Skip to main content
SpringerLink
Log in

Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms

  • Published:
Foundations of Science Aims and scope Submit manuscript

Abstract

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. A debate of long standing concerns the issue of whether Cauchy modified or clarified the hypothesis of the sum theorem of 1853 as compared to 1821. Our analysis of the 1853 text is independent of this debate and our main conclusions are compatible with either view.

  2. Some historians are fond of recycling the claim that Abraham Robinson used model theory to develop his system with infinitesimals. What they tend to overlook is not merely the fact that an alternative construction of the hyperreals via an ultrapower requires nothing more than a serious undergraduate course in algebra (covering the existence of a maximal ideal), but more significantly the distinction between procedures and foundations (see Sect. 1.2) which highlights the point that whether one uses Weierstrass’s foundations or Robinson’s is of little import, procedurally speaking.

  3. See http://u.cs.biu.ac.il/~katzmik/infinitesimals.html for a more detailed list.

  4. Here the equation numbers (1) and (3) are in Cauchy’s text as reprinted in Cauchy (1900).

  5. The equation number (2) is in Cauchy’s original text.

  6. The equation number (4) is in Cauchy’s original text.

  7. Historians often use the term punctiform continuum to refer to a continuum made out of points (as for example the traditional set-theoretic \({{\mathrm{{{\mathbb {R}}}}}}\)). Earlier notions of continuum are generally thought to be non-punctiform (i.e., not made out of points). The term punctiform also has a technical meaning in topology unrelated to the above distinction.

  8. Nelson’s syntactic recasting of Robinson’s framework is a good illustration, in that the logical procedures in Nelson’s framework are certainly up to modern standards of rigor and are expressed in the classical Zermelo–Fraenkel set theory (ZFC). With respect to an enriched set-theoretic language, infinitesimals in Nelson’s Internal Set Theory (IST) can be found within the real number system \({{\mathrm{{{\mathbb {R}}}}}}\) itself. The semantic/ontological issues are handled in an appendix to Nelson (1977), showing Nelson’s IST to be a conservative extension of ZFC.

  9. A similar criterion was formulated in Fraenkel (1928, pp.  116–117). For a discussion of the Klein–Fraenkel criterion see Kanovei et al. (2013, Section  6.1).

  10. For the role of these in a possible construction of the hyperreals see Sect. 5.2.

  11. Here Laugwitz is referring to Cauchy’s motto to the effect that “Mon but principal a été de concilier la rigueur, dont je m’étais fait une loi dans mon Cours d’analyse avec la simplicité que produit la consideration directe des quantités infiniment petites.”

  12. The fact that Laugwitz had published articles in leading periodicals does not mean that he couldn’t have said something wrong. However, it does suggest the existence of a strawman aspect of Schubring’s sweeping claims against him.

  13. The transfer principle is a type of theorem that, depending on the context, asserts that rules, laws or procedures valid for a certain number system (or more general mathematical structure), still apply (i.e., are transfered) to an extended number system (or more general mathematical structure). Thus, the familiar extension \({{\mathrm{{\mathbb {Q}}}}}\,\hookrightarrow\, {{\mathrm{{{\mathbb {R}}}}}}\) preserves the properties of an ordered field. To give a negative example, the extension \({{\mathrm{{{\mathbb {R}}}}}}\,\hookrightarrow \,{{\mathrm{{{\mathbb {R}}}}}}\cup \{\pm \infty \}\) of the real numbers to the so-called extended reals does not preserve the properties of an ordered field. The hyperreal extension \({{\mathrm{{{\mathbb {R}}}}}}\,\hookrightarrow \,{{}^{*}{{\mathrm{{{\mathbb {R}}}}}}}\) preserves all first-order properties. For example, the identity \(\sin ^2 x+\cos ^2 x=1\) remains valid for all hyperreal x, including infinitesimal and infinite values of \(x\in {{}^{*}{{\mathrm{{{\mathbb {R}}}}}}}\). In particular, the properties of the reciprocal function remain the same after it is extended to the hyperreal domain.

