was correct in ridiculing the views of the most extreme social contructivists. The basic issue of the controversy—to what extent are science and mathematics affected by the society in which they developed?—remains, however, unresolved. See also Noretta Koertge, ed., A House Built on Sand: Exposing Postmodernist Myths about Science (New York, 1998), especially the essays by Alan Sokal and Philip Kitcher.