, Volume 78, Issue 3, pp 571–625 | Cite as

Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

  • Mikhail G. KatzEmail author
  • David Sherry
Original Paper


Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.


Status Transitus Transfer Principle Null Sequence Infinite Divisibility Infinitesimal Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are grateful to H. Jerome Keisler for helpful remarks that helped improve an earlier version of the manuscript. The influence of Hilton Kramer (1928–2012) is obvious.


  1. Albeverio, S., Høegh-Krohn, R., Fenstad, J., & Lindstrøm, T. (1986). Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Orlando, FL: Academic Press, Inc.Google Scholar
  2. Andersen, K. (2011). One of Berkeley’s arguments on compensating errors in the calculus. Historia Mathematica, 38(2), 219–231.CrossRefGoogle Scholar
  3. Anderson, R. (1976). A non-standard representation for Brownian motion and Itô integration. Israel Journal of Mathematics, 25(1–2), 15–46.CrossRefGoogle Scholar
  4. Arkeryd, L. (1981). Intermolecular forces of infinite range and the Boltzmann equation. Archive for Rational Mechanics and Analysis, 77(1), 11–21.CrossRefGoogle Scholar
  5. Arkeryd, L. (2005). Nonstandard analysis. American Mathematical Monthly, 112(10), 926–928.CrossRefGoogle Scholar
  6. Arthur, R. (2007). Leibniz’s syncategorematic infinitesimals, Smooth Infinitesimal Analysis, and Newton’s Proposition 6. See∼rarthur/papers/LsiSiaNp6.rev.pdf.
  7. Bair, J., & Henry, V. (2010). Implicit Differentiation with Microscopes. The Mathematical Intelligencer, 32(1), 53–55.CrossRefGoogle Scholar
  8. Beeley, P. (2008). Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics. In Goldenbaum and Jesseph (2008) (pp. 31–52).Google Scholar
  9. Bell, E.T. (1945). The Development of Mathematics. New York: McGraw-Hill Book Company, Inc.Google Scholar
  10. Bell, J. L. (2008). A primer of infinitesimal analysis. 2 edn. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  11. Bell, J. L. (2009). Continuity and infinitesimals. Stanford Encyclopedia of philosophy.Google Scholar
  12. Berkeley, G. (1734). The Analyst, a Discourse Addressed to an Infidel Mathematician.Google Scholar
  13. Berkeley, G. (1948). Works, vol. 1, ed. Luce and Jessop. London: T. Nelson & Sons.Google Scholar
  14. Błasczcyk, P., Katz, M., & Sherry, D. (2012). Ten misconceptions from the history of analysis and their debunking. Foundations of Science. See and
  15. Borovik, A., Jin, R., & Katz, M. (2012) An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals. Notre Dame Journal of Formal Logic, 53(4).Google Scholar
  16. Borovik, A., & Katz, M. (2011) Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Foundations of Science, see and
  17. Bos, H. J. M. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.CrossRefGoogle Scholar
  18. Bourbaki, N. (1960). Eléments d’histoire des mathématiques. Histoire de la Pensée, IV. Paris: Hermann.Google Scholar
  19. Boyer, C. (1949). The concepts of the calculus. Hafner Publishing Company.Google Scholar
  20. Boyer, C. (1959). The history of the calculus and its conceptual development. New York: Dover Publications, Inc.Google Scholar
  21. Bråting, K. (2007). A new look at E. G. Björling and the Cauchy sum theorem. Archive for History of Exact Sciences, 61(5), 519–535.CrossRefGoogle Scholar
  22. Burgess, J. (1983). Why I am not a nominalist. Notre Dame Journal of Formal Logic, 24(1), 93–105.CrossRefGoogle Scholar
  23. Cajori, F. (1917). Discussion of Fluxions: From Berkeley to Woodhouse. American Mathematical Monthly, 24(4), 145–154.CrossRefGoogle Scholar
  24. Carchedi, G. (2008). Dialectics and Temporality in Marx’s Mathematical Manuscripts. Science & Society, 72(4), 415–426.CrossRefGoogle Scholar
  25. Cauchy, A. L. (1821). Cours d’Analyse de L’Ecole Royale Polytechnique. Première Partie. Analyse algébrique. Paris: Imprimérie Royale.Google Scholar
  26. Cauchy, A. L. (1853). Note sur les séries convergentes dont les divers termes sont des fonctions continues d’une variable réelle ou imaginaire, entre des limites données. In: Oeuvres complètes, Series 1, Vol. 12 (pp. 30–36). Paris: Gauthier–Villars (1900).Google Scholar