    Arsac tries to explain nonstandard analysis but he seems to be as unaware of the transfer principle as Viertel: “Une fonction continue au sens habituel est une fonction continue aux points standards, mais elle ne l’est pas obligatoirement aux points non standard” (Arsac 2013, p. 133). Contrary to his claim, the natural extension of a continuous function f will be continuous at all hyperreal points c in the sense of the standard definition \(\forall \epsilon>0\;\exists \delta >0:|x-c|<\delta\,{\implies}\, |f(x)-f(c)|<\epsilon \), by the transfer principle. Indeed, Arsac confused S-continuity and continuity...

  14. The symbol \(\delta \) is not used in reference to an integer tending to infinity.

  15. Grattan-Guinness acknowledges this point GG4 implicitly in his summary of Cauchy’s proof in (Grattan-Guinness 1970, p. 123). Meanwhile, an \(\varepsilon \) does occur in Cauchy on p. 32; see Sect. 2.2.

  16. This is a reference to Cauchy’s sum (3) namely \(u_n+u_{n+1} \ldots u_{n^{\prime }-1}\) discussed in Sect. 2.

References

  • Alexander, A. (2014). Infinitesimal: How a dangerous mathematical theory shaped the modern world. Straus and Giroux: Farrar.

    Google Scholar 

  • Arsac, G. (2013). Cauchy, Abel, Seidel, Stokes et la convergence uniforme. De la difficulté historique du raisonnement sur les limites. Pari: Hermann.

    Google Scholar 

  • 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. See http://www.ams.org/notices/201307/rnoti-p886.pdf and arXiv:1306.5973

  • Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., et al. (2017). Interpreting the infinitesimal mathematics of Leibniz and Euler. Journal for General Philosophy of Science, 48(2), 195–238. See doi:10.1007/s10838-016-9334-z and arXiv:1605.00455

  • Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., Nowik, T., Sherry, D., & Shnider, S. (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. See http://www.ams.org/notices/201408/rnoti-p848.pdf and arXiv:1407.0233

  • Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Schaps, D., et al. (2016). Leibniz versus 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. doi:10.1086/685645. arXiv:1603.07209.

    Google Scholar 

  • Bascelli, T., Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., Nowik, T., Schaps, D., & Sherry, D. (2017) Gregory’s sixth operation. Foundations of Science. See doi:10.1007/s10699-016-9512-9 and arXiv:1612.05944

  • Bell, J. (2006). The continuous and the infinitesimal in mathematics and philosophy. Milan: Polymetrica.

    Google Scholar 

  • Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.

    Article  Google Scholar 

  • Bishop, E. (1977). Book review: Elementary calculus. Bulletin of the American Mathematical Society 83, 205–208 (review of the first edition of Keisler 1986)

  • 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. doi:10.1007/s10699-011-9235-x. arXiv:1108.2885.

    Article  Google Scholar 

  • Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.

    Article  Google Scholar 

  • Bosmans, H. (1927). André Tacquet (S. J.) et son traité d’ ‘Arithmétique théorique et pratique’. Isis, 9(1), 66–82.

    Article  Google Scholar 

  • Bottazzini, U. (1986). The higher calculus: A history of real and complex analysis from Euler to Weierstrass. Translated from the Italian by Warren Van Egmond: Springer-Verlag, New York.

  • Boyer, C. (1949). The concepts of the calculus. New York, NY: Hafner Publishing.

    Google Scholar 

  • 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, NY: Springer.

    Google Scholar 

  • Błaszczyk, P., Borovik, A., Kanovei, V., Katz, M., Kudryk, T., Kutateladze, S., et al. (2016). A non-standard analysis of a cultural icon: The case of Paul Halmos. Logica Universalis, 10(4), 393–405. doi:10.1007/s11787-016-0153-0. arXiv:1607.00149.

    Article  Google Scholar 

  • Błaszczyk, P., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., & Sherry. D. (2017a). Toward a history of mathematics focused on procedures. Foundations of Science. See doi:10.1007/s10699-016-9498-3 and arXiv:1609.04531

  • Błaszczyk, P., Kanovei, V., Katz, M., & Sherry, D. (2017b). Controversies in the foundations of analysis: Comments on Schubring’s Conflicts. Foundations of Science. See doi:10.1007/s10699-015-9473-4 and arXiv:1601.00059

  • Cauchy, A. L. (1821). Cours d’Analyse de L’Ecole Royale Polytechnique Première Partie. Analyse algébrique. Paris: Imprimérie Royale.

