Foundations of Science

, Volume 18, Issue 1, pp 43–74

Ten Misconceptions from the History of Analysis and Their Debunking

Article

Abstract

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others.

Keywords

Abraham Robinson Adequality Archimedean continuum Bernoullian continuum Cantor Cauchy Cognitive bias Completeness Constructivism Continuity Continuum du Bois-Reymond Epsilontics Felix Klein Fermat-Robinson standard part Infinitesimal Leibniz–Łoś transfer principle Limit Mathematical rigor Nominalism Non-Archimedean Simon Stevin Stolz Sum theorem Weierstrass 

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References

  1. Albeverio S., Høegh-Krohn R., Fenstad J., Lindstrøm T. (1986) Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc, Orlando, FLGoogle Scholar
  2. Andersen K. (2011) One of Berkeley’s arguments on compensating errors in the calculus. Historia Mathematica 38(2): 219–231CrossRefGoogle Scholar
  3. Anderson R. (1976) A non-standard representation for Brownian motion and Itô integration. Israel Journal of Mathematics 25(1–2): 15–46CrossRefGoogle Scholar
  4. Arkeryd L. (1981) Intermolecular forces of infinite range and the Boltzmann equation. Archive for Rational Mechanics and Analysis 77(1): 11–21CrossRefGoogle Scholar
  5. Arkeryd L. (2005) Nonstandard analysis. American Mathematical Monthly 112(10): 926–928CrossRefGoogle Scholar
  6. Bacon, F. (1620). Novum Organum.Google Scholar
  7. Baltus C. (2004) D’Alembert’s proof of the fundamental theorem of algebra. Historia Mathematica 31(4): 414–428CrossRefGoogle Scholar
  8. Barany M. J. (2011) Revisiting the introduction to Cauchy’s Cours d’analyse. Historia Mathematica 38(3): 368–388CrossRefGoogle Scholar
  9. Bell E.T. (1945) The development of mathematics. McGraw-Hill Book Company, Inc, New YorkGoogle Scholar
  10. Bell J. L. (2008) A primer of infinitesimal analysis (2nd ed.). Cambridge University Press, CambridgeCrossRefGoogle Scholar
  11. Bell, J. L. (2009a). Continuity and infinitesimals. Stanford Encyclopedia of Philosophy. Revised 20 July 2009.Google Scholar
  12. Bell, J. L. (2009b). Le continu cohésif. Intellectica, 2009/1(51).Google Scholar
  13. Benci V., Di Nasso M. (2003) Alpha-theory: An elementary axiomatics for nonstandard analysis. Expositiones Mathematicae 21(4): 355–386CrossRefGoogle Scholar
  14. Berkeley, G. (1734). The analyst, a discourse addressed to an infidel Mathematician.Google Scholar
  15. Błaszczyk, P. (2009). Nonstandard analysis from a philosophical point of view. In Non-classical mathematics 2009 (Hejnice, 18–22 June 2009), pp. 21–24.Google Scholar
  16. Borel, E. (1902) Leçons sur les séries à termes positifs, ed. Robert d’Adhemar, Paris (Gauthier-Villars).Google Scholar
  17. 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
  18. 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.
