Journal for General Philosophy of Science

, Volume 48, Issue 2, pp 195–238 | Cite as

Interpreting the Infinitesimal Mathematics of Leibniz and Euler

  • Jacques Bair
  • Piotr Błaszczyk
  • Robert Ely
  • Valérie Henry
  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. KatzEmail author
  • Semen S. Kutateladze
  • Thomas McGaffey
  • Patrick Reeder
  • David M. Schaps
  • David Sherry
  • Steven Shnider


We apply Benacerraf’s distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves.


Archimedean axiom Infinite product Infinitesimal Law of continuity Law of homogeneity Principle of cancellation Procedure Standard part principle Ontology Mathematical practice Euler Leibniz 

Mathematics Subject Classification

Primary 01A50; Secondary 26E35 01A85 03A05 


  1. Arthur, R. (2008). Leery Bedfellows: Newton and Leibniz on the status of infinitesimals. In Ursula Goldenbaum & Douglas Jesseph (Eds.), Infinitesimal differences: Controversies between Leibniz and his contemporaries (pp. 7–30). Berlin and New York: de Gruyter.Google Scholar
  2. Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D. & Shnider, S. (2013). Is mathematical history written by the victors? Notices of the American Mathematical Society, 60(7), 886–904.Google Scholar
  3. Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., et al. (2014). Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow. Notices of the American Mathematical Society, 61(8), 848–864.CrossRefGoogle Scholar
  4. Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Schaps, D. & Sherry, D. (2016). Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania. HOPOS: The Journal of the International Society for the History of Philosophy of Science, 6(1), 117–147.Google Scholar
  5. Beckmann, F. (1967/1968). Neue Gesichtspunkte zum 5. Buch Euklids. Archive for History Exact Sciences, 4, 1–144.Google Scholar
  6. Bell, J. (2008). A primer of infinitesimal analysis (2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  7. Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74, 47–73.CrossRefGoogle Scholar
  8. Benci, V., & Di Nasso, M. (2003). Numerosities of labelled sets: A new way of counting. Advances in Mathematics, 173(1), 50–67.CrossRefGoogle Scholar
  9. Berkeley, G. (1734). The analyst, a discourse addressed to an infidel mathematician.Google Scholar
  10. Błaszczyk, P. (2013). A note on Otto Hölder’s treatise Die Axiome der Quantität und die Lehre vom Mass. Annales Academiae Paedagogicae Cracoviensis. Studia ad Didacticum Mathematicae V, 129–142 [In Polish].Google Scholar
  11. Błaszczyk, P., Katz, M., & Sherry, D. (2013). Ten misconceptions from the history of analysis and their debunking. Foundations of Science, 18(1), 43–74.Google Scholar
  12. Błaszczyk, P., Kanovei, V., Katz, M., & Sherry, D. (2016). Controversies in the foundations of analysis: Comments on Schubring’s conflicts. Foundations of Science. doi:  10.1007/s10699-015-9473-4. Online first.
  13. Błaszczyk, P., Mrówka, K. (2013). Euklides, Elementy, Ksiegi V-VI. Tłumaczenie i komentarz [Euclid, Elements, Books V–VI. Translation and commentary]. Krakow: Copernicus.Google Scholar
  14. Borovik, A., & Katz, M. (2012). Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Foundations of Science, 17(3), 245–276.CrossRefGoogle Scholar
  15. Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.CrossRefGoogle Scholar
  16. Bos, H. (2010). Private communication. Nov 2, 2010.Google Scholar
  17. Boyer, C. (1949). The concepts of the calculus. New York: Hafner.Google Scholar
  18. Bradley, R., & Sandifer, C. (2009). Cauchy’s Cours d’analyse. An annotated translation. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.Google Scholar
  19. Breger, H. (1992). Le continu chez Leibniz. Le labyrinthe du continu (Cerisy-la-Salle, 1990) (pp. 76–84). Paris: Springer.Google Scholar
  20. Carroll, M., Dougherty, S., & Perkins, D. (2013). Indivisibles, infinitesimals and a tale of seventeenth-century mathematics. Mathematics Magazine, 86(4), 239–254.CrossRefGoogle Scholar
