Educational Studies in Mathematics

, Volume 74, Issue 3, pp 259–273

Zooming in on infinitesimal 1–.9.. in a post-triumvirate era

Article

Abstract

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula \(\hbox{.999\ldots = 1}\). Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol “.999...”? We present some non-standard thoughts on the ambiguity of the ellipsis in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals.

Keywords

Decimal representation Generic limit Hypernatural number Infinitesimal Limit Unital evaluation 

References

  1. Abbott, E. A. (2008). The annotated Flatland. A romance of many dimensions. With an introduction and notes by Ian Stewart. New York: Basic Books (Original work published 2002).Google Scholar
  2. Artigue, M. (1994). Analysis. In D. Tall (Ed.), Advanced mathematical thinking (p. 172). New York: Springer (“The non-standard analysis revival and its weak impact on education”).Google Scholar
  3. Avigad, J., & Reck, E. (2001). Clarifying the nature of the infinite: The development of metamathematics and proof theory, 11 Dec 2001. Carnegie Mellon Technical Report CMU-PHIL-120.Google Scholar
  4. Bell, J. L. (2009). Continuity and infinitesimals. Stanford Encyclopedia of Philosophy (Revised 20 Jul 2009).Google Scholar
  5. Bishop, E. (1975). The crisis in contemporary mathematics. Historia Mathematica, 2(4), 507–517.CrossRefGoogle Scholar
  6. Bishop, E. (1977). Review: H. Jerome Keisler. Elementary calculus. Bulletin of the American Mathematical Society, 83, 205–208.CrossRefGoogle Scholar
  7. Bishop, E. (1985). Schizophrenia in contemporary mathematics. In M. Rosenblatt (Ed.), Errett Bishop: Reflections on him and his research (San Diego, Calif., 1983). Contemp. Math., Vol. 39 (p. 1–32). Providence: American Mathematical Society.Google Scholar
  8. Bos, H. J. M. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.CrossRefGoogle Scholar
  9. Boyer, C. (1949). The concepts of the calculus. New York: Hafner.Google Scholar
  10. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking. Mathematics Education Library, 11 (pp. 153–166). Dordrecht: Kluwer Academic.Google Scholar
  11. 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 (1988). New York: Wiley.Google Scholar
  12. Dauben, J. (1992). Appendix (1992): Revolutions revisited. In D. Gillies (Ed.), Revolutions in mathematics (pp. 72–82 ). New York: Clarendon.Google Scholar
  13. Dauben, J. (1996). Arguments, logic and proof: mathematics, logic and the infinite. History of mathematics and education: Ideas and experiences (Essen, 1992). Stud. Wiss. Soz. Bildungsgesch. Math., 11, 113–148 (Vandenhoeck & Ruprecht, Göttingen).Google Scholar
  14. Davis, M. (1977). Review: J. Donald Monk. Mathematical logic. Bulletin of the American Mathematical Society, 83, 1007–1011.CrossRefGoogle Scholar
  15. Ely, R. (2007). Student obstacles and historical obstacles to foundational concepts of calculus. Ph.D. thesis, The University of Wisconsin—Madison.Google Scholar
  16. Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117–146.Google Scholar
  17. Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci., 60(1), 1–121.CrossRefGoogle Scholar
  18. Feferman, S. (2000). Relationships between constructive, predicative and classical systems of analysis. Proof theory (Roskilde, 1997). Synthese Lib., Vol. 292 (pp. 221–236). Dordrecht: Kluwer Academic.Google Scholar
  19. Goldblatt, R. (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate texts in mathematics, 188. New York: Springer.Google Scholar
  20. Goldwurm, R’. H. (2001). The Rishonim. Biographical sketches of the prominent early rabbinic sages and leaders from the tenth–fifteenth centuries (2nd ed.). Artscroll History Series. Brooklyn: Mesorah.Google Scholar
  21. Gray, E., & Tall, D. (1991). Duality, ambiguity and flexibility in successful mathematical thinking. Proceedings of PME 15, 2, 72–79 (Assisi).Google Scholar
  22. Halmos, P. (1985). I want to be a mathematician. An automathography. New York: SpringerGoogle Scholar