  19. Bos H. J. M. (1974) Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences 14: 1–90CrossRefGoogle Scholar
  20. Boyer C. (1949) The concepts of the calculus. Hafner Publishing Company, New YorkGoogle 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): 519–535CrossRefGoogle Scholar
  22. Cajori. (1993). A history of mathematical notations (Vol. 1–2). La Salle, IL: The Open Court Publishing Company, 1928/1929. Reprinted by Dover.Google Scholar
  23. Cauchy A. L. (1821) Cours d’Analyse de L’Ecole Royale Polytechnique. Première Partie. Analyse algébrique. Imprimérie Royale, ParisGoogle Scholar
  24. Cauchy A. L. (1823) Résumé des Leçons données à l’Ecole Royale Polytechnique sur le Calcul Infinitésimal. Imprimérie Royale, ParisGoogle Scholar
  25. Cauchy A.L. (1815) Théorie de la propagation des ondes à la surface d’un fluide pesant d’une profondeur indéfinie (published 1827, with additional Notes). Oeuvres 1(1): 4–318Google Scholar
  26. Cauchy, A. L. (1829). Leçons sur le calcul différentiel. Paris: Debures. In Oeuvres complètes (Ser. 2, Vol. 4, pp. 263–609). Paris: Gauthier-Villars, 1899.Google Scholar
  27. 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 (Ser. 1, Vol. 12, pp. 30–36). Paris: Gauthier–Villars, 1900.Google Scholar
  28. Connes A. (1995) Noncommutative geometry and reality. Journal of Mathematical Physics 36(11): 6194–6231CrossRefGoogle Scholar
  29. Crossley J. (1987) The emergence of number (2nd ed.). World Scientific Publishing Co, SingaporeCrossRefGoogle Scholar
  30. Crowe, M. (1988). Ten misconceptions about mathematics and its history. History and philosophy of modern mathematics (Minneapolis, MN, pp. 260–277, 1985). Minnesota studies in philosophical science, XI. Minneapolis, MN: University of Minnesota Press.Google Scholar
  31. 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): 375–378CrossRefGoogle Scholar
  32. D’Alembert, J. (1754). Differentiel. In Encyclopédie, 4.Google Scholar
  33. Dauben J. (1995) Abraham Robinson. The creation of nonstandard analysis. A personal and mathematical odyssey. With a foreword by Benoit B. Mandelbrot. Princeton University Press, PrincetonGoogle Scholar
  34. Dauben, J. (1996). Arguments, logic and proof: Mathematics, logic and the infinite. History of mathematics and education: Ideas and experiences (Essen, 1992; Vol. 11, pp. 113–148), Stud. Wiss. Soz. Bildungsgesch. Mathematics. Göttingen: Vandenhoeck & Ruprecht.Google Scholar
  35. Davis, M. (1977). Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience: New York. Reprinted: Dover, NY, 2005, see http://store.doverpublications.com/0486442292.html.
  36. Dedekind R. (1872) Stetigkeit und irrationale Zahlen, Freid. Vieweg & Sohn, BraunschweigGoogle Scholar
  37. Dedekind R. (1963) Essays on the theory of numbers. I: Continuity and irrational numbers. II: The nature and meaning of numbers. Authorized translation by Wooster Woodruff Beman Dover Publications, Inc, New YorkGoogle Scholar
  38. Dehn M. (1900) Die Legendre’schen Sätze über die Winkelsumme im Dreieck. Mathematische Annalen 53(3): 404–439CrossRefGoogle Scholar
  39. Dossena R., Magnani L. (2007) Mathematics through diagrams: Microscopes in non-standard and smooth analysis. Studies in Computational Intelligence (SCI) 64: 193–213CrossRefGoogle Scholar
  40. Ehrlich, P. (2005). Continuity. In D. M. Borchert (Ed.), Encyclopedia of philosophy (2nd ed., pp. 489–517). Farmington Hills, MI: Macmillan Reference USA (The online version contains some slight improvements).Google Scholar
  41. 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): 1–121CrossRefGoogle Scholar
  42. Ehrlich P. (2012) The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic 18(1): 1–45CrossRefGoogle Scholar
  43. Ely R. (2010) Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education 41(2): 117–146Google Scholar
  44. Euclid. (2007). Euclid’s elements of geometry. Edited, and provided with a modern English translation, by Richard Fitzpatrick, http://farside.ph.utexas.edu/euclid.html.
  45. Euler, L. (1770). Elements of algebra (3rd English edition). English translated by John Hewlett and Francis Horner. Orme Longman, 1822.Google Scholar
  46. Faltin F., Metropolis N., Ross B., Rota G.-C. (1975) The real numbers as a wreath product. Advances in Mathematics 16: 278–304CrossRefGoogle Scholar
  47. Fearnley-Sander D. (1979) Hermann Grassmann and the creation of linear algebra. The American Mathematical Monthly 86(10): 809–817CrossRefGoogle Scholar
  48. Feferman, S. (2009). Conceptions of the continuum [Le continu mathématique. Nouvelles conceptions, nouveaux enjeux]. Intellectica, 51, 169–189. See also http://math.stanford.edu/~feferman/papers/ConceptContin.pdf.