  21. Cauchy, A.L. (1823). Résumé des Leçons données à l’Ecole Royale Polytechnique sur le Calcul Infinitésimal. Paris, Imprimérie Royale, 1823. In Oeuvres complètes, Series 2, Vol. 4, pp. 9–261. Paris: Gauthier-Villars (1899).Google Scholar
  22. Ciesielski, K., & Miller, D. (2016). A continuous tale on continuous and separately continuous functions. Real Analysis Exchange, 41(1), 19–54.CrossRefGoogle Scholar
  23. Clavius, C. (1589). Euclidis elementorum. Libri XV, Roma.Google Scholar
  24. Dauben, J. (1980). The development of Cantorian set theory. In I. Grattan-Guinnes (Ed.), From the calculus to set theory, 1630–1910 (pp. 181–219). Princeton: Princeton University Press.Google Scholar
  25. De Risi, V. (2016). The development of euclidean axiomatics. The systems of principles and the foundations of mathematics in editions of the elements from antiquity to the eighteenth century. Archive for History of Exact Science, online first. doi: 10.1007/s00407-015-0173-9.
  26. Di Nasso, M., & Forti, M. (2010). Numerosities of point sets over the real line. Transactions of the American Mathematical Society, 362(10), 5355–5371.CrossRefGoogle Scholar
  27. Dijksterhuis, D. (1987). Archimedes 1987. Princeton: Princeton University Press.Google Scholar
  28. Edwards, C. H, Jr. (1979). The historical development of the calculus. New York/Heidelberg: Springer.CrossRefGoogle Scholar
  29. Edwards, H. (2007). Euler’s definition of the derivative. Bulletin of the American Mathematical Society (NS), 44(4), 575–580.CrossRefGoogle Scholar
  30. Edwards, H. (2015). Euler’s conception of the derivative. The Mathematical Intelligencer, 47(4), 52–53.CrossRefGoogle Scholar
  31. Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences, 60(1), 1–121.CrossRefGoogle Scholar
  32. Euclid. (1660). Euclide’s Elements, The whole Fifteen Books, compendiously Demonstrated. By Mr. Isaac Barrow Fellow of Trinity College in Cambridge. And Translated out of the Latin. London.Google Scholar
  33. Euclid. (2007). Euclid’s Elements of Geometry. Edited, and provided with a modern English translation, by Richard Fitzpatrick. See
  34. Euler, L. (1730–1731). De progressionibus trascendentibus seu quarum termini generales algebraice dari nequeunt. In Euler, Opera omnia (Series I, Opera Mathematica, Berlin, Bern, Leipzig, 1911–...), 14(1), 1–24.Google Scholar
  35. Euler, L. (1748). Introductio in Analysin Infinitorum, Tomus primus. Saint Petersburg and Lausana.Google Scholar
  36. Euler, L. (1755). Institutiones Calculi Differentialis. Saint Petersburg.Google Scholar
  37. Euler, L. (1768–1770). Institutionum calculi integralis.Google Scholar
  38. Euler, L. (1771). Vollständige Anleitung zur Algebra. St. Petersburg: Kaiserliche Akademie der Wissenschaften.Google Scholar
  39. Euler, L. (1807). Éléments d’algèbre. Paris: Courcier.Google Scholar
  40. Euler, L. (1810). Elements of Algebra. Translated form the French with additions of La Grange, Johson and Co., London.Google Scholar
  41. Euler, L. (1988). Introduction to analysis of the infinite. Book I. New York: Springer.Google Scholar
  42. Euler, L. (2000). Foundations of Differential Calculus. New York: Springer.Google Scholar
  43. Ferraro, G. (1998). Some aspects of Euler’s theory of series: Inexplicable functions and the Euler–Maclaurin summation formula. Historia Mathematica, 25(3), 290–317.CrossRefGoogle Scholar
  44. Ferraro, G. (2004). Differentials and differential coefficients in the Eulerian foundations of the calculus. Historia Mathematica, 31(1), 34–61.CrossRefGoogle Scholar
  45. Ferraro, G. (2007). Euler’s treatises on infinitesimal analysis: Introductio in analysin infinitorum, institutiones calculi differentialis, institutionum calculi integralis. In R. Baker (Ed.), Euler reconsidered (pp. 39–101). Heber City: Kendrick Press.Google Scholar
  46. Ferraro, G. (2008). The rise and development of the theory of series up to the early 1820s. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.Google Scholar
  47. Ferraro, G. (2012). Euler, infinitesimals and limits. Manuscript (Feb 2012). See
