## Abstract

Statistical reasoning is an integral part of modern scientific practice. In *The Seven Pillars of Statistical Wisdom* Stephen Stigler presents seven core ideas, or pillars, of statistical thinking and the historical developments of each of these pillars, many of which were concurrent with developments in biology. Here we focus on Stigler’s fifth pillar, regression, and his discussion of how regression to the mean came to be thought of as a solution to a challenge for the theory of natural selection. Stigler argues that the purely mathematical phenomenon of regression to the mean provides a resolution to a problem for Darwin’s evolutionary theory. Thus, he argues that the resolution to the problem for Darwin’s theory is purely mathematical, rather than causal. We show why this argument is problematic.

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## Notes

An important debate concerning the relation between mathematical and causal explanations in the philosophy of biology literature is the debate over whether explanations of population change that refer to natural selection and random drift are causal or mathematical explanations (e.g. Walsh et al. 2002; Shapiro and Sober 2007). We will not discuss how our analysis of Stigler's views relates to this debate.

An alternative interpretation of this chapter is that Stigler describes two distinct problems: "Darwin's problem" being the maintenance of sufficient variation in natural populations and "Galton's version of Darwin's problem" being the problem of the apparent conflict between intergenerational variability and stable variation. We thank an anonymous referee for this suggestion. We prefer our reading of the chapter and note that in the section "The Solution to Darwin's Problem", Stigler describes only one problem, which is the problem we focus on here. The alternative interpretation does not fundamentally affect our argument, which is concerned with Galton's solution for the problem of the apparent conflict between intergenerational variability and stable variation in natural populations.

It is important to note that this is Stigler's argument, not necessarily Galton's. We are not providing an interpretation of Galton's work in this paper, but rather analyze Stigler's interpretation in his analysis of Galton's work.

From this point on, whenever we speak of mathematical explanations we mean pure mathematical explanations. That is, explanations that are mathematical rather than causal.

These authors take explanations for population changes that appeal to natural selection or drift to be mathematical and not causal because these changes "can be explained by referring to the deductive consequences of statistical models, independent of considerations of causation" (Ariew et al. 2015, p. 636). They note that to be explanatory, it is not sufficient that the explanandum will simply be deduced from the mathematical facts. The mathematical facts need to also provide counterfactual information, by telling us "how things would have been different in various counterfactual situations" (Ariew et al. 2015, p. 655).

The argument made here by Stigler, namely that regression to the mean solved Darwin's problem, did not appear in his earlier works on Galton and Darwin's problem (e.g. Stigler 2010). We thank an anonymous referee for pointing this out.

This line of thought is compatible with recent work by Andersen (forthcoming) who suggests that mathematical and causal descriptions should be thought of as complementary explanations of phenomena.

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## Acknowledgements

Adam Krashniak’s work was kindly supported by The Interuniversitry Ph.D. Program in the History and Philosophy of the Life Sciences, supported by the Humanities Fund of the Israeli Council of Higher Education. The work of Ehud Lamm is supported by Israeli Science Foundation Grant 1128/15.

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Krashniak, A., Lamm, E. Was regression to the mean really the solution to Darwin’s problem with heredity?.
*Biol Philos* **32**, 749–758 (2017). https://doi.org/10.1007/s10539-017-9582-2

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DOI: https://doi.org/10.1007/s10539-017-9582-2