Experimental Mathematics

Abstract

The rise of the field of “experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization.

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Notes

  1. 1.

    E.g., The Journal of Experimental Mathematics (established 1992).

  2. 2.

    E.g., the Institute for Experimental Mathematics at the University of Essen.

  3. 3.

    E.g., the Experimental Mathematics Colloquium at Rutgers University.

  4. 4.

    E.g., Borwein and Bailey (2004a, b).

  5. 5.

    Van Bendegem (1998, p. 172).

  6. 6.

    A good source for more details on Plateau’s experiments is Courant and Robbins (1941, pp. 386–397).

  7. 7.

    Borwein and Bailey (2004b, Chap. 4).

  8. 8.

    http://www.expmath.org/expmath/philosophy.html.

  9. 9.

    Gallian and Pearson (2007, p. 14).

  10. 10.

    Note that this objection is independent of the issue of whether the boundary between experimental and non-experimental mathematics is vague.

  11. 11.

    Franklin (1987, p. 1).

  12. 12.

    For more on issues concerning the role and status of enumerative induction in mathematics, see Baker (2007).

  13. 13.

    Mathematician George Andrews has described a computer as “a pencil with power-steering.”

  14. 14.

    See Appel and Haken (1978) for more details of their proof.

  15. 15.

    Part of the ambiguity here between type and token stems from the fact that it is only the topological features of maps which matter for the purposes of the Four-Color Conjecture. Hence there are many features of individual maps which are irrelevant, so one such map can ‘stand in’ for a whole subclass.

  16. 16.

    See, e.g., Tymoczko (1979).

  17. 17.

    One exception is work by Mark Steiner in the late 1970s. See e.g., Steiner (1978).

  18. 18.

    See Mancosu (2001) for a useful overview of contemporary work on mathematical explanation.

  19. 19.

    For more on this debate, see Baker (2005).

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Correspondence to Alan Baker.

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Baker, A. Experimental Mathematics. Erkenn 68, 331–344 (2008). https://doi.org/10.1007/s10670-008-9109-y

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Keywords

  • General Hypothesis
  • Empirical Science
  • Experimental Mathematic
  • Indispensability Argument
  • Deductive Proof