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Magicicada, Mathematical Explanation and Mathematical Realism

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Abstract

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

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Notes

  1. For example, Malament (1985) expresses skepticism about the possibility of extending Field’s results to quantum mechanics, while Balaguer (1996) and Balaguer (1998) sketch a way of doing it.

  2. This possibility for the realist to pursue this strategy is already outlined in Field (1989, p. 14–20).

  3. That explanatory power warrants ontological commitment is assumed by Baker for the sake of argument. I intend to show in the next sections that this is not the case.

  4. In what follows, lcm and gcd are, respectively, the least common multiple and greatest common divisor of X, Y.

  5. A justification for the choice of bounds can be found in Goles et al. (2001, p. 35).

  6. Goles et al. assume that a mutant predator will supersede an existing predator only if it is fitter.

  7. According to Baker ‘[…] number theory deals with intrinsic mathematical properties of the natural number’ (Baker 2005, p. 236), so the use of number-theoretical facts within an explanation involves references to the natural numbers as abstract mathematical entities.

  8. This amounts to saying that the explanandum is satisfied by mathematical objects. For details, see Bangu (2008, p. 18).

  9. However it is not conceded that the mathematical references in the explanandum should be interpreted realistically. The explanandum refers to a property of the length of Magicicada’s life-cycles measured in years, not to a purely number-theoretical property. It will be seen below that this property can in fact be formulated non-numerically and represented numerically by primeness.

  10. ‘I do not see how one can coherently deny that mathematical objects play a part in the explanation’ (Baker 2005, p. 234).

  11. Note that one could dismiss a geometrical language and describe congruence, divisibility and juxtaposition as empirical relations between time-intervals.

  12. A ‘meaningful’ use of numbers is one that can be explicitly related to some appropriate empirical content. Meaningfulness is required if numerical results are to be applied to investigate the structure of empirical phenomena. I am not assuming that meaningfulness is only established by proving a representation theorem, although this is what happens in my discussion of the Magicicada case.

  13. The relation D can be seen as an ordering of intervals whose endpoints are in A.

  14. For a proof, see Suppes (1972, pp. 56–58).

  15. This operation is not supposed to correspond to arithmetical multiplication in a numerical structure like N, isomorphic to the given non-numerical structure.

  16. The bounds on life-cycles and their lengths are then expressed using the endpoints of the relevant intervals.

  17. This is confirmed by the fact that Goles et al. (2001) go on to generalize their prey-predator model so that ‘prime numbers are selected by competition between neighboring residents in a spatially extended system’ (Goles et al. 2001, p. 35).

  18. Otherwise gcd(jY, Y) would decrease if Y changed to Y–1, in which case the prey, whose life-cycle is Y–1, would be fitter than the prey with life-cycle Y. But if Y is prime, Y–1 is an even number, so primality is no longer related to fitness maximization.

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Acknowledgments

I wish to thank Alan Baker, Sorin Bangu, Mary Leng and David Liggins for helpful comments on previous drafts of this paper. Its clarity and structure have also benefited from valuable suggestions and remarks from two anonymous referees.

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  1. University of East Anglia, School of Philosophy, Norwich, Norfolk, NR4 7TJ, UK

    Davide Rizza

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Rizza, D. Magicicada, Mathematical Explanation and Mathematical Realism. Erkenn 74, 101–114 (2011). https://doi.org/10.1007/s10670-010-9261-z

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