  48. Ferraro, G., & Panza, M. (2003). Developing into series and returning from series: A note on the foundations of eighteenth-century analysis. Historia Mathematica, 30(1), 17–46.CrossRefGoogle Scholar
  49. Fraenkel, A. (1928). Einleitung in die Mengenlehre. Berlin: Springer.Google Scholar
  50. Fraser, C. (1999). Book review: Paolo Mancosu. Philosophy of mathematics and mathematical practice in the seventeenth century. Notre Dame Journal of Formal Logic, 40(3), 447–454.CrossRefGoogle Scholar
  51. Fraser, C. (2015). Nonstandard analysis, infinitesimals, and the history of calculus. In D. Row & W. Horng (Eds.), A delicate balance: Global perspectives on innovation and tradition in the history of mathematics (pp. 25–49). Birkhäuser: Springer.CrossRefGoogle Scholar
  52. Gerhardt, C. I. (Ed.). (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann.Google Scholar
  53. Gödel, K. (1990). Collected Works. In Feferman, S., Dawson, J., Kleene, S., Moore, G., Solovay, R., & van Heijenoort, J. (Eds). Vol. II. New York: Oxford University Press.Google Scholar
  54. Goldblatt, R. (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics 188. New York: Springer.Google Scholar
  55. Gordon, E., Kusraev, A.&, Kutateladze, S. (2002). Infinitesimal analysis. Dordrecht: Kluwer.Google Scholar
  56. Grant, E. (1974). The definitions of Book V of Euclid’s Elements in thirteenth-century version, and commentary. Campanus of Novara. In E. Grant (Ed.), A source book in medieval science (pp. 136–149). Cambridge, MA: Harvard University Press.Google Scholar
  57. Gray, J. (2008a). Plato’s ghost. The modernist transformation of mathematics. Princeton: Princeton University Press.Google Scholar
  58. Gray, J. (2008b). A short life of Euler. BSHM Bulletin, 23(1), 1–12.CrossRefGoogle Scholar
  59. Hacking, I. (2014). Why is there philosophy of mathematics at all? Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  60. Heiberg, J. (1881). Archimedis Opera Omnia cum Archimedis Opera Omnia cum Commentariis Eutocii (Vol. I). Leipzig: Teubner.Google Scholar
  61. Heiberg, J. (1883–1888). Euclidis Elementa, Vol. I–V. Leipzig: Teubner.Google Scholar
  62. Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-physikalische Classe, 53, 1–64.Google Scholar
  63. Hölder, O. (1996). The axioms of quantity and the theory of measurement. Translated from the 1901 German original and with notes by Joel michell & Catherine Ernst. With an introduction by Michell. The Journal of Mathematical Psychology, 40(3), 235–252.Google Scholar
  64. Ishiguro, H. (1990). Leibniz’s philosophy of logic and language (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  65. Jesseph, D. (2015). Leibniz on the elimination of infinitesimals. In N. B. Goethe, P. Beeley & D. Rabouin (Eds.), G.W. Leibniz, Interrelations between mathematics and philosophy. Archimedes Series 41 (pp. 189–205). Berlin: Springer.Google Scholar
  66. Kanovei, V. (1988). The correctness of Euler’s method for the factorization of the sine function into an infinite product. Russian Mathematical Surveys, 43, 65–94.CrossRefGoogle Scholar
  67. Kanovei, V., Katz, K., Katz, M., Nowik, T. (2016). Small oscillations of the pendulum, Euler’s method, and adequality. Quantum Studies: Mathematics and Foundations.Google Scholar
  68. Kanovei, V., Katz, M., Mormann, T. (2013). Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics. Foundations of Science 18(2), 259–296.Google Scholar
  69. Kanovei, V., Katz, K., Katz, M., & Schaps, M. (2015). Proofs and retributions, Or: Why Sarah can’t take limits. Foundations of Science, 20(1), 1–25.CrossRefGoogle Scholar
  70. Kanovei, V., Katz, K., Katz, M., Sherry, D. (2015). Euler’s lute and Edwards’ oud. The Mathematical Intelligencer, 37(4), 48–51.Google Scholar
  71. Kanovei, V., & Reeken, M. (2004). Nonstandard analysis, axiomatically. Springer Monographs in Mathematics. Berlin: Springer.CrossRefGoogle Scholar
  72. Katz, K., Katz, M. (2011). Cauchy’s continuum. Perspectives on Science, 19(4), 426–452.Google Scholar
  73. Katz, M., Schaps, D., Shnider, S. (2013). Almost equal: The method of adequality from diophantus to fermat and beyond. Perspectives on Science, 21(3), 283–324.Google Scholar