  23. HaYisraeli, Y. (1310). Yesod Olam. In Poel hashem. Bnei Braq.Google Scholar
  24. 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
  25. Hodgson, B. R. (1994). Le calcul infinitésimal. In D. F. Robitaille, D. H. Wheeler, & C. Kieran (Eds.), Choix de conférence du 7e Congrès international sur l’enseignement des mathématiques (ICME-7) (pp. 157–170). Sainte-Foy: Presses de l’Université Laval.Google Scholar
  26. Jesseph, D. (1998). Leibniz on the foundations of the calculus: The question of the reality of infinitesimal magnitudes. Leibniz and the sciences. Perspect. Sci., 6(1-2), 6–40.Google Scholar
  27. Katz, K., & Katz, M. (2010). When is .999... less than 1? The Montana Mathematics Enthusiast, 7(1), 3–30.Google Scholar
  28. Keisler, H. J. (1986). Elementary Calculus: An Infinitesimal Approach (2nd ed.). Boston: Prindle, Weber & Schimidt.Google Scholar
  29. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  30. Lakatos, I. (1978). Cauchy and the continuum: The significance of nonstandard analysis for the history and philosophy of mathematics. Mathematical Intelligencer 1(3), 151–161 (originally published in 1966).CrossRefGoogle Scholar
  31. Lightstone, A. H. (1972). Infinitesimals. American Mathematical Monthly, 79, 242–251.CrossRefGoogle Scholar
  32. Luxemburg, W. (1964). Nonstandard analysis. Lectures on A. Robinson’s Theory of infinitesimals and infinitely large numbers (2nd ed.). Pasadena: Mathematics Department, California Institute of Technology.Google Scholar
  33. Luzin, N. N. (1931). Two letters by N. N. Luzin to M. Ya. Vygodskiĭ. With an introduction by S. S. Demidov. Translated from the 1997 Russian original by A. Shenitzer. American Mathematical Monthly, 107(1), 64–82, (2000).Google Scholar
  34. Medvedev, F. A. (1998). Nonstandard analysis and the history of classical analysis. Translated by Abe Shenitzer. American Mathematical Monthly, 105(7), 659–664.CrossRefGoogle Scholar
  35. Monaghan, J. (2001). Young peoples’ ideas of infinity. Educational Studies in Mathematics 48(2–3), 239–257.CrossRefGoogle Scholar
  36. Robinson, A. (1975). Concerning progress in the philosophy of mathematics. Logic Colloquium ’73 (Bristol, 1975). Studies in Logic and the Foundations of Mathematics, Vol. 80 (pp. 41–52). Amsterdam: North-Holland.Google Scholar
  37. Robinson, A. (1979). Selected papers of Abraham Robinson. In W. A. J. Luxemburg & S. Krner (Eds.), Nonstandard analysis and philosophy, Vol. II. New Haven: Yale University Press.Google Scholar
  38. Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69, 217–233.CrossRefGoogle Scholar
  39. Roquette, P. (2008). Numbers and Models, standard and nonstandard. Algebra days, May 2008, Antalya.Google Scholar
  40. Sad, L. A., Teixeira, M. V., & Baldino, R. B. (2001). Cauchy and the problem of point-wise convergence. Archives Internationales d’histoire des Sciences, 51(147), 277–308.Google Scholar
  41. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th Century France and Germany. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.Google Scholar
  42. Stewart, I. (2009). Professor Stewart’s hoard of mathematical treasures. London: Profile Books.Google Scholar
  43. Sullivan, K. (1976). The teaching of elementary calculus using the nonstandard analysis approach. American Mathematical Monthly, 83, 370–375.CrossRefGoogle Scholar
  44. Tall, D. (1980). Looking at graphs through infinitesimal microscopes, windows and telescopes, Mathematical Gazette, 64, 22–49.CrossRefGoogle Scholar
  45. Tall, D. (1991). The psychology of advanced mathematical thinking. In David Tall (Ed.), Advanced mathematical thinking. Mathematics Education Library, 11. Dordrecht: Kluwer Academic.Google Scholar
  46. Tall, D. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. In Transforming mathematics education through the use of dynamic mathematics (pp. 1–11). ZDMGoogle Scholar
  47. Tall, D. (2010). How humans learn to think mathematically. Princeton: Princeton University Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Bar Ilan UniversityRamat GanIsrael

Personalised recommendations