  27. Child, J. M. (Ed.). (1920). The early mathematical manuscripts of Leibniz. (Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child). Chicago-London: The Open Court Publishing Co.Google Scholar
  28. Cutland, N., Kessler, C., Kopp, E., & Ross, D. (1988). On Cauchy’s notion of infinitesimal. The British Journal for the Philosophy of Science, 39(3), 375–378.CrossRefGoogle Scholar
  29. De Morgan, A. (1852). On the early history of infinitesimals in England. Philosophical Magazine, Series 4, 4(26), 321–330. See
  30. Dossena, R., & Magnani, L. (2007). Mathematics through diagrams: Microscopes in non-standard and smooth analysis. Studies in Computational Intelligence (SCI), 64, 193–213.CrossRefGoogle Scholar
  31. Earman, J. (1975). Infinities, infinitesimals, and indivisibles: The Leibnizian labyrinth. Studia Leibnitiana, 7(2), 236–251.Google Scholar
  32. Edwards Jr., C.H. (1979). The historical development of the calculus. New York: Springer.CrossRefGoogle Scholar
  33. 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(1), 1–121.CrossRefGoogle Scholar
  34. Ehrlich, P. (2012). The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic, 18(1), 1–45.CrossRefGoogle Scholar
  35. Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117–146.Google Scholar
  36. Euler, L. (1770). Elements of algebra. Translated from the German by John Hewlett. Reprint of the 1840 edition. With an introduction by C. Truesdell. New York: Springer.Google Scholar
  37. Fearnley-Sander, D. (1979). Hermann Grassmann and the creation of linear algebra. American Mathematical Monthly, 86(10), 809–817.CrossRefGoogle Scholar
  38. 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
  39. Fraenkel, A. (1946). Einleitung in die Mengenlehre. New York, N.Y.: Dover (originally published by Springer, Berlin, 1928).Google Scholar
  40. Fraenkel, A. (1966). Abstract set theory. Third revised edition. Amsterdam: North-Holland.Google Scholar
  41. Freudenthal, H. (1971). Cauchy, Augustin-Louis. In: C. C. Gillispie (Ed.). Dictionary of scientific biography, vol. 3 (pp. 131–148). New York: Charles Scribner’s sons.Google Scholar
  42. Fowler, D. (1992). Dedekind’s theorem: \(\sqrt 2\times\sqrt 3=\sqrt 6\). American Mathematical Monthly, 99(8), 725–733.CrossRefGoogle Scholar
  43. Gerhardt, C. I. (Ed.). (1846). Historia et Origo calculi differentialis a G. G. Leibnitio conscripta. In C. I. Gerhardt (Ed.), Hannover.Google Scholar
  44. Gerhardt, C. I. (Ed.). (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann.Google Scholar
  45. Gerhardt C. I. (Ed.). (1875–1990). G. W. Leibniz: Philosophische Schriften. In C. I. Gerhardt (Ed.). 7 vols. Reprint, Hildesheim: Georg Olms Verlag, 1962.Google Scholar
  46. Gillispie, C. C. (1971). Lazare Carnot, savant. A monograph treating Carnot’s scientific work, with facsimile reproduction of his unpublished writings on mechanics and on the calculus, and an essay concerning the latter by A. P. Youschkevitch. Princeton, N.J.: Princeton University Press.Google Scholar
  47. Giordano, P., & Katz, M. (2011). Two ways of obtaining infinitesimals by refining Cantor’s completion of the reals. Preprint, see
  48. Goldenbaum, U., & Jesseph, D. (Eds.). (2008). Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin: Walter de Gruyter (reviewed by Monnoyeur-Broitman 2010).Google Scholar
  49. Grabiner, J. (1981). The origins of Cauchy’s rigorous calculus. Cambridge, MA: MIT Press.Google Scholar
  50. Grattan-Guinness, I. (2004). The mathematics of the past: Distinguishing its history from our heritage. Historia Mathematica, 31, 163–185.CrossRefGoogle Scholar
  51. Gray, J. (2008). Plato’s ghost. The modernist transformation of mathematics. Princeton, NJ: Princeton University Press.Google Scholar
  52. Hawking, S. (Ed.). (2007). The mathematical breakthroughs that changed history., Philadelphia, PA: Running Press. (originally published 2005).Google Scholar
  53. Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions on American Mathematical Society, 64, 45–99.CrossRefGoogle Scholar
