Foundations of Science

, Volume 20, Issue 1, pp 1–25 | Cite as

Proofs and Retributions, Or: Why Sarah Can’t Take Limits

  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. Katz
  • Mary Schaps
Article

Abstract

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history.

Keywords

Errett Bishop Abraham Robinson Infinitesimals Hyperreals Intuitionists Petard 

References

  1. 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. http://www.ams.org/notices/201307/rnoti-p886, http://arxiv.org/abs/1306.5973.
  2. 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. http://arxiv.org/abs/1407.0233.
  3. Beeson, M. (1985). Foundations of constructive mathematics. Metamathematical studies. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Vol. 6). Berlin: Springer.Google Scholar
  4. Berkeley, G. (1734). The analyst: A discourse addressed to an infidel mathematician.Google Scholar
  5. Billinge, H. (2003). Did Bishop have a philosophy of mathematics? Philosophia Mathematica (3), 11(2), 176–194.CrossRefGoogle Scholar
  6. Birkhoff, G. (Ed.) (1975). Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, MA, 1974). Historia Mathematica, 2(4).Google Scholar
  7. Bishop, E. (1965a). Differentiable manifolds in complex Euclidean space. Duke Mathematical Journal, 32, 1–21.CrossRefGoogle Scholar
  8. Bishop, E. (1965b). Colloquium lecture at Stanford University, Circa.Google Scholar
  9. Bishop, E. (1967). Foundations of constructive analysis. New York-Toronto, Ontario-London: McGraw-Hill.Google Scholar
  10. Bishop, E. (1968). Mathematics as a numerical language. 1970 Intuitionism and Proof Theory (Proc. Conf., Buffalo, New York) pp. 53–71. Amsterdam: North-Holland.Google Scholar
  11. Bishop, E. (1973). Schizophrenia in contemporary mathematics. [Published posthumously; originally distributed in 1973]. In Errett Bishop: reflections on him and his research (San Diego, CA, 1983), 1–32, Contemp. Math., 39. Providence, RI: American Mathematical Society (1985).Google Scholar
  12. Bishop, E. (1975). The crisis in contemporary mathematics. Historia Mathematica, 2(4), 507–517.CrossRefGoogle Scholar
  13. Bishop, E. (1977). Review: H. J. Keisler, elementary calculus. Bulletin of the American Mathematical Society, 83, 205–208.Google Scholar
  14. Błaszczyk, P., Katz, M., & Sherry, D. (2013). Ten misconceptions from the history of analysis and their debunking. Foundations of Science, 18(1), 43–74. doi:10.1007/s10699-012-9285-8. 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), 557–570. doi:10.1215/00294527-1722755. http://arxiv.org/abs/1210.7475.
  16. 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. http://arxiv.org/abs/1108.2885.
  17. Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.CrossRefGoogle Scholar
  18. Bos, H. (2010). Private communication, 2 Nov (2010).Google Scholar
  19. Cantor, G. (1892). Ueber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutsche Mathematiker-Vereinigung, 1, 75–78.Google Scholar
  20. Connes, A. (1995). Noncommutative geometry and reality. Journal of Mathematical Physics, 36(11), 6194–6231.CrossRefGoogle Scholar
  21. Connes, A., Lichnerowicz, A., & Schützenberger, M. (2001). Triangle of thoughts. Translated from the 2000 French original by Jennifer Gage. Providence, RI: American Mathematical Society.Google Scholar
  22. Courant, R. (1937). Differential and integral calculus (Vol. I). Translated from the German by E. J. McShane. Reprint of the second edition. Wiley Classics Library. A Wiley-Interscience Publication. New York: Wiley (1988).Google Scholar
  23. Dauben, J. (1995). Abraham Robinson. The creation of nonstandard analysis. A personal and mathematical odyssey. With a foreword by Benoit B. Mandelbrot. Princeton, NJ: Princeton University Press.Google Scholar
  24. Dauben, J. (1996). Arguments, logic and proof: mathematics, logic and the infinite. History of mathematics and education: Ideas and experiences (Essen, 1992), pp. 113–148, Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik, 11, Vandenhoeck and Ruprecht, Göttingen.Google Scholar
  25. Dauben, J. (2003) Abraham Robinson. Biographical memoirs V. 82. http://www.nap.edu/catalog/10683.html. National Academy of Sciences.