    Google Scholar 

  • Cauchy, A. L. (1823). Résumé des Leçons données à l’Ecole Royale Polytechnique sur le Calcul Infinitésimal. Paris: Imprimérie Royale.

    Google Scholar 

  • Cauchy, A. L. (1829). Leçons sur le calcul différentiel. Oeuvres Complètes, Series, 2, 4.

    Google Scholar 

  • 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. Comptes Rendus de l’ Academie Royale des Sciences, 36, 454–459. Reprinted as (Cauchy 1900).

  • Cauchy, A. L. (1900). Note sur les séries convergentes dont les divers termes sont des fonctions continues d’une variable réelle ou imaginaire, entre des limites données. Oeuvres complètes, Series 1, Vol. 12, pp. 30–36. Paris, Gauthier–Villars.

  • Child, J., 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. The Open Court Publishing, Chicago-London. Reprinted by Dover in 2005.

  • Cleave, J. (1971). Cauchy, convergence and continuity. The British Journal for the Philosophy of Science, 22, 27–37.

    Article  Google Scholar 

  • Connes, A., Lichnerowicz, A., & Schützenberger, M. (2001). Triangle of thoughts. Translated from the 2000 French original by Jennifer Gage. American Mathematical Society, Providence, RI.

  • 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.

    Article  Google Scholar 

  • Dauben, J. (1988). Abraham Robinson and nonstandard analysis: history, philosophy, and foundations of mathematics. History and philosophy of modern mathematics (Minneapolis, MN, 1985), 177–200, Minnesota Stud. Philos. Sci., XI, Univ. Minnesota Press, Minneapolis, MN.

  • Davis, M. (1977). Applied nonstandard analysis. Pure and applied mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. Reprinted by Dover, NY, 2005. See http://store.doverpublications.com/0486442292.html

  • Earman, J. (1975). Infinities, infinitesimals, and indivisibles: The Leibnizian labyrinth. Studia Leibnitiana, 7(2), 236–251.

    Google Scholar 

  • Easwaran, K. (2014). Regularity and hyperreal credences. Philosophical Review, 123(1), 1–41.

    Article  Google Scholar 

  • Ebbinghaus, H.-D., Hermes, H., Hirzebruch, F., Koecher, M., Mainzer, K., Neukirch, J., Prestel, A., & Remmert, R. (1990). Numbers. With an introduction by K. Lamotke. Translated from the second German edition by H. L. S. Orde. Translation edited and with a preface by J. H. Ewing. Graduate Texts in Mathematics, 123. Readings in Mathematics. Springer-Verlag, New York.

  • Edwards, H. (2007). Euler’s definition of the derivative. Bulletin of the American Mathematical Society (N.S.), 44(4), 575–580.

    Article  Google Scholar 

  • Ferraro, G. (2004). Differentials and differential coefficients in the Eulerian foundations of the calculus. Historia Mathematica, 31(1), 34–61.

    Article  Google Scholar 

  • Festa, E. (1990). La querelle de l’atomisme: Galilée, Cavalieri et les Jésuites. La Recherche (sept. 1990), 1038–1047.

  • Festa, E. (1992). Quelques aspects de la controverse sur les indivisibles. Geometry and Atomism in the Galilean School, 193–207, Bibl. Nuncius Studi Testi, X, Olschki, Florence.

  • Fisher, G. (1978). Cauchy and the infinitely small. Historia Mathematica, 5(3), 313–331.

    Article  Google Scholar 

  • Fletcher, P., Hrbacek, K., Kanovei, V., Katz, M., Lobry, C., & Sanders, S. (2017). Approaches to analysis with infinitesimals following Robinson, Nelson, and others. Real Analysis Exchange, 42(2). See arXiv:1703.00425

  • Fraenkel, A. (1928). Einleitung in die Mengenlehre. Dover Publications, New York, NY, 1946 (originally published by Springer, Berlin, 1928).