  54. Horváth, M. (1986). On the attempts made by Leibniz to justify his calculus. Studia Leibnitiana, 18(1), 60–71.Google Scholar
  55. Ishiguro, H. (1990). Leibniz’s philosophy of logic and language, 2 edn. Cambridge: Cambridge University Press.Google Scholar
  56. Jesseph, D. (1993). Berkeley’s philosophy of mathematics. Science and its Conceptual Foundations. Chicago, IL: University of Chicago Press.CrossRefGoogle Scholar
  57. Jesseph, D. (2005). George Berkeley, the analyst (1734). In: I. Grattan-Guinness (Ed.), Landmark writings in western mathematics (pp. 1640–1940). Elsevier B. V., Amsterdam.Google Scholar
  58. Jesseph, D. (2011). Leibniz on the elimination of infinitesimals: Strategies for finding truth in fiction, 27 pages. In N. B. Goethe, P. Beeley, & D. Rabouin (Eds.), Leibniz on the interrelations between mathematics and philosophy, Archimedes Series. Springer (to appear).Google Scholar
  59. Jorgensen, L. (2009). The principle of continuity and Leibniz’s theory of consciousness. Journal of the History of Philosophy, 47(2), 223–248.CrossRefGoogle Scholar
  60. Katz, K., & Katz, M. (2010a). Zooming in on infinitesimal 1−9. in a post-triumvirate era. Educational Studies in Mathematics, 74(3), 259–273. See
  61. Katz, K., & Katz, M. (2010b). When is .999… less than 1? The Montana Mathematics Enthusiast, 7(1), 3–30.Google Scholar
  62. Katz, K., & Katz, M. (2012). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51–89. See and
  63. Katz, K., & Katz, M. (2011a). Cauchy’s continuum. Perspectives on Science, 19(4), 426–452. See and
  64. Katz, K., & Katz, M. (2011b) Stevin numbers and reality. Foundations of Science, see and
  65. Katz, K., & Katz, M. (2011c). Meaning in classical mathematics: Is it at odds with Intuitionism? Intellectica, 56(2), 223–302. See
  66. Katz, M., & Leichtnam, E. Commuting and non-commuting infinitesimals. American Mathematical Monthly. (to appear).Google Scholar
  67. Katz, M., & Tall, D. (2011). The tension between intuitive infinitesimals and formal mathematical analysis. In B. Sriraman (Ed.) Crossroads in the history of mathematics and mathematics education. The Montana Mathematics Enthusiast Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC. See and
  68. Keisler, H. J. (1976). Foundations of infinitesimal Calculus. Instructor’s manual. Prindle, Weber & Schmidt. See∼keisler/foundations.html.
  69. Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach. (2nd ed). Boston: Prindle, Weber & Schimidt. See∼keisler/calc.html.
  70. Kennedy, H. (1977). Karl Marx and the foundations of differential calculus. Historia Mathematica, 4(3), 303–318.CrossRefGoogle Scholar
  71. Klein, F. Elementary Mathematics from an advanced standpoint. Vol. I. Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick and C. A. Noble (Macmillan, New York, 1932) from the third German edition (Springer, Berlin, 1924). Originally published as Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908).Google Scholar
  72. Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.Google Scholar
  73. Knobloch, E. (2000). Archimedes, Kepler, and Guldin: The role of proof and analogy. pp. 82–100 in Thiele (2000).Google Scholar
  74. Knobloch, E. (2002). Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums. Foundations of the formal sciences, 1 (Berlin, 1999). Synthese, 133(1–2), 59–73.CrossRefGoogle Scholar
  75. Laugwitz, D. (1987). Infinitely small quantities in Cauchy’s textbooks. Historia Mathematica, 14(3), 258–274.CrossRefGoogle Scholar
  76. 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
  77. Laugwitz, D. (1992a). Leibniz’ principle and omega calculus. In [A] Le labyrinthe du continu, Colloq., Cerisy-la-Salle/Fr. 1990, pp. 144–154.Google Scholar
  78. Laugwitz, D. (1992b). Early delta functions and the use of infinitesimals in research. Revue d’Histoire des Sciences, 45(1), 115–128.CrossRefGoogle Scholar
  79. Lawvere, F.W. (1980). Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body Third Colloquium on Categories (Amiens, 1980) Part I. Cahiers Topologie Géom. Différentielle, 21(4), 377–392.Google Scholar
