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

Erkenntnis

Volumn 78, Issue 3, 2013, Pages 571-625

  Katz, Mikhail G ^a   Sherry, David ^b  

^a BAR ILAN UNIVERSITY   (Israel)

^b NORTHERN ARIZONA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84876923301     PISSN: 01650106     EISSN: 15728420     Source Type: Journal    
DOI: 10.1007/s10670-012-9370-y     Document Type: Article

References (153)

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.
2. Andersen, K. (2011). One of Berkeley's arguments on compensating errors in the calculus. Historia Mathematica, 38(2), 219-231.
3. Anderson, R. (1976). A non-standard representation for Brownian motion and Itô integration. Israel Journal of Mathematics, 25(1-2), 15-46.
4. Arkeryd, L. (1981). Intermolecular forces of infinite range and the Boltzmann equation. Archive for Rational Mechanics and Analysis, 77(1), 11-21.
5. Arkeryd, L. (2005). Nonstandard analysis. American Mathematical Monthly, 112(10), 926-928.
6. Arthur, R. (2007). Leibniz's syncategorematic infinitesimals, Smooth Infinitesimal Analysis, and Newton's Proposition 6. See http://www. humanities. mcmaster. ca/∼rarthur/papers/LsiSiaNp6. rev. pdf.
7. Bair, J., & Henry, V. (2010). Implicit Differentiation with Microscopes. The Mathematical Intelligencer, 32(1), 53-55.
8. Beeley, P. (2008). Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics. In Goldenbaum and Jesseph (2008) (pp. 31-52).
9. Bell, E. T. (1945). The Development of Mathematics. New York: McGraw-Hill Book Company, Inc.
10. Bell, J. L. (2008). A primer of infinitesimal analysis. 2 edn. Cambridge: Cambridge University Press.
11. Bell, J. L. (2009). Continuity and infinitesimals. Stanford Encyclopedia of philosophy.
12. Berkeley, G. (1734). The Analyst, a Discourse Addressed to an Infidel Mathematician.
13. Berkeley, G. (1948). Works, vol. 1, ed. Luce and Jessop. London: T. Nelson & Sons.
14. Błasczcyk, P., Katz, M., & Sherry, D. (2012). Ten misconceptions from the history of analysis and their debunking. Foundations of Science. See http://dx. doi. org/10. 1007/s10699-012-9285-8 and http://arxiv. org/abs/1202. 4153.
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).
16. Borovik, A., & Katz, M. (2011) Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus. Foundations of Science, see http://dx. doi. org/10. 1007/s10699-011-9235-x and http://arxiv. org/abs/1108. 2885.
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.
18. Bourbaki, N. (1960). Eléments d'histoire des mathématiques. Histoire de la Pensée, IV. Paris: Hermann.
19. Boyer, C. (1949). The concepts of the calculus. Hafner Publishing Company.
20. Boyer, C. (1959). The history of the calculus and its conceptual development. New York: Dover Publications, Inc.
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.
22. Burgess, J. (1983). Why I am not a nominalist. Notre Dame Journal of Formal Logic, 24(1), 93-105.
23. Cajori, F. (1917). Discussion of Fluxions: From Berkeley to Woodhouse. American Mathematical Monthly, 24(4), 145-154.
24. Carchedi, G. (2008). Dialectics and Temporality in Marx's Mathematical Manuscripts. Science & Society, 72(4), 415-426.
25. Cauchy, A. L. (1821). Cours d'Analyse de L'Ecole Royale Polytechnique. Première Partie. Analyse algébrique. Paris: Imprimérie Royale.
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).
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.
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.
29. De Morgan, A. (1852). On the early history of infinitesimals in England. Philosophical Magazine, Series 4, 4(26), 321-330. See http://www. tandfonline. com/doi/abs/10. 1080/14786445208647134.
30. Dossena, R., & Magnani, L. (2007). Mathematics through diagrams: Microscopes in non-standard and smooth analysis. Studies in Computational Intelligence (SCI), 64, 193-213.
31. Earman, J. (1975). Infinities, infinitesimals, and indivisibles: The Leibnizian labyrinth. Studia Leibnitiana, 7(2), 236-251.
32. Edwards Jr., C. H. (1979). The historical development of the calculus. New York: Springer.
