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Optimisation and mathematical explanation: doing the Lévy Walk

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Abstract

The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra-mathematical explanation (the explanation of physical facts by mathematical facts). In this paper, I identify a new case of extra-mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra-mathematical explanation in science.

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Notes

  1. This is the version offered by Quine (1948, 1951, 1960) and Putnam (1979a, b).

  2. Following Colyvan (2001, p. 77) an entity \(E\) is dispensable to a scientific theory \(T\) iff:

    1. (1)

      There exists a modification of \(T\) resulting in a second theory \(T^{*}\) with exactly the same observational consequences as \(T\) in which \(E\) is neither mentioned nor predicted.

    2. (2)

      \(T^{*}\) is preferable to \(T\).

    And thus \(E\) is indispensable to \(T\) when either condition (1) or (2) fails to hold.

  3. For further discussion of this argument see Colyvan (2001), Field (1989), Maddy (1992), Sober (1993) and Melia (2000).

  4. I.e. the strategy Field (1980) pursues.

  5. See e.g. Azzouni (2004), Melia (2000, 2002), Leng (2010) and Yablo (2009).

  6. Following Baker (2012, p. 9) the Honeycomb Theorem can be stated as follows:

    • (Honeycomb Theorem) Any Partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

    The honeycomb theorem was proved by Hales (2001).

  7. See e.g. Daly and Langford (2009), Bangu (2008) and Rizza (2011). For responses, see Baker (2009).

  8. Named for the mathematician Paul Lévy (Mandelbrot 1983).

  9. I am grateful to an anonymous referee for raising this issue.

  10. My thanks go to an anonymous referee for pressing me on this point.

  11. I am grateful to an anonymous referee for this suggestion.

  12. On Baker’s (Baker 2009) view, E3 is treated as a tentative hypothesis. As a tentative hypothesis, E3 is not assumed to be true but, rather, is confirmed by the number-theoretic explanation that explains E4. The charge of question-begging is thus avoided because (i) E4 is nominalistically acceptable and (ii) although E3 is not, the truth of that claim is never assumed. In offering this response to Bangu, however, Baker claims that E3 is part of the explanandum in the cicada case, even though that claim is a tentative hypothesis. This is troubling: a hypothesis is usually an explanation for some physical phenomenon. It is not typically an explanandum. So if E3 is part of the explanandum, it’s not clearly a hypothesis. Because E3’s status as a hypothesis is crucial to Baker’s response, it would be better if E3 were absorbed into the explanans. Perhaps this is what Baker had in mind. If so, then the response to Bangu tabled here is a version of Baker’s.

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Acknowledgments

The author gratefully acknowledges the assistance of David Braddon-Mitchell, Mark Colyvan, Arnon Levy, Maureen O’Malley and three anonymous referees of this journal. The author is also indebted to audiences at the 2012 Australasian Association of Philosophy Conference where he received many constructive comments on an earlier draft of this paper. This research was supported under the Australian Research Council’s Discovery Projects funding scheme (project number DP120102871) and by a John Templeton Foundation grant held by Huw Price, Alex Holcombe, Kristie Miller, and Dean Rickles, entitled: New Agendas for the Study of Time: Connecting the Disciplines.

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  1. School of Philosophical and Historical Inquiry, University of Sydney, Quadrangle A14, Sydney, NSW, 2006, Australia

    Sam Baron

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Baron, S. Optimisation and mathematical explanation: doing the Lévy Walk. Synthese 191, 459–479 (2014). https://doi.org/10.1007/s11229-013-0284-2

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