  80. Levey, S. (2008). Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism. In Goldenbaum and Jesseph (2008) pp. 107–134.Google Scholar
  81. Leibniz, G. W. (2001). The Labyrinth of the Continuum (trans: Arthur, R., and ed.). New Haven: Yale University Press.Google Scholar
  82. Leibniz, G. W. (1993). De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis, kritisch herausgegeben und kommentiert von Eberhard Knobloch, Göttingen (Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-physikalische Klasse 3; 43).Google Scholar
  83. Leibniz, G. W. (2004). Quadrature arithmétique du cercle, de l’ellipse et de l’hyperbole, Marc Parmentier (trans: and Ed.)/Latin text Eberhard Knobloch (Ed.), Paris: J. Vrin.Google Scholar
  84. Leibniz, G. W. (1680). Elementa calculi novi … ab, in Gerhardt (1846).Google Scholar
  85. Leibniz, G. W. (1684). Nova methodus pro maximis et minimis … in Acta Erud. See Gerhardt 1850–1863, V, pp. 220–226.Google Scholar
  86. Leibniz, G. W. (1687). Letter of Mr. Leibniz on a general principle useful in explaining the laws of nature through a consideration of the divine wisdom; to serve as a reply to the response of the rev. father Malebranche. In Loemker (1956) pp. 351–354. The Latin original version in the Akademie edition (Leibniz 1999, series VI, Vol. 4C, pp. 2031–2039).Google Scholar
  87. Leibniz, G. W. (1695a). Letter to l’Hospital. In C. I. Gerhardt (1850–1863, Vol. II, pp. 287–289).Google Scholar
  88. Leibniz, G. W. (1695b). Letter to Huygens. In C. I. Gerhardt (1850–1863, Vol. II, pp. 205–208).Google Scholar
  89. Leibniz, G. W. (1695c). Letter to l’Hospital. In C. I. Gerhardt (1850–1863, Vol. II, pp. 302).Google Scholar
  90. Leibniz, G. W. (1698). Letter to Wallis. In C. I. Gerhardt (1850–1863, Vol. IV, p. 54).Google Scholar
  91. Leibniz, G. W. (1699). Letter to Wallis. In C. I Gerhardt (1850–1863, Vol. IV, pp. 62–65).Google Scholar
  92. Leibniz, G. W. (1700). Defense du calcul des differences, ab. LH XXXV, VoI. 22. (still unpublished).Google Scholar
  93. Leibniz, G. W. to Pinson, 29 Aug., 1701. In C. I. Gerhardt (1850–1863, IV, pp. 95–96).Google Scholar
  94. Leibniz, G. W. (1701a). Justification du Calcul des infinitesimales…. In Gerhardt (1850–1863, IV, pp. 104–106).Google Scholar
  95. Leibniz, G. W. (1701b) Cum Prodiisset mss “Cum prodiisset atque increbuisset Analysis mea infinitesimalis …” in Gerhardt (1846, pp. 39–50). Online at
  96. Leibniz, G. W. (1702). Letter to Varignon, 2 Feb. 1702. In Gerhardt (1850–1863, vol. IV, pp. 91–95).Google Scholar
  97. Leibniz, G. W. Enclosure to letter to Varignon, 2 Feb., 1702. In Gerhardt (1850–1863, IV, pp. 104–105).Google Scholar
  98. Leibniz, G. W. (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
  99. Leibniz, G. W. (2008). Sämtliche Schriften und Briefe. Reihe VII. Mathematische Schriften. Band 4. 1670-1673. Infinitesimalmathematik. [Collected works and letters. Series VII. Mathematical writings. Vol. 4. 1670–1673. Infinitesimal mathematics] Edited by Walter S. Contro and Eberhard Knobloch. Berlin: Akademie-Verlag.Google Scholar
  100. Leibniz, G. W. (1999) Sämtliche Schriften und Briefe. Reihe VI. [Collected works and letters. Series VI] Philosophische Schriften. Vierter Band 1677–Juni 1690, Teil C. [Philosophical writings. Vol. 4, 1677–June 1690, Part C] Bearbeiter dieses Bandes: Heinrich Schepers, Martin Schneider, Gerhard Biller, Ursula Franke, Herma Kliege-Biller. Akademie-Verlag, Berlin, 1999. xvii+ pp. 1957–2949. See
  101. L’Huilier, S. (1786). Exposition élémentaire des principes des calculs supérieurs, qui a remporté le prix proposé par l’Académie Royale des Sciences et Belles-Lettres pour l’année. Berlin [1787].Google Scholar
  102. Lightstone, A. H. (1972). Infinitesimals. American Mathematical Monthly, 79, 242–251.CrossRefGoogle Scholar
  103. Loemker, L. (Ed.). (1956). Leibniz: Philosophical papers and letters. Chicago: Chicago University Press [Reprinted as Leibniz, Gottfried Wilhelm: Philosophical papers and letters. Second edition. Synthese Historical Library. New York:Humanities Press, 1970].Google Scholar