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.
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.
35. Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117-146.
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.
37. Fearnley-Sander, D. (1979). Hermann Grassmann and the creation of linear algebra. American Mathematical Monthly, 86(10), 809-817.
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.
39. Fraenkel, A. (1946). Einleitung in die Mengenlehre. New York, N. Y.: Dover (originally published by Springer, Berlin, 1928).
40. Fraenkel, A. (1966). Abstract set theory. Third revised edition. Amsterdam: North-Holland.
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.
42. Fowler, D. (1992). Dedekind's theorem: √2 × √3 = √6. American Mathematical Monthly, 99(8), 725-733.
43. Gerhardt, C. I. (Ed.). (1846). Historia et Origo calculi differentialis a G. G. Leibnitio conscripta. In C. I. Gerhardt (Ed.), Hannover.
44. Gerhardt, C. I. (Ed.). (1850-1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann.
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.
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.
47. Giordano, P., & Katz, M. (2011). Two ways of obtaining infinitesimals by refining Cantor's completion of the reals. Preprint, see http://arxiv. org/abs/1109. 3553.
48. Goldenbaum, U., & Jesseph, D. (Eds.). (2008). Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin: Walter de Gruyter (reviewed by Monnoyeur-Broitman 2010).
49. Grabiner, J. (1981). The origins of Cauchy's rigorous calculus. Cambridge, MA: MIT Press.
50. Grattan-Guinness, I. (2004). The mathematics of the past: Distinguishing its history from our heritage. Historia Mathematica, 31, 163-185.
51. Gray, J. (2008). Plato's ghost. The modernist transformation of mathematics. Princeton, NJ: Princeton University Press.
52. Hawking, S. (Ed.). (2007). The mathematical breakthroughs that changed history., Philadelphia, PA: Running Press. (originally published 2005).
53. Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions on American Mathematical Society, 64, 45-99.
54. Horváth, M. (1986). On the attempts made by Leibniz to justify his calculus. Studia Leibnitiana, 18(1), 60-71.
55. Ishiguro, H. (1990). Leibniz's philosophy of logic and language, 2 edn. Cambridge: Cambridge University Press.
56. Jesseph, D. (1993). Berkeley's philosophy of mathematics. Science and its Conceptual Foundations. Chicago, IL: University of Chicago Press.
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.
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).
59. Jorgensen, L. (2009). The principle of continuity and Leibniz's theory of consciousness. Journal of the History of Philosophy, 47(2), 223-248.
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 http://arxiv. org/abs/1003. 1501.
61. Katz, K., & Katz, M. (2010b). When is. 999. less than 1? The Montana Mathematics Enthusiast, 7(1), 3-30.
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 http://dx. doi. org/10. 1007/s10699-011-9223-1 and http://arxiv. org/abs/1104. 0375.
63. Katz, K., & Katz, M. (2011a). Cauchy's continuum. Perspectives on Science, 19(4), 426-452. See http://arxiv. org/abs/1108. 4201 and http://www. mitpressjournals. org/doi/abs/10. 1162/POSC_a_00047.
64. Katz, K., & Katz, M. (2011b) Stevin numbers and reality. Foundations of Science, see http://arxiv. org/abs/1107. 3688 and http://dx. doi. org/10. 1007/s10699-011-9228-9.
65. Katz, K., & Katz, M. (2011c). Meaning in classical mathematics: Is it at odds with Intuitionism? Intellectica, 56(2), 223-302. See http://arxiv. org/abs/1110. 5456.
66. Katz, M., & Leichtnam, E. Commuting and non-commuting infinitesimals. American Mathematical Monthly. (to appear).
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 http://arxiv. org/abs/1110. 5747 and http://www. infoagepub. com/products/Crossroads-in-the-History-of-Mathematics.