  74. Katz, M., Sherry, D. (2012). “Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550–1558.Google Scholar
  75. Katz, M. & Sherry, D. (2013). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78(3), 571–625.Google Scholar
  76. Katz, V. (2014). Review of “Bair et al., Is mathematical history written by the victors?” Notices American Mathematical Society, 60(7), 886–904.Google Scholar
  77. Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach. Second Edition. Prindle, Weber & Schimidt, Boston. See online version at
  78. Klein, F. (1932). Elementary Mathematics from an advanced standpoint. Vol. I.: Arithmetic, algebra, analysis. New York: Macmillan.Google Scholar
  79. Kock, A. (2006). Synthetic differential geometry, 2nd edition. London Mathematical Society Lecture Note Series, 333. Cambridge: Cambridge University Press.Google Scholar
  80. Kuhn, T. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press.Google Scholar
  81. Lakatos, I. (1978). Cauchy and the continuum: The significance of nonstandard analysis for the history and philosophy of mathematics. Mathematical Intelligencer, 1(3), 151–161.Google Scholar
  82. Laugwitz, D. (1987a). Hidden lemmas in the early history of infinite series. Aequationes Mathematicae, 34, 264–276.CrossRefGoogle Scholar
  83. Laugwitz, D. (1987b). Infinitely small quantities in Cauchy’s textbooks. Historia Mathematica, 14, 258–274.CrossRefGoogle Scholar
  84. Laugwitz, D. (1989). Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820. Archive for History of Exact Sciences, 39(3), 195–245.CrossRefGoogle Scholar
  85. Laugwitz, D. (1992). Leibniz’ principle and omega calculus. Le labyrinthe du continu (Cerisy-la-Salle, 1990) (pp. 144–154). Paris: Springer.Google Scholar
  86. Laugwitz, D. (1999). Bernhard Riemann (1826–1866). Turning points in the conception of mathematics. Boston: Birkhäuser Boston.Google Scholar
  87. Leibniz, G. (1684). “Nova methodus pro maximis et minimis ...” in Acta Erud., Oct. 1684. See [Gerhardt 1850–1863], V, pp. 220–226.Google Scholar
  88. Leibniz, G. (1695). To l’Hospital, 21 june 1695, in [Gerhardt 1850–1863], I, pp. 287–289.Google Scholar
  89. Leibniz, G. (1702). To Varignon, 2 febr., 1702, in [Gerhardt 1850–1863] IV, pp. 91–95.Google Scholar
  90. Leibniz, G. (1710). Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. In [Gerhardt 1850–1863, vol. V, pp. 377–382].Google Scholar
  91. l’Hôpital, G. (1696). Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes.Google Scholar
  92. Luxemburg, W. (1973). What is nonstandard analysis? Papers in the foundations of mathematics. American Mathematical Monthly, 80(6, part II), 38–67.Google Scholar
  93. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York: The Clarendon Press.Google Scholar
  94. Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? The Review of Symbolic Logic, 2(4), 612–646.CrossRefGoogle Scholar
  95. McKinzie, M., & Tuckey, C. (1997). Hidden lemmas in Euler’s summation of the reciprocals of the squares. Archive for History of Exact Sciences, 51, 29–57.CrossRefGoogle Scholar
  96. Mueller, I. (1981). Philosophy of mathematics and deductive structure in Euclid’s Elements. Cambridge, MA: MIT Press.Google Scholar
  97. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
  98. Nieuwentijt, B. (1695). Analysis infinitorum, seu curvilineorum proprietates ex polygonorum natura deductae. Amsterdam.Google Scholar
  99. Nowik, T., Katz, M. (2015). Differential geometry via infinitesimal displacements. Journal of Logic and Analysis, 7(5), 1–44.Google Scholar
  100. Panza, M. (2007). Euler’s Introductio in analysin infinitorum and the program of algebraic analysis. In R. Backer (Ed.), Euler reconsidered (pp. 119–166). Heber City: Kendrick Press.Google Scholar
  101. Pulte, H. (1998). Jacobi’s criticism of Lagrange: The changing role of mathematics in the foundations of classical mechanics. Historia Mathematica, 25(2), 154–184.CrossRefGoogle Scholar
  102. Pulte, H. (2012). Rational mechanics in the eighteenth century. On structural developments of a mathematical science. Berichte zur Wissenschaftsgeschichte, 35(3), 183–199.CrossRefGoogle Scholar
