Inconsistencies in the History of Mathematics

The Case of Infinitesimals
Part of the Origins book series (ORIN, volume 2)


In this paper I will not confine myself exclusively to historical considerations. Both philosophical and technical matters will be raised, all with the purpose of trying to understand (better) what Newton, Leibniz and the many precursors (might have) meant when they talked about infinitesimals. The technical part will consist of an analysis why apparently infinitesimals have resisted so well to be formally expressed. The philosophical part, actually the most important part of this paper, concerns a discussion that has been going on for some decennia now. After the Kuhnian revolution in philosophy of science, notwithstanding Kuhn’s own suggestion that mathematics is something quite special, the question was nevertheless asked how mathematics develops. Are there revolutions in mathematics? If so, what do we have to think of? If not, why do they not occur? Is mathematics the so often claimed totally free creation of the human spirit? As usual, there is a continuum of positions, but let me sketch briefly the two extremes: the completists (as I call them) on the one hand, and the contingents (as I call them as well) on the other hand.


Paraconsistent Logic Vague Predicate Quantifier Talk Modest Proposal Arithmetical Truth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Altmann, S. L. (1992), Icons and Symmetries. Oxford: Clarendon Press.Google Scholar
  2. Bell, J. L. (1986), From Absolute to Local Mathematics. Synthese 69, 409–426.CrossRefGoogle Scholar
  3. Bell, J. L. (1988), Infinitesimals. Synthese 75, 285–315.CrossRefGoogle Scholar
  4. Bell, J. L. (1995), Infinitesimals and the Continuum. The Mathematical Intelligencer 17, 55–57.CrossRefGoogle Scholar
  5. Bloor, D. (1976), Knowledge and Social Imagery. London: RKP. Second edition: Chicago: University of Chicago Press, 1991.Google Scholar
  6. Boyer, C. B. (1959), The History of the Calculus and its Conceptual Development. New York: Dover Books.Google Scholar
  7. Dauben, J. W. (1979), Georg Cantor. His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press.Google Scholar
  8. Dauben, J. W. (1995), Abraham Robinson. The Creation of Nonstandard Analysis. A Personal and Mathematical Odyssey. Princeton: Princeton University Press.Google Scholar
  9. Gillies, D. (1992), (ed.), Revolutions in Mathematics. Oxford: Clarendon Press.Google Scholar
  10. Grattan-Guinness, I. (1992), (ed.), Encyclopedia of the History and Philosophy of the Mathematical Sciences. London: Routledge.Google Scholar
  11. Grattan-Guinness, I. (1997), The Fontana History of the Mathematical Sciences. London: Fontana Press.Google Scholar
  12. Henle, J. M. (1999), Non-nonstandard Analysis: Real Infinitesimals. The Mathematical Intelligencer 21, 67–73.CrossRefGoogle Scholar
  13. Hersh, R. (1997), What is Mathematics, Really? London: Jonathan Cape.Google Scholar
  14. Ishiguro, H. (1990), Leibniz’s Philosophy ofLogic and Language. Cambridge: Cambridge University Press.Google Scholar
  15. Knuth, D. (1998), Teach Calculus with Big O (Letter to the Editor), Notices of the AMS 45, 687–688.Google Scholar
  16. Lakatos, I. (1976), Proofs and Refutations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. MacLane, S. (1986), Mathematics: Form and Function. Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  18. McLarty, C. (1992), Elementary Categories, Elementary Toposes. Oxford: Oxford University Press.Google Scholar
  19. Moerdijk, I. and G. E. Reyes (1991), Models for Smooth Infinitesimal Analysis. Heidelberg: SpringerVerlag.Google Scholar
  20. Naisse, J. P. (1992), L’approximation analytique. Vers une théorie empirique, constructive et finie. Brussels: Editions de l’Université de Bruxelles.Google Scholar
  21. Newton, I. (1962), (Motte’s Translation. Revised by Cajori), Principia. Vol. I: The Motion of Bodies. Vol. II: The System of the World. Berkeley: University of California Press.Google Scholar
  22. Priest, G., R. Routley and J. Norman (eds.) (1989), Paraconsistent Logic. Essays on the Inconsistent. München: Philosophia Verlag.Google Scholar
  23. Restivo, S., J. P. Van Bendegem and R. Fischer (eds.), (1993), Math Worlds: New Directions in the Social Studies and Philosophy of Mathematics. New York: State University New York Press.Google Scholar
  24. Van Bendegem, J. P. (1993), Real-Life Mathematics versus Ideal Mathematics: The Ugly Truth. In Empirical Logic and Public Debate. Essays in Honour of Else M. Barth. E. C. W. Krabbe, R. J. Dalitz and P. A. Smit (eds.), 1993, Amsterdam: Rodopi, pp. 263–272.Google Scholar
  25. Van Bendegem, J. P. (1999), The Creative Growth of Mathematics. Philosophica 63, 119–152.Google Scholar
  26. Van Bendegem, J. P. (2000), Alternative Mathematics: The Vague Way. In Festschrift in Honour of Newton C. A. da Costa on the Occasion of his Seventieth Birthday. D. Krause, S. French and F. A. Doria (eds.), Synthese 125, pp. 19–31.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  1. 1.Department of PhilosophyFree University of BrusselsBelgium

Personalised recommendations