  • Fraser, C. (2008). Cauchy. The new dictionary of scientific biography (Vol. 2). New York, NY: Scribners and Sons.

    Google Scholar 

  • 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). Heidelberg: Springer: Birkhäuser.

    Chapter  Google Scholar 

  • Gerhardt, C. (Ed.). (1846). Historia et Origo calculi differentialis a G. Hannover: G. Leibnitio conscripta.

  • Gerhardt, C. (Ed.). (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle, Eidmann.

  • Gilain, C. (1989). Cauchy et le cours d’analyse de l’Ecole polytechnique. With an editorial preface by Emmanuel Grison. Bulletin de la Sabix. Société des amis de la Bibliothèque et de l’Histoire de l’École Polytechnique, no. 5.

  • Giusti, E. (1984). Cauchy’s ‘errors’ and the foundations of analysis. (Italian) Bollettino di Storia delle Scienze Matematiche, 4(2), 24–54.

    Google Scholar 

  • Grabiner, J. (1883). The Changing concept of change: The derivative from Fermat to Weierstrass. Mathematics Magazine, 56(4), 195–206.

    Article  Google Scholar 

  • Grabiner, J. (1981). The origins of Cauchy’s rigorous calculus. Cambridge, MA: MIT Press.

    Google Scholar 

  • Grabiner, J. (1983). Who gave you the epsilon? Cauchy and the origins of rigorous calculus. The American Mathematical Monthly, 90(3), 185–194.

    Article  Google Scholar 

  • Grabiner, J. (2006). Review of Schubring (2005). SIAM Review, 48(2), 413–416 (jun., 2006).

  • Grattan-Guinness, I. (1970). The development of the foundations of mathematical analysis from Euler to Riemann. Cambridge, MA: The MIT Press.

    Google Scholar 

  • Gray, J. (2015). The real and the complex: A history of analysis in the nineteenth century., Springer Undergraduate Mathematics Series Cham: Springer.

    Google Scholar 

  • Gutman, A., Katz, M., Kudryk, T., & Kutateladze, S. (2017). The mathematical intelligencer flunks the olympics. Foundations of Science. See doi:10.1007/s10699-016-9485-8 and arXiv:1606.00160

  • Halmos, P. (1985). I want to be a mathematician. An automathography. New York, NY: Springer.

    Book  Google Scholar 

  • Hellyer, M. (1996). ‘Because the authority of my superiors commands’: Censorship, physics and the German Jesuits. Early Science and Medicine, 3, 319–354.

    Article  Google Scholar 

  • Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions of the American Mathematical Society, 64, 45–99.

    Article  Google Scholar 

  • Ishiguro, H. (1990). Leibniz’s philosophy of logic and language (2nd ed.). Cambridge: Cambridge University Press.

    Google Scholar 

  • 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. doi:10.1007/s10699-012-9316-5. arXiv:1211.0244.

    Article  Google Scholar 

  • Kanovei, V., Katz, K., Katz, M., & Sherry, D. (2015). Euler’s lute and Edwards’ oud. The Mathematical Intelligencer, 37(4), 48–51. doi:10.1007/s00283-015-9565-6. arXiv:1506.02586.

    Article  Google Scholar 

  • Katz, V. (2014). Review of “Bair et al., Is mathematical history written by the victors? Notices of the American Mathematical Society (2013) 607, 886–904.” See http://www.ams.org/mathscinet-getitem?mr=3086638

  • Katz, K., & Katz, M. (2011a). Meaning in classical mathematics: Is it at odds with intuitionism? Intellectica, 56(2), 223–302. arXiv:1110.5456.

    Google Scholar 

  • Katz, K., & Katz, M. (2011b). Cauchy’s continuum. Perspectives on Science, 19(4), 426–452. doi:10.1162/POSC_a_00047. arXiv:1108.4201.

    Article  Google Scholar 

  • Katz, M., & Leichtnam, E. (2013). Commuting and noncommuting infinitesimals. American Mathematical Monthly, 120(7), 631–641. doi:10.4169/amer.math.monthly.120.07.631. arXiv:1304.0583.