  26. Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences, 60(1), 1–121.CrossRefGoogle Scholar
  27. Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117–146.Google Scholar
  28. Euler, L. (1748). Introductio in Analysin Infinitorum, Tomus primus. SPb and Lausana.Google Scholar
  29. Euler, L. (1988). Introduction to analysis of the infinite. Book I. Translated from the Latin and with an introduction by John D. Blanton. New York: Springer, 1988 [translation of (Euler 1748)].Google Scholar
  30. Ewing, J. (2009). Private communication, 27 January.Google Scholar
  31. Feferman, S. (2000). Relationships between constructive, predicative and classical systems of analysis. Proof theory (Roskilde. (1997). 221–236, Synthese Library, 292. Dordrecht: KluwerGoogle Scholar
  32. Fraenkel, A. (1967). Lebenskreise. Aus den Erinnerungen eines jüdischen Mathematikers. Stuttgart: Deutsche Verlags-Anstalt.Google Scholar
  33. Gerhardt, C. I. (Ed.) (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann)Google Scholar
  34. Goodstein, R. L. (1970) Review of (Bishop 1968). See http://www.ams.org/mathscinet-getitem?mr=270894.
  35. Halmos, P. (1985). I want to be a mathematician. An automathography. New York: Springer.CrossRefGoogle Scholar
  36. Heijting, A. (1973). Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof. A. Robinson on the 26th April 1973. Nieuw Archief voor Wiskunde (3), 21, 134–137.Google Scholar
  37. Hellman, G. (1993). Constructive mathematics and quantum mechanics: Unbounded operators and the spectral theorem. Journal of Philosophical Logic, 12, 221–248.CrossRefGoogle Scholar
  38. Hellman, G. (1998). Mathematical constructivism in spacetime. The British Journal for the Philosophy of Science, 49(3), 425–450.CrossRefGoogle Scholar
  39. Herzberg, F. (2013). Stochastic calculus with infinitesimals. Lecture notes in mathematics (Vol. 2067). Heidelberg: Springer.Google Scholar
  40. Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions of the American Mathematical Society, 64, 45–99.CrossRefGoogle Scholar
  41. Heyting, A. (1956). Intuitionism. An introduction. Amsterdam: North-Holland.Google Scholar
  42. Hill, D. (2009). Personal communication, 21 January 2009.Google Scholar
  43. Hill, D. (2013). Personal communication, 12 March 2013.Google Scholar
  44. Hrbáček, K. (1978). Axiomatic foundations for nonstandard analysis. Fundamenta Mathematicae, 98(1), 1–19.Google Scholar
  45. Hrbacek, K. (2005). Remarks on nonstandard class theory. (Russian) Fundam. Prikl. Mat. 11(5), 233–255; translation in Journal of Mathematical Science (N.Y.) 146(1), 5608–5621 (2007).Google Scholar
  46. Kanovei, V. (1988). The correctness of Euler’s method for the factorization of the sine function into an infinite product. Russian Mathematical Surveys, 43, 65–94.CrossRefGoogle Scholar
  47. 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. See doi:10.1007/s10699-012-9316-5 and http://arxiv.org/abs/1211.0244.
  48. Kanovei, V., & Lyubetskii, V. (2007). Problems of set-theoretic nonstandard analysis. Uspekhi Mat. Nauk, 62(1) (373), 51–122 (Russian); translation in Russian Math. Surveys, 62(1), 45–111.Google Scholar
  49. Kanovei, V., & Reeken, M. (2004). Nonstandard analysis, axiomatically. Springer monographs in mathematics Berlin: Springer.Google Scholar
  50. Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. Journal of Symbolic Logic, 69(1), 159–164.CrossRefGoogle Scholar
  51. Katz, K., & Katz, M. (2010a). When is.999... less than 1? The Montana Mathematics Enthusiast 7(1), 3–30. http://arxiv.org/abs/arXiv:1007.3018
  52. Katz, K., & Katz, M. (2010b). Zooming in on infinitesimal \(1-.9..\) in a post-triumvirate era. Educational Studies in Mathematics, 74(3), 259–273. http://arxiv.org/abs/arXiv:1003.1501.
  53. Katz, K., & Katz, M. (2011). Meaning in classical mathematics: Is it at odds with Intuitionism? Intellectica, 56(2), 223–302.Google Scholar
  54. Katz, K., & Katz, M. (2012a). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51–89. doi:10.1007/s10699-011-9223-1. http://arxiv.org/abs/1104.0375.
  55. Katz, K., & Katz, M. (2012b). Stevin numbers and reality. Foundations of Science, 17(2), 109–123. http://arxiv.org/abs/1107.3688. doi:10.1007/s10699-011-9228-9.