  49. Felscher W. (1979) Naive Mengen und abstrakte Zahlen. III. [Naive sets and abstract numbers. III] Transfinite Methoden. With an erratum to Vol. II. Bibliographisches Institut, MannheimGoogle Scholar
  50. Felscher W. (2000) Bolzano, Cauchy, epsilon, delta. American Mathematical Monthly 107(9): 844–862CrossRefGoogle Scholar
  51. Fisher G. (1979) Cauchy’s variables and orders of the infinitely small. The British Journal for the Philosophy of Science 30(3): 261–265CrossRefGoogle Scholar
  52. Fourier, J. (1822). Théorie analytique de la chaleur.Google Scholar
  53. Fowler D. H. (1992) Dedekind’s theorem: \({\sqrt 2\times\sqrt 3=\sqrt6}\) . American Mathematical Monthly 99(8): 725–733CrossRefGoogle Scholar
  54. Fraenkel A. (1967) Lebenskreise. Aus den Erinnerungen eines jüdischen Mathematikers. Deutsche Verlags-Anstalt, StuttgartGoogle Scholar
  55. Freudenthal H. (1971) Did Cauchy plagiarise Bolzano?. Archive for History of Exact Sciences 7: 375–392CrossRefGoogle Scholar
  56. Gerhardt, C. I. (ed.) (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann.Google Scholar
  57. Gilain, C. (1989). Cauchy et le cours d’analyse de l’Ecole polytechnique. With an editorial preface by Emmanuel Grison. Bulletin Society des amis de la Bibliothèque, Ecole Polytechnique, Vol. 5Google Scholar
  58. Giordano P. (2010) Infinitesimals without logic. Russian Journal of Mathematical Physics 17(2): 159–191CrossRefGoogle Scholar
  59. Giordano P. (2010) The ring of Fermat reals. Advances in Mathematics 225(4): 2050–2075CrossRefGoogle Scholar
  60. Giordano, P., & Katz, M. (2011). Two ways of obtaining infinitesimals by refining Cantor’s completion of the reals. See http://arxiv.org/abs/1109.3553.
  61. Gordon, E. I., Kusraev, A. G., & Kutateladze, S. S. (2002). Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its applications (Vol. 544). Dordrecht: Kluwer.Google Scholar
  62. Grabiner J. (1983) Who gave you the epsilon? Cauchy and the origins of rigorous calculus. American Mathematics Monthly 90(3): 185–194CrossRefGoogle Scholar
  63. Grattan-Guinness I. (2004) The mathematics of the past: Distinguishing its history from our heritage. Historia Mathematica 31: 163–185CrossRefGoogle Scholar
  64. Gray J. (2008) Plato’s ghost. The modernist transformation of mathematics. Princeton University Press, Princeton, NJGoogle Scholar
  65. Guicciardini, N. (2011). Private communication, 27 november.Google Scholar
  66. Hawking, S., (Ed.) (2007). The mathematical breakthroughs that changed history. Philadelphia, PA: Running Press (originally published 2005).Google Scholar
  67. Hewitt E. (1948) Rings of real-valued continuous functions. I. Transactions of the American Mathematical Society 64: 45–99CrossRefGoogle Scholar
  68. Hoborski, A. (1923). Aus der theoretischen Arithmetik (German). Opusc. math. Kraków, 2 11–12 (1938).Google Scholar
  69. Hrbáček K. (1978) Axiomatic foundations for nonstandard analysis. Fundamenta Mathematicae 98(1): 1–19Google Scholar
  70. Hrbacek K. (2007) Stratified analysis? The strength of nonstandard analysis. Springer, Vienna, pp 47–63CrossRefGoogle Scholar
  71. Hrbacek K., Lessmann O., O’Donovan R. (2010) Analysis with ultrasmall numbers. American Mathematics Monthly 117(9): 801–816CrossRefGoogle Scholar
  72. R., R. (1940) OEuvres complètes. Martinus Nijhoff, La HayeGoogle Scholar
  73. 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/arXiv:1003.1501.