    Article  Google Scholar 

  • Katz, M., & Polev, L. (2017). From Pythagoreans and Weierstrassians to true infinitesimal calculus. Journal of Humanistic Mathematics, 7(1), 87–104. doi:10.5642/jhummath.201701.07. arXiv:1701.05187.

    Article  Google Scholar 

  • 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. doi:10.1162/POSC_a_00101. arXiv:1210.7750.

    Article  Google Scholar 

  • 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. doi:10.1007/s10670-012-9370-y. arXiv:1205.0174.

    Article  Google Scholar 

  • Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach, 2nd edn. Boston, MA: Prindle, Weber and Schmidt. See http://www.math.wisc.edu/~keisler/calc.html

  • Klein, F. (1908). 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).

  • Kock, A. (2006). Synthetic differential geometry (2nd ed.)., London Mathematical Society Lecture Note Series, 333 Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Kuhn, T. (1996). The structure of scientific revolutions (3rd ed.). Chicago, IL: University of Chicago.

    Book  Google Scholar 

  • Laugwitz, D. (1987). Infinitely small quantities in Cauchy’s textbooks. Historia Mathematica, 14, 258–274.

    Article  Google Scholar 

  • Laugwitz, D. (1989). Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820. Archive for the History of Exact Sciences, 39, 195–245.

    Article  Google Scholar 

  • Laugwitz, D. (1990). Das mathematisch Unendliche bei Cauchy und bei Euler. ed. Gert König, Konzepte des mathematisch Unendlichen im 19. Jahrhundert (Göttingen, Vandenhoeck u. Ruprecht, 1990), 9–33.

  • Leibniz, G. (1695). Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas. Act. Erudit. Lips. In Gerhardt (1850–1863), vol. V, 320–328. A French translation is in (Leibniz 1989, pp. 316–334).

  • Leibniz, G. (1701). 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_s

  • Leibniz, G. (1989). La naissance du calcul différentiel. 26 articles des Acta Eruditorum. Translated from the Latin and with an introduction and notes by Marc Parmentier. With a preface by Michel Serres. Mathesis. Librairie Philosophique J. Vrin, Paris.

  • Lindstrøm, T. (1988). An invitation to nonstandard analysis. Nonstandard analysis and its applications (Hull, 1986), 1–105, London Math. Soc. Stud. Texts, 10, Cambridge University Press, Cambridge.

  • Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York, NY: The Clarendon Press, Oxford University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • McKinzie, M., & Tuckey, C. (2001). Higher trigonometry, hyperreal numbers, and Euler’s analysis of infinities. Mathematics Magazine, 74(5), 339–368.

    Article  Google Scholar 

  • Nakane, M. (2014). Did Weierstrass’s differential calculus have a limit-avoiding character? His definition of a limit in \(\epsilon \)-\(\delta \) style. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 29(1), 51–59.

    Article  Google Scholar 

  • Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.

    Article  Google Scholar 

  • Planck, M. (1950). Scientific autobiography and other papers (F. Gaynor, Trans.). London.

  • Prestel, A. (1990). Chapter 12. Nonstandard analysis. Pp. 305–327 in Ebbinghaus et al. (1990).

  • Quine, W. (1968). Ontological relativity. The Journal of Philosophy, 65(7), 185–212.

    Article  Google Scholar 

  • Ramis, J.-P. (2012). Poincaré et les développements asymptotiques (première partie). Gazette Mathematique No., 133, 33–72.

    Google Scholar 

  • Redondi, P. (1987). Galileo: heretic. Translated from the Italian by Raymond Rosenthal: Princeton University Press, Princeton, NJ.

  • Robinson, A. (1961). Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A, 64 = Indag. Math., 23 (1961), 432–440. Reprinted in Selected Papers (Robinson 1979), pp. 3–11.

  • Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing.

    Google Scholar 

  • 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. Yale University Press, New Haven, Conn.

