## 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.
E.g.,

*The Journal of Experimental Mathematics*(established 1992). - 2.
E.g., the Institute for Experimental Mathematics at the University of Essen.

- 3.
E.g., the Experimental Mathematics Colloquium at Rutgers University.

- 4.
- 5.
Van Bendegem (1998, p. 172).

- 6.
A good source for more details on Plateau’s experiments is Courant and Robbins (1941, pp. 386–397).

- 7.
Borwein and Bailey (2004b, Chap. 4).

- 8.
- 9.
Gallian and Pearson (2007, p. 14).

- 10.
Note that this objection is independent of the issue of whether the boundary between experimental and non-experimental mathematics is vague.

- 11.
Franklin (1987, p. 1).

- 12.
For more on issues concerning the role and status of enumerative induction in mathematics, see Baker (2007).

- 13.
Mathematician George Andrews has described a computer as “a pencil with power-steering.”

- 14.
See Appel and Haken (1978) for more details of their proof.

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

- 17.
One exception is work by Mark Steiner in the late 1970s. See e.g., Steiner (1978).

- 18.
See Mancosu (2001) for a useful overview of contemporary work on mathematical explanation.

- 19.
For more on this debate, see Baker (2005).

## References

Appel, K., & Haken, W. (1978). The four-color problem. In L. Steen (Ed.),

*Mathematics today: Twelve informal essays*(pp. 153–180). New York: Springer-Verlag.Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena?

*Mind, 114*, 223–238.Baker, A. (2007). Is there a problem of induction for mathematics? In M. Potter (Ed.),

*Mathematical knowledge*(pp. 59–73). Oxford: Oxford University Press.Borwein, J., & Bailey, D. (2004a).

*Experimentation in mathematics: Computational paths to discovery*. Natick, MA: AK Peters.Borwein, J., & Bailey, D. (2004b).

*Mathematics by experiment: Plausible reasoning for the 21st century*. Natick, MA: AK Peters.Courant, R., & Robbins, H. (1941).

*What is mathematics?*Oxford: Oxford University Press.Franklin, J. (1987). Non-deductive logic in mathematics.

*British Journal for the Philosophy of Science, 38*, 1–18.Gallian, J., & Pearson, M. (2007). An interview with Doron Zeilberger.

*MAA Focus, May/June*, 14–17.Mancosu, P. (2001). Mathematical explanation: Problems and prospects.

*Topoi, 20*, 97–117.Steiner, M. (1978). Mathematical explanation.

*Philosophical Studies, 34*, 135–151.Tymoczko, T. (1979). The four-color problem and its philosophical significance.

*Journal of Philosophy, 76*, 57–83.van Bendegem, J.-P. (1998). What, if anything, is an experiment in mathematics? In D. Anapolitanos, A. Baltas, & S. Tsinorema (Eds.),

*Philosophy and the many faces of science*(pp. 172–182). London: Rowman & Littlefield.

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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