Abstract
Biology has proved to be a rich source of examples in which mathematics plays a role in explaining some physical phenomena. In this paper, two examples from evolutionary biology, one involving periodical cicadas and one involving bee honeycomb, are examined in detail. I discuss the use of such examples to defend platonism about mathematical objects, and then go on to distinguish several different varieties of mathematical explanation in biology. I also connect these discussions to issues concerning generality in biological explanation, and to the question of how to pick out which mathematical properties are explanatorily relevant.
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Notes
- 1.
Colyvan (2003) gives a good overview of the Indispensability Argument.
- 2.
Field (1980).
- 3.
- 4.
- 5.
For example, that “periodical cicadas are among the most unusual insects in the world” (Yoshimura 1997, p. 112).
- 6.
Goles et al. (2001, p. 33).
- 7.
- 8.
Presuming that period length is a heritable trait, which is a presupposition of both candidate explanations.
- 9.
For proofs of these lemmas, see Landau (1958).
- 10.
Note that we are, by assumption, restricting attention to periodical predators, i.e., predators that have life-cycles that are greater than 1 year. Prime periods remain optimal even if annual predators are included. However, they are no better (or worse) than non-prime periods with respect to annual predators, since the lcm is n in both cases.
- 11.
Clearly a parallel constraint may be formulated for 13-year cicadas, in which the ecosystem limits potential periods to the range from 12 to 15 years.
- 12.
- 13.
- 14.
- 15.
The first discussion of this example in the philosophical literature, as far as I am aware, is in Lyon and Colyvan (2008), although their remarks on it are relatively brief.
- 16.
The constraints here might include conditions such as it being energetically costly to produce the material to build the walls of the cells, that the cells be contiguous, and that the cells be of uniform area.
- 17.
Steiner (1978).
- 18.
Hales (2001). The full proof runs to 18 pages, so this will be of necessity no more than a brief overview.
- 19.
This in known in the mathematics literature as the isoperimetric problem.
- 20.
Since every vertex in a finite graph corresponds to at least three half edges, \( e\ge \left(3/2\right)v \), so (by substitution into Euler’s formula), \( \left(2/3\right)e\hbox{--} e+f=2 \). Hence \( f=\left(1/3\right)e+2 \), from which it follows that \( e<3f \). Since an edge borders two faces, the average number of edges per face cannot be greater than 6.
- 21.
Carroll et al. (2006, p. 1). Note the implication that both simplicity and purity will tend to enhance the explanatoriness of a proof.
- 22.
To be clear, Hales’s proof is fully rigorous. In other words, the ‘mysterious’ coefficient works to establish the theorem that regular hexagons are optimal. What is not clear is why this coefficient works.
- 23.
A distinctive feature of numerical analysis is the use of algorithms, and other methods of numerical approximation. This is in contrast to the symbol manipulation characteristic of purely analytic approaches. Hence there is a greater likelihood of numerical analysis producing methods and results that ‘work’, in some specified domain, but are such that it is not clear why they work.
- 24.
ibid., p. 7.
- 25.
See e.g., Baker (2008).
- 26.
Note that nothing I have said here denies the importance of mathematical explanatoriness to scientific explanatoriness. I am simply arguing that the latter is not a sufficient condition for the former.
- 27.
For further discussion and references, see Pincock (2012, 51–54).
- 28.
Colyvan (2010).
- 29.
See Gould (2002, pp. 648–9).
- 30.
Tóth (1964).
- 31.
- 32.
More controversially, some physicists have argued for optimization explanations in cosmology, in which the values of the basic physical constants are ‘explained’ in terms of producing universes that are more conducive to the formation of black holes, which in turn (according to certain theories) spawn further universes as ‘offspring.’ For more on such explanations, see Smolin (1999).
- 33.
Bangu (2008).
- 34.
This approach is explored further in Baker (2012), where the concept of science-driven mathematical explanation is introduced.
- 35.
West and Brown (2004).
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Baker, A. (2015). Mathematical Explanation in Biology. In: Explanation in Biology. History, Philosophy and Theory of the Life Sciences, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9822-8_10
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