  74. Katz K., Katz M. (2010) When is 0.999…less than 1?. The Montana Mathematics Enthusiast 7(1): 3–30Google Scholar
  75. Katz, K., & Katz, M. (2011a). Cauchy’s continuum. Perspectives on Science, 19(4), 426–452. See http://www.mitpressjournals.org/toc/posc/19/4 and http://arxiv.org/abs/1108.4201.Google Scholar
  76. Katz, K., & Katz, M. (2011b). Stevin numbers and reality. Foundations of Science. See http://dx.doi.org/10.1007/s10699-011-9228-9 and http://arxiv.org/abs/1107.3688.
  77. 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.
  78. 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.
  79. Katz, K., & Sherry, D. (2012). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis (to appear).Google Scholar
  80. 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 (Vol. 12). Charlotte, NC: Information Age Publishing, Inc. See http://arxiv.org/abs/1110.5747 and http://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics.
  81. Katz V. J. (1993) Using the history of calculus to teach calculus. Contributions from History, Philosophy and Sociology of Science and Education. Science & Education 2(3): 243–249CrossRefGoogle Scholar
  82. Keisler H. J. (1986) Elementary calculus: An infinitesimal approach (2nd Ed.). Prindle, Weber & Schimidt, BostonGoogle Scholar
  83. Keisler, H. J. (1994). The hyperreal line. Real numbers, generalizations of the reals, and theories of continua, Synthese Lib. (Vol. 242, pp. 207–237). Dordrecht: Kluwer.Google Scholar
  84. Keisler, H. J. (2008a). The ultraproduct construction. In Proceedings of the ultramath conference, Pisa, Italy.Google Scholar
  85. Keisler, H. J. (2008b). Quantifiers in limits. Andrzej Mostowski and Foundational Studies (pp. 151–170). Amsterdam: IOS.Google Scholar
  86. Klein, F. (1908). (Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis). Translation by E. R. Hedrick & 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
  87. Kleiner I., Movshovitz-Hadar N. (1994) The role of paradoxes in the evolution of mathematics. American Mathematical Monthly 101(10): 963–974CrossRefGoogle Scholar
  88. 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): 59–73CrossRefGoogle Scholar
  89. Koetsier T. (2009) Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity. In: Hanna G., Jahnke H., Pulte H. (eds) Explanation and proof in mathematics. Philosophical and Educational Perspectives. Springer, BerlinGoogle Scholar
  90. Lagrange, J.-L. (2009). (1811) Mécanique Analytique. Courcier. Reissued by Cambridge University Press.Google Scholar
  91. Lakatos I. (1976) Proofs and refutations. The logic of mathematical discovery. Edited by John Worrall and Elie Zahar. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  92. Lakatos, I. (1978). Cauchy and the continuum: The significance of nonstandard analysis for the history and philosophy of mathematics. The Mathematical Intelligencer, 1(3), 151–161 (paper originally presented in 1966).Google Scholar
  93. Laugwitz D. (1987) Infinitely small quantities in Cauchy’s textbooks. Historia Mathematica 14(3): 258–274CrossRefGoogle Scholar
  94. 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): 195–245CrossRefGoogle Scholar
  95. Laugwitz D. (1992) Early delta functions and the use of infinitesimals in research. Revue d’histoire des sciences 45(1): 115–128CrossRefGoogle Scholar
  96. Laugwitz D. (1997) On the historical developement of infinitesimal mathematics. Part II. The conceptual thinking of Cauchy. Translated from the German by Abe Shenitzer with the editorial assistance of Hardy Grant. American Mathematical Monthly 104(7): 654–663CrossRefGoogle Scholar
  97. 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): 377–392Google Scholar
  98. Leibniz G. (2001) The Labyrinth of the continuum (Arthur, R., translator and editor). Yale University Press, New HavenGoogle Scholar
  99. Leibniz. (1701). Mémoire de M.G.G. Leibniz touchant son sentiment sur le calcul differéntiel. Mémoires de Trévoux, 1701(November), 270–272. Reprinted in (Gerhardt, 1850–1863, Vol. 5, p. 350).Google Scholar