  • Sanders, S. (2017a). Reverse Formalism 16. Synthese, Special issue for SOTFOMIII. See doi:10.1007/s11229-017-1322-2 and arXiv:1701.05066

  • Sanders, S. (2017b). To be or not to be constructive. Indagationes Mathematicae and the Brouwer volume L.E.J. Brouwer, fifty years later, 2017–2018, p. 68. See arXiv:1704.00462

  • Schmieden, C., & Laugwitz, D. (1958). Eine Erweiterung der infinitesimalrechnung. Mathematische Zeitschrift, 69, 1–39.

    Article  Google Scholar 

  • 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-Verlag, New York.

  • Sergeyev, Y. (2015). The olympic medals ranks, lexicographic ordering, and numerical infinities. The Mathematical Intelligencer, 37(2), 4–8.

    Article  Google Scholar 

  • Sherry, D., & Katz, M. (2014) Infinitesimals, imaginaries, ideals, and fictions. Studia Leibnitiana, 44(2) (2012), 166–192. See http://www.jstor.org/stable/43695539 and arXiv:1304.2137 (Article was published in 2014 even though the journal issue lists the year as 2012)

  • Spalt, D. (2002). Cauchys kontinuum. Eine historiografische Annäherung via Cauchys summensatz. Archive for History of Exact Sciences, 56(4), 285–338.

    Article  Google Scholar 

  • Sørensen, H. (2005). Exceptions and counterexamples: Understanding Abel’s comment on Cauchy’s theorem. Historia Mathematica, 32(4), 453–480.

    Article  Google Scholar 

  • Tao, T. (2014). Hilbert’s fifth problem and related topics., Graduate Studies in Mathematics Providence, RI: American Mathematical Society.

    Book  Google Scholar 

  • Tao, T. & Vu, V. (2016). Sum-avoiding sets in groups. Discrete Analysis 2016:15, 31 pp, see doi:10.19086/da.887 and arXiv:1603.03068

  • Tarski, A. (1930). Une contribution à la théorie de la mesure. Fundamenta Mathematicae, 15, 42–50.

    Article  Google Scholar 

  • Unguru, S. (1976). Fermat revivified, explained, and regained. Francia, 4, 774–789.

    Google Scholar 

  • Viertel, K. (2014). Geschichte der gleichmäßigen Konvergenz. Ursprünge und Entwicklungen des Begriffs in der Analysis des 19. Jahrhunderts. UK: Springer Spektrum.

    Google Scholar 

  • 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 (Vol. XXXIX, pp. 717–737). Boston Studies in the Philosophy of Science Dordrecht: D. Reidel Publishing.

    Chapter  Google Scholar 

Download references

Acknowledgements

V. Kanovei was supported in part by the RFBR Grant Number 17-01-00705. M.  Katz was partially funded by the Israel Science Foundation Grant Number 1517/12. We are grateful to Dave L.  Renfro for helpful suggestions.

Author information

Authors and Affiliations

  1. Lyceum Gymnasium “F. Corradini”, Thiene, Italy

    Tiziana Bascelli

  2. Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland

    Piotr Błaszczyk

  3. School of Mathematics, University of Manchester, Oxford Street, Manchester, M13 9PL, United Kingdom

    Alexandre Borovik

  4. IPPI, Moscow, and MIIT, Moscow, Russia

    Vladimir Kanovei

  5. Department of Mathematics, Bar Ilan University, Ramat Gan, 5290002, Israel

    Karin U. Katz & Mikhail G. Katz

  6. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    Semen S. Kutateladze

  7. Rice University, Houston, USA

    Thomas McGaffey

  8. Department of Classical Studies, Bar Ilan University, Ramat Gan, 5290002, Israel

    David M. Schaps

  9. Department of Philosophy, Northern Arizona University, Flagstaff, AZ 86011, US

    David Sherry

Authors
  1. Tiziana Bascelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr Błaszczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandre Borovik
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vladimir Kanovei
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Karin U. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Mikhail G. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Semen S. Kutateladze
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Thomas McGaffey
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. David M. Schaps
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. David Sherry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail G. Katz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bascelli, T., Błaszczyk, P., Borovik, A. et al. Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms. Found Sci 23, 267–296 (2018). https://doi.org/10.1007/s10699-017-9534-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10699-017-9534-y

Keywords

Search

Navigation