  104. Łoś, J. (1955). Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. In: Mathematical interpretation of formal systems (pp. 98–113). Amsterdam: North-Holland.Google Scholar
  105. Luxemburg, W. (1964). Nonstandard analysis. Lectures on A. Robinson’s Theory of infinitesimals and infinitely large numbers. Pasadena: Mathematics Department, California Institute of Technology’ second corrected ed.Google Scholar
  106. Magnani, L., & Dossena, R. (2005). Perceiving the infinite and the infinitesimal world: unveiling and optical diagrams in mathematics. Foundations of Science, 10(1), 7–23.CrossRefGoogle Scholar
  107. Mal’tsev, A.I. [Malcev, Mal’cev] (1936). Untersuchungen aus dem Gebiete der mathematischen Logik. [J] Rec. Math. Moscou (Matematicheskii Sbornik), 1(43), 323–335.Google Scholar
  108. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York: The Clarendon Press/Oxford University Press.Google Scholar
  109. Mancosu, P. (Ed.).. (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.Google Scholar
  110. Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? Reviews in Symbols and Logistics, 2(4), 612–646.CrossRefGoogle Scholar
  111. Mancosu, P., & Vailati, E. (1991). Detleff Clüver: An early opponent of the Leibnizian differential calculus. Centaurus, 33(4), 325–344.Google Scholar
  112. Marx, K. (1968). Matematicheskie rukopisi. Edited by S. A. Yanovskaya Moscow (Izd. “Nauka”). Translation in: Marx, K.: Mathematical manuscripts of Karl Marx. Translated from the Russian. With additional material by Ernst Kol’man, S. A. Yanovskaya and C. Smith. London: New Park Publications Ltd., 1983.Google Scholar
  113. McClenon, R. B. (1923). A contribution of Leibniz to the history of complex numbers. American Mathematical Monthly, 30(7), 369–374.CrossRefGoogle Scholar
  114. Monnoyeur-Broitman, F. (2010). Review of “Infinitesimal Differences” (see Goldenbaum and Jesseph 2008). Journal of the History of Philosophy, 48(4), 527–528.Google Scholar
  115. Narens, L. (1976). Utility-uncertainty trade-off structures. Journal of Mathematical Psychology, 13(3), 296–322.CrossRefGoogle Scholar
  116. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
  117. Newton, I. (1946). Sir Isaac Newton’s mathematical principles of natural philosophy and his system of the world, a revision by F. Cajori of A. Motte’s 1729 translation. Berkeley: University of California Press.Google Scholar
  118. Newton, I. (1999). The Principia: Mathematical principles of natural philosophy, translated by I. B. Cohen & A. Whitman, preceded by A guide to Newton’s Principia by I. B. Cohen. Berkeley: University of California Press.Google Scholar
  119. Poincaré, H. (2008). The Foundations of Science: Science and Hypothesis, the Value of Science Science and Methods. Introduction by George Bruce Halsted. USA: Bibliobazaar.Google Scholar
  120. Pourciau, B. (2001). Newton and the notion of limit. Historia Mathematica, 28(1), 18–30.CrossRefGoogle Scholar
  121. Probst, S. Indivisibles and Infinitesimals in Early Mathematical Texts of Leibniz. In: “Infinitesimal Differences”, see (Goldenbaum and Jesseph 2008).Google Scholar
  122. 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) , pp. 3–11].Google Scholar
  123. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.Google Scholar
  124. Robinson, A. (1967). The metaphysics of the calculus. In Problems in the Philosophy of Mathematics, ed. Lakatos (Amsterdam: North Holland), pp. 28–46, Reprinted in Robinson (1979).Google Scholar
  125. Robinson, A. (1970). From a formalist’s point of view. Dialectica, 23, 45–49.CrossRefGoogle Scholar
  126. Robinson, A. (1979). Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. Edited and with introductions by W. A. J. Luxemburg and S. Körner. New Haven, CT: Yale University Press.Google Scholar
  127. Roquette, P. (2010). Numbers and models, standard and nonstandard. Mathematische Semesterberichte, 57, 185–199.CrossRefGoogle Scholar
  128. Rothman, T. (1982). Genius and biographers: The fictionalization of Evariste Galois. American Mathematical Monthly, 89(2), 84–106.CrossRefGoogle Scholar