  100. Leibniz. (1710). GM V, pp. 381-2 (in Gerhardt (1850–1863)).Google Scholar
  101. Levy S. H. (1991) Charles S. Peirce’s Theory of Infinitesimals. International Philosophical Quarterly 31: 127–140CrossRefGoogle Scholar
  102. Lightstone A. H. (1972) Infinitesimals. American Mathematical Monthly 79: 242–251CrossRefGoogle Scholar
  103. 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 Publishing Co.Google Scholar
  104. Lutz R., Albuquerque L. (2003) Modern infinitesimals as a tool to match intuitive and formal reasoning in analysis. Logic and mathematical reasoning (Mexico City, 1997). Synthese 134(1–2): 325–351CrossRefGoogle Scholar
  105. Lützen J. (1982) The prehistory of the theory of distributions. Studies in the history of mathematics and physical sciences Vol. 7. Springer, New YorkCrossRefGoogle Scholar
  106. Lützen, J. (2003). The foundation of analysis in the 19th century. A history of analysis. History of Mathematics (Vol. 24, pp. 155–195). Providence, RI: American Mathematical Society.Google Scholar
  107. 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
  108. Madison E., Stroyan K. (1977) Reviews: Elementary calculus (Review of first edition of Keisler (1986)).. American Mathematical Monthly 84(6): 496–500CrossRefGoogle Scholar
  109. Magnani L., Dossena R. (2005) Perceiving the infinite and the infinitesimal world: Unveiling and optical diagrams in mathematics. Foundations of Science 10(1): 7–23CrossRefGoogle Scholar
  110. Malet A. (2006) Renaissance notions of number and magnitude. Historia Mathematica 33(1): 63–81CrossRefGoogle Scholar
  111. Mancosu P. (1996) Philosophy of mathematics and mathematical practice in the seventeenth century. The Clarendon Press/Oxford University Press, New YorkGoogle Scholar
  112. Mancosu, P. (eds) (2008) The philosophy of mathematical practice. Oxford University Press, OxfordGoogle Scholar
  113. 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): 612–646CrossRefGoogle Scholar
  114. Mancosu, P., & Vailati, E. (1990). Detleff Clüver: An early opponent of the Leibnizian differential calculus. Centaurus, 33(4), 325–344.Google Scholar
  115. Marsh, J. (1742). Decimal arithmetic made perfect. London.Google Scholar
  116. McClenon R. B. (1923) A contribution of Leibniz to the history of complex numbers. American Mathematical Monthly 30(7): 369–374CrossRefGoogle Scholar
  117. Mormann T. (2008) Idealization in Cassirer’s philosophy of mathematics. Philosophia Mathematica (3) 16(2): 151–181CrossRefGoogle Scholar
  118. Naets J. (2010) How to define a number? A general epistemological account of Simon Stevin’s art of defining. Topoi 29(1): 77–86CrossRefGoogle Scholar
  119. Nelson E. (1977) Internal set theory: A new approach to nonstandard analysis. Bulletin of American Mathematical Society 83(6): 1165–1198CrossRefGoogle Scholar
  120. Newton, I. (1671). A treatise on the methods of series and fluxions. In Whiteside (1969) (Vol. III), pp. 33–35.Google Scholar
  121. 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
  122. 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
  123. Peirce, C. S. (1976). The new elements of mathematics, Vol. III/1. Mathematical miscellanea. Edited by Carolyn Eisele. Atlantic Highlands, NJ: Mouton Publishers/The Hague-Paris/Humanities Press.Google Scholar
  124. Pourciau B. (2001) Newton and the notion of limit. Historia Mathematica 28(1): 18–30CrossRefGoogle Scholar
  125. Putnam H. (1975) What is mathematical truth? Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Mathematica 2(4): 529–533CrossRefGoogle Scholar
  126. Robinson A. (1966) Non-standard analysis. North-Holland Publishing Co, AmsterdamGoogle Scholar
  127. Roquette P. (2010) Numbers and models, standard and nonstandard. Math Semesterber 57: 185–199CrossRefGoogle Scholar
  128. Russell B. (1903) The principles of mathematics. Routledge, LondonGoogle Scholar
  129. 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