  56. Katz, K., Katz, M., & Kudryk, T. (2014). Toward a clarity of the extreme value theorem. Logica Universalis, 8(2), 193–214. http://arxiv.org/abs/1404.5658. doi:10.1007/s11787-014-0102-8. http://www.ams.org/mathscinet-getitem?mr=3210286.
  57. 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. http://arxiv.org/abs/1304.0583.
  58. 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. http://www.mitpressjournals.org/doi/abs/10.1162/POSC_a_00101. http://arxiv.org/abs/1210.7750.
  59. Katz, M., & Sherry, D. (2012). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550–1558. http://www.ams.org/notices/201211/. http://arxiv.org/abs/1211.7188.
  60. 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. http://arxiv.org/abs/1205.0174.
  61. Katz, M., & Tall, D. (2013). A Cauchy–Dirac delta function. Foundations of Science, 18(1), 107–123. doi:10.1007/s10699-012-9289-4. http://www.ams.org/mathscinet-getitem?mr=3031797, http://arxiv.org/abs/1206.0119.
  62. Keisler, H. J. (1977). Letter to the editor. Notices of the American Mathematical Society, 24, 269.Google Scholar
  63. Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach (2nd ed.). Boston: Prindle, Weber and Schimidt.Google Scholar
  64. Keisler, H. J. (1994). The hyperreal line. In Real numbers, generalizations of the reals, and theories of continua (pp. 207–237). Synthese Lib., 242, Dordrecht: Kluwer.Google Scholar
  65. Keisler, H. J. (2007). Foundations of infinitesimal calculus. On-line Edition. https://www.math.wisc.edu/keisler/foundations.html.
  66. Kiepert, L. (1912). Grundriss der Differential- und Integralrechnung. I. Teil: Differentialrechnung. Helwingsche Verlagsbuchhandlung. Hannover, 12th edn. XX, 863 S.Google Scholar
  67. 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).Google Scholar
  68. Kolmogorov, A. (2006). Modern debates on the nature of mathematics. With a commentary by V. A. Uspenskii. Reprinted from Nauchnoe Slovo 1929, no. 6, 41–54. Problemy Peredachi Informatsii, 42(4), 129–141; translation in Problems of Information Transmission, 42(4), 379–389 (2006).Google Scholar
  69. Lakatos, I. (1976). Proofs and refutations. The logic of mathematical discovery. Edited by John Worrall and Elie Zahar. Cambridge University Press, Cambridge-New York-Melbourne.Google Scholar
  70. Leibniz, G. Letter to Varignon, 2 Feb 1702. In Gerhardt (see item (1850–1863)) IV, pp. 91–95.Google Scholar
  71. Leibniz, G. (1710). Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. In Gerhardt [(1850–1863), vol. V, pp. 377–382]Google Scholar
  72. Lightstone, A. H. (1972). Infinitesimals. American Mathematical Monthly, 79, 242–251.CrossRefGoogle Scholar
  73. 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
  74. Maddy, P. (1989). The roots of contemporary Platonism. Journal of Symbolic Logic, 54(4), 1121–1144.CrossRefGoogle Scholar
  75. Mancosu, P. (Ed). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.Google Scholar
  76. McKinzie, M., & Tuckey, C. (1997). Hidden lemmas in Euler’s summation of the reciprocals of the squares. Archive for History of Exact Sciences, 51, 29–57.CrossRefGoogle Scholar
  77. McLarty, C. (2010). What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory. Bulletin of Symbolic Logic, 16(3), 359–377.CrossRefGoogle Scholar
  78. McLarty, C. (2011). A finite order arithmetic foundation for cohomology, preprint. http://arxiv.org/abs/1102.1773.
  79. Meschkowski, H. (1965). Aus den Briefbuchern Georg Cantors. Archive for History of Exact Sciences, 2, 503–519.CrossRefGoogle Scholar
  80. Mormann, T., & Katz, M. (2013). Infinitesimals as an issue of neo-Kantian philosophy of science. HOPOS: The Journal of the International Society for the History of Philosophy of Science, 3(2), 236–280. http://www.jstor.org/stable/10.1086/671348. http://arxiv.org/abs/1304.1027.
  81. Nabokov, V. (1962). Pale fire. USA: G. P. Putnam’s Sons.Google Scholar
  82. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
  83. Netz, R., Saito, K., & Tchernetska, N. (2001). A new reading of method proposition 14: Preliminary evidence from the Archimedes Palimpsest (Part 1). In SCIAMVS (Vol. 2, pp. 9–29).Google Scholar