  129. Russell, B. (1903). The principles of mathematics. Vol. I. Cambridge: Cambridge University Press.Google Scholar
  130. Rust, H. (2005). Operational semantics for timed systems. Lecture Notes in Computer Science, 3456, 23–29, doi: 10.1007/978-3-540-32008-1_4.CrossRefGoogle Scholar
  131. Schmieden, C., & Laugwitz, D. (1958). Eine Erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift, 69, 1–39.CrossRefGoogle Scholar
  132. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th Century France and Germany. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.Google Scholar
  133. Serfati, M. (Ed.).. (2002). De la méthode. Recherches en histoire et philosophie des mathématiques. USA: Presses universitaires de Franche-Comté (PuFC) Besançon.Google Scholar
  134. Serfati, M. (2010). The principle of continuity and the ‘paradox’ of Leibnizian mathematics. In M. Dascal (Ed.), The practice of reason: Leibniz and his controversies. Controversies 7, John Benjamins.Google Scholar
  135. Sherry, D. (1987). The wake of Berkeley’s Analyst: rigor mathematicae? Studies in History and Philosophy of Science, 18(4), 455–480.CrossRefGoogle Scholar
  136. Sherry, D. (1995). Book review: Berkeley’s philosophy of mathematics, by Douglas M. Jesseph. The Philosophical review, 104(1), 126–128.CrossRefGoogle Scholar
  137. Skolem, T. (1934). Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae, 23, 150–161.Google Scholar
  138. Stevin, S. (1958). The principal works of Simon Stevin. Vols. IIA, IIB: Mathematics. In D. J. Struik, C. V. Swets, & Zeitlinger. Amsterdam, Vol. IIA: v+pp. 1–455 (1 plate), Vol. IIB: 1958 iv+pp, 459–976.Google Scholar
  139. Strong, E. (1951). Newton’s “mathematical way”. Journal of the History of Ideas, 12(12), 90–110.CrossRefGoogle Scholar
  140. Stroyan, K. (1972). Uniform continuity and rates of growth of meromorphic functions. Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), pp. 47–64. Studies in Logic and Foundations of Math., Vol. 69. North-Holland, Amsterdam.Google Scholar
  141. Struik, D. (1948). Marx and mathematics. A Centenary of Marxism. In S. Bernstein and the Editors of Science and Society (pp. 181–196). New York: Science and Society.Google Scholar
  142. Struik, D. (Ed.). (1969). In: D. J. Struik (Ed.), A source book in mathematics (pp. 1200–1800). Cambridge, MA: Harvard University Press.Google Scholar
  143. Tall, D. (1980). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematics Gazette, 64, 22–49.CrossRefGoogle Scholar
  144. Tall, D. (1991). Advanced mathematical thinking. In: D. Tall (Ed.), Mathematics education library (pp. 11–155). Dordrecht: Kluwer.Google Scholar
  145. Tall, D. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. Transforming Mathematics Education through the use of Dynamic Mathematics. ZDM Mathematics Education, 41(4), 481–492.CrossRefGoogle Scholar
  146. Tao, T. (2008). Structure and randomness. Pages from year one of a mathematical blog. Providence, RI: American Mathematical Society.Google Scholar
  147. Tarski, A. (1930). Une contribution à la théorie de la mesure. Fundamenta Mathematicae, 15, 42–50.Google Scholar
  148. Thiele, R. (2000). Mathesis. Festschrift zum siebzigsten Geburtstag von Matthias Schramm. [Festschrift for the 70th birthday of Matthias Schramm] Edited by Rüdiger Thiele. Verlag für Geschichte der Naturwissenschaften und der Technik, Berlin.Google Scholar
  149. Urquhart, A. (2008). Mathematics and physics: Strategies of assimilation. In: Mancosu, pp. 417–440.Google Scholar
  150. van der Waerden, B. L. (1985). A history of algebra. From al-Khwarizmi to Emmy Noether. Berlin: Springer.Google Scholar
  151. Wisdom, J. (1953). Berkeley’s criticism of the infinitesimal. The British Journal for the Philosophy of Science, 4(13), 22–25.CrossRefGoogle Scholar
  152. Wittgenstein, L. (1953/2001). Philosophical investigations. New York: Blackwell.Google Scholar
  153. Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59(4), 514–516.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  2. 2.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

Personalised recommendations