  103. Quine, W. (1968). Ontological relativity. The Journal of Philosophy, 65(7), 185–212.CrossRefGoogle Scholar
  104. Reeder, P. (2012). A ‘non-standard analysis’ of Euler’s Introductio in Analysin Infitorum. MWPMW 13. University of Notre Dame, October 27-28, 2012. See
  105. Reeder, P. (2013). Internal set theory and Euler’s Introductio in analysin infinitorum. M.Sc. Thesis, Ohio State University.Google Scholar
  106. Robinson, A. (1961). “Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A, 64 = Indag. Math., 23, 432–440. (reprinted in Selected Works, see Robinson 1979).Google Scholar
  107. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.Google Scholar
  108. Robinson, A. (1969). From a formalist’s points of view. Dialectica, 23, 45–49.CrossRefGoogle Scholar
  109. Robinson, A. (1979). Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. Edited and with introductions by W. A. J. Luxemburg & S. Körner. New Haven: Yale University Press.Google Scholar
  110. Sandifer, C. (2007). Euler’s solution of the Basel problem–the longer story. In Euler at 300, 105–117. MAA Spectrum, Math. Assoc. America, Washington, DC.Google Scholar
  111. Schubring, G. (2016). Comments on a Paper on Alleged Misconceptions Regarding the History of Analysis: Who Has Misconceptions? Foundations of Science. Online first. doi: 10.1007/s10699-015-9424-0.
  112. Sherry, D. (1987). The wake of Berkeley’s Analyst: Rigor mathematicae? Studies in History and Philosophy of Science, 18(4), 455–480.CrossRefGoogle Scholar
  113. Sherry, D., Katz, M. (2014). Infinitesimals, imaginaries, ideals, and fictions. Studia Leibnitiana, 44(2), 166–192.Google Scholar
  114. Stolz, O. (1883). Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes. Mathematische Annalen, 22(4), 504–519.CrossRefGoogle Scholar
  115. Stolz, O. (1885). Vorlesungen über Allgemeine Arithmetik. Leipzig: Teubner.Google Scholar
  116. Tao, T. (2014). Hilbert’s fifth problem and related topics. Graduate Studies in Mathematics, 153. Providence: American Mathematical Society.Google Scholar
  117. Tho, T. (2012). Equivocation in the foundations of Leibniz’s infinitesimal fictions. Society and Politics, 6(2), 70–98.Google Scholar
  118. Vermij, R. (1989). Bernard Nieuwentijt and the Leibnizian calculus. Studia Leibnitiana, 21(1), 69–86.Google Scholar
  119. Wallis, J. (1656/2004). The arithmetic of infinitesimals. Translated from the Latin and with an introduction by Jaequeline A. Stedall. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.Google Scholar
  120. Wallis, J. (1685). Treatise of Algebra. Oxford.Google Scholar
  121. Wartofsky, M. (1976). The relation between philosophy of science and history of science. In R. S. Cohen, P. K. Feyerabend, & M. W. Wartofsky (Eds.), Essays in Memory of Imre Lakatos (pp. 717–737). Dordrecht: Reidel.CrossRefGoogle Scholar
  122. Weber, H. (1895). Lehrbuch der Algebra. Braunschweig: Vieweg.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Jacques Bair
    • 1
  • Piotr Błaszczyk
    • 2
  • Robert Ely
    • 3
  • Valérie Henry
    • 4
  • Vladimir Kanovei
    • 5
  • Karin U. Katz
    • 6
  • Mikhail G. Katz
    • 6
    Email author
  • Semen S. Kutateladze
    • 7
  • Thomas McGaffey
    • 8
  • Patrick Reeder
    • 9
  • David M. Schaps
    • 10
  • David Sherry
    • 11
  • Steven Shnider
    • 6
  1. 1.HEC-ULGUniversity of LiegeLiègeBelgium
  2. 2.Institute of MathematicsPedagogical University of CracowKrakówPoland
  3. 3.Department of MathematicsUniversity of IdahoMoscowUSA
  4. 4.Department of MathematicsUniversity of NamurNamurBelgium
  5. 5.IPPI, Moscow, and MIITMoscowRussia
  6. 6.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  7. 7.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  8. 8.Rice UniversityHoustonUSA
  9. 9.Kenyon CollegeGambierUSA
  10. 10.Department of Classical StudiesBar Ilan UniversityRamat GanIsrael
  11. 11.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

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