  84. Netz, R., Saito, K., & Tchernetska, N. (2002). A new reading of method proposition 14: Preliminary evidence from the Archimedes Palimpsest (Part 2). In SCIAMVS 3 (pp. 109–125).Google Scholar
  85. Novikov, S. P. (2002a). The second half of the 20th century and its conclusion: crisis in the physics and mathematics community in Russia and in the West. Translated from Istor.-Mat. Issled. (2), 7(42), 326–356, 369; by A. Sossinsky. American Mathematical Society Transl. Ser. 2, 212, Geometry, topology, and mathematical physics, 1–24, Providence, RI: American Mathematical Society (2004).Google Scholar
  86. Novikov, S. P. (2002b). The second half of the 20th century and its results: The crisis of the society of physicists and mathematicians in Russia and in the West. Istor.-Mat. Issled. (2) 7(42), 326–356, 369 (Russian).Google Scholar
  87. Pourciau, B. (1999). The education of a pure mathematician. American Mathematical Monthly, 106(8), 720–732.CrossRefGoogle Scholar
  88. Reeder, P. (2013). Internal set theory and Euler’s Introductio in Analysin Infinitorum. M.Sc. Thesis, Ohio State University.Google Scholar
  89. Ribenboim, P. (1999). Fermat’s last theorem for amateurs. New York: Springer.Google Scholar
  90. Richard, J. (1964). Les principes des mathématiques et le problème des ensembles. Revue gńérale des sciences pures et appliquées, 16 (1905), 541–543. Translated in Heijenoort, J. van, (Eds.) Source book in mathematical logic 1879–1931 (pp. 142–144). Cambridge, MA: Harvard University Press.Google Scholar
  91. Richman, F. (1987). The last croak. The Mathematical Intelligencer, 9(3), 25–26.CrossRefGoogle Scholar
  92. Richman, F. (1996). Interview with a constructive mathematician. Modern Logic, 6(3), 247–271.Google Scholar
  93. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland.Google Scholar
  94. Robinson, A. (1968). Reviews: Foundations of constructive analysis. American Mathematical Monthly, 75(8), 920–921.CrossRefGoogle Scholar
  95. Robinson, A. (1969). From a formalist’s points of view. Dialectica, 23, 45–49.CrossRefGoogle Scholar
  96. Roquette, P. (2010). Numbers and models, standard and nonstandard. Mathematische Semesterberichte, 57, 185–199.CrossRefGoogle Scholar
  97. Sanders, S. (2014). Algorithm and proof as \(\Omega \)-invariance and transfer: A new model of computation in nonstandard analysis. http://arxiv.org/abs/1404.0080.
  98. Shakespeare, W. (1623). Timon of Athens.Google Scholar
  99. Sherry, D. (1987). The wake of Berkeley’s Analyst: Rigor mathematicae? Studies in History and Philosophy of Science, 18(4), 455–480.CrossRefGoogle Scholar
  100. Sherry, D., & Katz, M. (2012). Infinitesimals, imaginaries, ideals, and fictions. Studia Leibnitiana, 44(2), 166–192. http://arxiv.org/abs/1304.2137.
  101. Simpson, S. (2009). Subsystems of second order arithmetic, 2nd edn. Perspectives in logic. Cambridge: Cambridge University Press; Poughkeepsie, NY: Association for Symbolic Logic.Google Scholar
  102. Stewart, I. (1996). From here to infinity. A retitled and revised edition of The problems of mathematics [New York: Oxford University Press (1992). With a foreword by James Joseph Sylvester. New York: The Clarendon Press, Oxford University Press.Google Scholar
  103. Stewart, I. (2009). Professor Stewart’s Hoard of mathematical treasures. Profile Books.Google Scholar
  104. Tarski, A. (1936). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405.Google Scholar
  105. Tho, T. (2012). Equivocation in the foundations of Leibniz’s infinitesimal fictions. Society and Politics, 6(2), 70–98.Google Scholar
  106. Urquhart, A. (2006). Mathematics and physics: Strategies of assimilation. In Mancosu (see item (2008)) (pp. 417–440).Google Scholar
  107. Weber, H. (1893). Leopold Kronecker. Mathematische Annalen, 43(1), 1–25.CrossRefGoogle Scholar
  108. Weyl, H. (1921). Über die neue Grundlagenkrise der Mathematik. (Vorträge, gehalten im mathematischen Kolloquium Zürich.). [J] Mathematische Zeitschrift, 10, 39–79.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Vladimir Kanovei
    • 1
    • 2
  • Karin U. Katz
    • 3
  • Mikhail G. Katz
    • 3
  • Mary Schaps
    • 3
  1. 1.IPPIMoscowRussia
  2. 2.MIITMoscowRussia
  3. 3.Department of MathematicsBar Ilan UniversityRamat GanIsrael

Personalised recommendations