68. Keisler, H. J. (1976). Foundations of infinitesimal Calculus. Instructor's manual. Prindle, Weber & Schmidt. See http://www. math. wisc. edu/∼keisler/foundations. html.
69. Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach. (2nd ed). Boston: Prindle, Weber & Schimidt. See http://www. math. wisc. edu/∼keisler/calc. html.
70. Kennedy, H. (1977). Karl Marx and the foundations of differential calculus. Historia Mathematica, 4(3), 303-318.
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).
72. Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.
73. Knobloch, E. (2000). Archimedes, Kepler, and Guldin: The role of proof and analogy. pp. 82-100 in Thiele (2000).
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.
75. Laugwitz, D. (1987). Infinitely small quantities in Cauchy's textbooks. Historia Mathematica, 14(3), 258-274.
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.
77. Laugwitz, D. (1992a). Leibniz' principle and omega calculus. In [A] Le labyrinthe du continu, Colloq., Cerisy-la-Salle/Fr. 1990, pp. 144-154.
78. Laugwitz, D. (1992b). Early delta functions and the use of infinitesimals in research. Revue d'Histoire des Sciences, 45(1), 115-128.
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.
80. Levey, S. (2008). Archimedes, Infinitesimals and the Law of Continuity: On Leibniz's Fictionalism. In Goldenbaum and Jesseph (2008) pp. 107-134.
81. Leibniz, G. W. (2001). The Labyrinth of the Continuum (trans: Arthur, R., and ed.). New Haven: Yale University Press.
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).
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.
84. Leibniz, G. W. (1680). Elementa calculi novi. ab, in Gerhardt (1846).
85. Leibniz, G. W. (1684). Nova methodus pro maximis et minimis. in Acta Erud. See Gerhardt 1850-1863, V, pp. 220-226.
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).
87. Leibniz, G. W. (1695a). Letter to l'Hospital. In C. I. Gerhardt (1850-1863, Vol. II, pp. 287-289).
88. Leibniz, G. W. (1695b). Letter to Huygens. In C. I. Gerhardt (1850-1863, Vol. II, pp. 205-208).
89. Leibniz, G. W. (1695c). Letter to l'Hospital. In C. I. Gerhardt (1850-1863, Vol. II, pp. 302).
90. Leibniz, G. W. (1698). Letter to Wallis. In C. I. Gerhardt (1850-1863, Vol. IV, p. 54).
91. Leibniz, G. W. (1699). Letter to Wallis. In C. I Gerhardt (1850-1863, Vol. IV, pp. 62-65).
92. Leibniz, G. W. (1700). Defense du calcul des differences, ab. LH XXXV, VoI. 22. (still unpublished).
93. Leibniz, G. W. to Pinson, 29 Aug., 1701. In C. I. Gerhardt (1850-1863, IV, pp. 95-96).
94. Leibniz, G. W. (1701a). Justification du Calcul des infinitesimales. In Gerhardt (1850-1863, IV, pp. 104-106).
95. Leibniz, G. W. (1701b) Cum Prodiisset. mss "Cum prodiisset atque increbuisset Analysis mea infinitesimalis." in Gerhardt (1846, pp. 39-50). Online at http://books. google. co. il/books?id=UOM3AAAAMAAJ& source=gbs_navlinks_.
96. Leibniz, G. W. (1702). Letter to Varignon, 2 Feb. 1702. In Gerhardt (1850-1863, vol. IV, pp. 91-95).
97. Leibniz, G. W. Enclosure to letter to Varignon, 2 Feb., 1702. In Gerhardt (1850-1863, IV, pp. 104-105).
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).
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.
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 http://www. leibniz-edition. de/Baende/ReiheVI. htm.
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].
102. Lightstone, A. H. (1972). Infinitesimals. American Mathematical Monthly, 79, 242-251.
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].
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.
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.
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.
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.
108. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York: The Clarendon Press/Oxford University Press.
109. Mancosu, P. (Ed.). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.
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.
111. Mancosu, P., & Vailati, E. (1991). Detleff Clüver: An early opponent of the Leibnizian differential calculus. Centaurus, 33(4), 325-344.