  130. 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. Springer, New YorkGoogle Scholar
  131. Sepkoski D. (2005) Nominalism and constructivism in seventeenth-century mathematical philosophy. Historia Mathematica 32(1): 33–59CrossRefGoogle Scholar
  132. Sherry D. (1987) The wake of Berkeley’s Analyst: Rigor mathematicae?. Studies in History and Philosophical Science 18(4): 455–480CrossRefGoogle Scholar
  133. Sherry D. (1993) Don’t take me half the way: On Berkeley on mathematical reasoning. Studies in History and Philosophical Science 24(2): 207–225CrossRefGoogle Scholar
  134. Sinaceur H. (1973) Cauchy et Bolzano. Reviews of Histoire in Science and Applications 26((2): 97–112CrossRefGoogle Scholar
  135. Skolem Th. (1934) Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae 23: 150–161Google Scholar
  136. Smale S. (1981) The fundamental theorem of algebra and complexity theory. Bulletin of the American Mathematical Society (N.S.) 4(1): 1–36CrossRefGoogle Scholar
  137. Smith D. E. (1920) Source book in mathematics. Mc Grow-Hill, New YorkGoogle Scholar
  138. Stevin, S. (1585). L’Arithmetique. In A. Girard (Ed.), Les Oeuvres Mathematiques de Simon Stevin (Leyde, 1634), part I (pp. 1–101).Google Scholar
  139. Stevin, S. L’Arithmetique. In A. Girard (Ed.), 1625, part II. Online at http://www.archive.org/stream/larithmetiqvedes00stev#page/353/mode/1up.
  140. Stevin, S. (1958). The principal works of Simon Stevin (Vols. IIA, IIB). In D. J. Struik, C. V. Swets, & Zeitlinger (Eds.), Mathematics (Vols. IIA: v+pp. 1–455 (1 plate), IIB: 1958 iv+pp, pp. 459–976). Amsterdam.Google Scholar
  141. Strømholm P. (1968) Fermat’s methods of maxima and minima and of tangents. A reconstruction. Archive for History of Exact Sciences 5(1): 47–69CrossRefGoogle Scholar
  142. Stroyan, K. (1972). Uniform continuity and rates of growth of meromorphic functions. Contributions to non-standard analysis (Sympos., Oberwolfach, 1970; Vol. 69, pp. 47–64). Studies in Logic and Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar
  143. Tall D. (1980) Looking at graphs through infinitesimal microscopes, windows and telescopes. The Mathematical Gazette 64: 22–49CrossRefGoogle Scholar
  144. Tall D. (1991) The psychology of advanced mathematical thinking. In: Tall D. O. (eds) Advanced mathematical thinking mathematics education library, 11. Kluwer, DordrechtCrossRefGoogle Scholar
  145. Tall, D. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. In Transforming Mathematics Education through the use of Dynamic Mathematics. ZDM (pp. 1–11) (June 2009).Google Scholar
  146. Tao T. (2010) An epsilon of room, II. Pages from year three of a mathematical blog. American Mathematical Society, Providence, RIGoogle Scholar
  147. Tarski A. (1930) Une contribution à la théorie de la mesure. Fundamenta Mathematicae 15: 42–50Google Scholar
  148. Urquhart, A. Mathematics and physics: Strategies of assimilation. In Mancosu (2008), pp. 417–440.Google Scholar
  149. van der Waerden B. L. (1985) A history of algebra. From al-Khwarizmi to Emmy Noether. Springer, BerlinGoogle Scholar
  150. Weil A. (1973) Book review: The mathematical career of Pierre de Fermat. Bulletin of the American Mathematical Society 79(6): 1138–1149CrossRefGoogle Scholar
  151. Weil A. (1984) Number theory. An approach through history. From Hammurapi to Legendre. Birkhäuser Boston, Inc, BostonGoogle Scholar
  152. Whiteside, D. T. (Ed.) (1969). The mathematical papers of Isaac Newton. In D. T. Whiteside with the assistance in publication of M. A. Hoskin and A. Prag (Eds.) (Vol. III, pp. 1670–1673). London: Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Piotr Błaszczyk
    • 1
  • Mikhail G. Katz
    • 2
  • David Sherry
    • 3
  1. 1.Institute of MathematicsPedagogical University of CracowKrakówPoland
  2. 2.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  3. 3.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

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