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Mathematical Experiments on Paper and Computer

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

We propose a characterization of mathematical experiments in terms of a setup, a process with an outcome, and an interpretation. Using a broad notion of process, this allows us to consider arithmetic calculations and geometric constructions as components of mathematical experiments. Moreover, we argue that mathematical experiments should be considered within a broader context of an experimental research project. Finally, we present a particular case study of the genesis of a geometric construction to illustrate the experimental use of hand drawings and computers in mathematical practice.

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Notes

  1. 1.

    As examples for the rich taxonomies of scientific experimentation developed by philosophers of science, see Hacking (1988, 508) and Galison (1988, 525), in which many different relations between experimentation, theory, and instrumentation are discussed. For a more recent overview, see Feest and Steinle (2016). In light of the turn in philosophy of science towards research processes and knowledge generation, one can see the present chapter as a contribution towards bringing together analyses of scientific and mathematical experimental practices.

  2. 2.

    In philosophy of science, we find this distinction under the labels of “theory-driven” and “exploratory” experiments (Steinle 1997, S70); for a discussion of the epistemological significance of exploratory experiments, see Steinle (1997, S72).

  3. 3.

    If the experiment is carried out by hand or as a thought experiment, the experimenter is directly involved in the process, but this should not affect the outcome.

  4. 4.

    Notice that a computation, understood as a process, is only a component of a mathematical experiment; this might answer in part Van Bendegem’s worry that “it seems that it is very difficult to make any sense out of the idea of considering a computation (whether aiming for a numerical result or a visual image) as a form of mathematical experiment” (Van Bendegem 1998, 178).

  5. 5.

    See Starikova and Giaquinto (2018) and Starikova (2023) for a discussion of mathematical thought experiments.

  6. 6.

    See Morgan’s discussion of different “materialities” of experiments in Morgan (2003): material, semi-material, nonmaterial, and mathematical; for a discussion of computer simulations, see Barberousse et al. (2009).

  7. 7.

    Van Bendegem considers such “real-world-experiments” but concludes that “they are quite uninteresting” for mathematics, “rather trivial,” and that “no serious implications should be drawn from them” (Van Bendegem 1998, 179).

  8. 8.

    This view about the context-dependency of experiments is not shared by everyone; Van Bendegem, for example, considers it “slightly, if not simply, odd” (Van Bendegem 1998, 175).

  9. 9.

    See Ferreirós (2015) and Wagner (2017) for general accounts of mathematical practices.

  10. 10.

    In the case of investigations directed at the solution of specific questions or problems, Detlefsen and Arana speak of “directed investigations” (Detlefsen and Arana 2011, 9); see also the notion of a “trial” in a mathematical thought experiment (Starikova and Giaquinto 2018, 263).

  11. 11.

    The iterative nature of experimental research in science has been emphasized by Chang (2004); see also Elliott (2012) for a discussion.

  12. 12.

    These passages are also referred to in Lakatos (1976, 7).

  13. 13.

    Similarly, Dedekind’s contemporary Cantor also speaks of his “experiments [Versuche] which have lasted many years” (Cantor 1883, quoted from the translation in Ewald 1996, 890).

  14. 14.

    See, e.g., Giardino (2018), Carter (2019), and Vold and Schlimm (2020). Because of the predominance of presenting mathematical reasoning in terms of external representations, some philosophers, such as Peirce and Wittgenstein, have claimed that all mathematical reasoning is ultimately based on observations.

  15. 15.

    Heath warns against interpreting the object of the postulates purely as limiting possible constructions (Heath 1968, 124).

  16. 16.

    The background of the development of this result is presented in the form of an interview in Fernández González and Schlimm (2023). Some of the reproductions from Juan’s notebooks shown in that article are also used below with permission from the authors.

  17. 17.

    As reported by Martin Gardner (1983, 82–83), Stanislaw Ulam’s discovery of particular patterns in the distribution of prime numbers (Stein et al. 1964) began in a similar, absent-minded kind of way: During a “long and very boring” conference talk, Ulam “doodled a grid of horizontal and vertical lines on a sheet of paper,” then numbered them in a specific way, circled the prime numbers, and “to his surprise the primes seemed to have an uncanny tendency to crowd into straight lines.” We are thankful to David Waszek for pointing us to this example.

  18. 18.

    Gilbert Labelle, emeritus professor at the Department of Mathematics at the Université du Québec à Montréal, helped Juan with the writing of the published article.

  19. 19.

    These results are presented in Fernández González (2016).

  20. 20.

    For more information about this freeware software package, see https://www.geogebra.org

  21. 21.

    See Starikova and Giaquinto (2018, 273–274) for a discussion of the importance of zooming in and out in mathematical research.

  22. 22.

    This last remark echoes a sentiment about mathematical proofs expressed in Hume’s Treatise (Norton and Norton 2007, 121).

  23. 23.

    This is in line with the application of Lakatos’ account of scientific research programs to mathematics, as discussed in Hallett (1979).

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Acknowledgments

The authors would like to thank Jessica Carter, Valeria Giardino, Fenner Tanswell, and David Waszek for comments on a previous version of this chapter. This work was supported in part by the Social Sciences and Humanities Research Council of Canada.

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Authors and Affiliations

  1. Department of Philosophy, McGill University, Montreal, QC, Canada

    Dirk Schlimm

  2. Graduate School of Design, Harvard University, Cambridge, MA, USA

    Juan Fernández González

Authors
  1. Dirk Schlimm
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  2. Juan Fernández González
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Correspondence to Dirk Schlimm .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. No affiliation provided

    Jessica Carter

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Schlimm, D., Fernández González, J. (2024). Mathematical Experiments on Paper and Computer. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-031-40846-5_112

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