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.
113. McClenon, R. B. (1923). A contribution of Leibniz to the history of complex numbers. American Mathematical Monthly, 30(7), 369-374.
114. Monnoyeur-Broitman, F. (2010). Review of "Infinitesimal Differences" (see Goldenbaum and Jesseph 2008). Journal of the History of Philosophy, 48(4), 527-528.
115. Narens, L. (1976). Utility-uncertainty trade-off structures. Journal of Mathematical Psychology, 13(3), 296-322.
116. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of American Mathematical Society, 83(6), 1165-1198.
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.
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.
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.
120. Pourciau, B. (2001). Newton and the notion of limit. Historia Mathematica, 28(1), 18-30.
121. Probst, S. Indivisibles and Infinitesimals in Early Mathematical Texts of Leibniz. In: "Infinitesimal Differences", see (Goldenbaum and Jesseph 2008).
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].
123. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.
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).
125. Robinson, A. (1970). From a formalist's point of view. Dialectica, 23, 45-49.
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.
127. Roquette, P. (2010). Numbers and models, standard and nonstandard. Mathematische Semesterberichte, 57, 185-199.
128. Rothman, T. (1982). Genius and biographers: The fictionalization of Evariste Galois. American Mathematical Monthly, 89(2), 84-106.
129. Russell, B. (1903). The principles of mathematics. Vol. I. Cambridge: Cambridge University Press.
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.
131. Schmieden, C., & Laugwitz, D. (1958). Eine Erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift, 69, 1-39.
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.
133. Serfati, M. (Ed.). (2002). De la méthode. Recherches en histoire et philosophie des mathématiques. USA: Presses universitaires de Franche-Comté (PuFC) Besançon.
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.
135. Sherry, D. (1987). The wake of Berkeley's Analyst: rigor mathematicae? Studies in History and Philosophy of Science, 18(4), 455-480.
136. Sherry, D. (1995). Book review: Berkeley's philosophy of mathematics, by Douglas M. Jesseph. The Philosophical review, 104(1), 126-128.
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.
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.
139. Strong, E. (1951). Newton's "mathematical way". Journal of the History of Ideas, 12(12), 90-110.
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.
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.
142. Struik, D. (Ed.). (1969). In: D. J. Struik (Ed.), A source book in mathematics (pp. 1200-1800). Cambridge, MA: Harvard University Press.
143. Tall, D. (1980). Looking at graphs through infinitesimal microscopes, windows and telescopes. Mathematics Gazette, 64, 22-49.
144. Tall, D. (1991). Advanced mathematical thinking. In: D. Tall (Ed.), Mathematics education library (pp. 11-155). Dordrecht: Kluwer.
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.
146. Tao, T. (2008). Structure and randomness. Pages from year one of a mathematical blog. Providence, RI: American Mathematical Society.
147. Tarski, A. (1930). Une contribution à la théorie de la mesure. Fundamenta Mathematicae, 15, 42-50.
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.
149. Urquhart, A. (2008). Mathematics and physics: Strategies of assimilation. In: Mancosu, pp. 417-440.
150. van der Waerden, B. L. (1985). A history of algebra. From al-Khwarizmi to Emmy Noether. Berlin: Springer.
151. Wisdom, J. (1953). Berkeley's criticism of the infinitesimal. The British Journal for the Philosophy of Science, 4(13), 22-25.
152. Wittgenstein, L. (1953/2001). Philosophical investigations. New York: Blackwell.
153. Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59(4), 514-516.