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. KatzEmail author
  • Mary Schaps


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.


Errett Bishop Abraham Robinson Infinitesimals Hyperreals Intuitionists Petard 



The work of V. Kanovei was partially supported by RFBR grant 13-01-00006. M. Katz was partially funded by the Israel Science Foundation grant no. 1517/12. The authors are grateful to Antonio Montalban for expert advice in matters of note 115. The influence of Hilton Kramer (1928–2012) is obvious.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

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

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