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One equation to rule them all: a philosophical analysis of the Price equation

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Abstract

This paper provides a philosophical analysis of the Price equation and its role in evolutionary theory. Traditional models in population genetics postulate simplifying assumptions in order to make the models mathematically tractable. On the contrary, the Price equation implies a very specific way of theorizing, starting with assumptions that we think are true and then deriving from them the mathematical rules of the system. I argue that the Price equation is a generalization-sketch, whose main purpose is to provide a unifying framework for researchers, helping them to develop specific models. The Price equation plays this role because, like other scientific principles, shows features as abstractness, unification and invariance. By underwriting this special role for the Price equation some recent disputes about it could be diverted.

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Notes

  1. Kuhn also called “law-sketch” (Kuhn 2000). Structuralists (Diez and Lorenzano 2015) use the term “guiding principle”.

  2. The allele and genotype frequencies must add to 1, respectively: p + q = 1, and p 2 + 2pq + q 2 = 1.

  3. The model makes the subsequent assumptions: there are non-overlapping generations; the population size is constant; there is no selection, mutation or migration; adults make an infinite number of gametes and every parent contributes equally to the gamete pool; all members breed; all members mate randomly.

  4. There are ways to separate the actions of natural selection and drift. Averaging over uncertainty eliminates drift, so the action of natural selection can be taken to be the expectation of this covariance (Gardner and Grafen 2009; Gardner 2015). Okasha (2006, pp. 32–33) also argues that if we separate the realized fitness w i in two parts—the expected fitness \(w_{i}^{*}\) and its deviation δ i —we can add it in the Price equation, assuming that there is no transmission bias for simplicity, as \(\bar{w}\Delta\bar{z} = Cov(w^{*},\,z^{{\prime }} ) + Cov(\delta ,\,z^{{\prime }} )\), being the first part of the right side of the equation the change due to selection and the second part the change due to drift.

  5. Notice that the Price equation is a categorization of the ancestors, connecting all the descendants to their ancestors through this categorization, i.e. “The focus is entirely on the categories of ancestors, not on which categories the descendants are in” (Walsh and Lynch 2013, p. 123).

  6. Here I follow Walsh and Lynch (2013). For another source, see Michod (1999, p. 57).

  7. Nevertheless, the exact version of Fisher’s Fundamental Theorem only applies to the partial evolutionary response caused by natural selection, \(\partial R_{\omega } = \sigma^{2} (A_{\omega } )\), (Walsh and Lynch 2013). See Frank (2012a), and Walsh and Lynch (2013), for detailed derivations of Robertson and Fisher’s theorems, and the breeder’s equation.

  8. The most relevant bibliography is reviewed but not intended to exhaust it.

  9. Other works are: Gardner et al. (2007), relating multilocus population genetics and social evolution); Barfield et al. (2011), Coulson and Tuljapurkar (2008) extending the Price equation for stage- and age-structured; and Gardner (2015), Grafen (2015), Taylor (2009), Rebke (2012), for study populations composition (class-structured populations, decomposition, etc.) expressed with the Price equation.

  10. I am grateful to Jun Otsuka for drawing my attention to this objection.

  11. This is not entirely accurate. There are trajectories that Newton’s second law does not aim to apply to; “for instance the movement of a pen in somebody’s hand at will” (Diez and Lorenzano 2015, p. 802). Likewise, there are biological problems that the Price equation may not be applied.

  12. Van Veelen (2005) argues that the covariance term in the Price equation is not a real covariance because there is no sample measure (i.e. sample statistics). Nevertheless, as Frank (2012a) has stressed, Price (1972) was considering the total population and not a sample population (i.e. the covariance it is not an estimate but a mathematical function (Rice 2004), so there is no statistical corrections associated with sample statistics (Rice and Papadopoulos 2009). There are different, but legitimate, uses of the term “covariance” (Frank 2012a; Gardner et al 2011).

  13. Hamilton’s rule is an inequality inside kin selection theory. Its aim is to explain the evolution of social behaviour in populations. Hamilton’s rule states that a social behaviour will be favoured by natural selection if and only if \(rb - c > 0\), where r represents the genetic relatedness of the recipient to the actor, b the benefits to the recipient, and c the costs to the actor (Davies et al. 2012). Hamilton derived his rule in two different ways, so there are two possible versions of it. The first version (Hamilton 1964) is characterized by its simplifying assumptions, and as a consequence of these simplifications, the applicability of this version is constrained to very specific cases and cannot handle more complicated ones (for example, when the frequency of cooperators matters). The other version comes from the Price equation (Hamilton 1970; Frank 1998), it is not tied to any simplifying assumption, making it a general statement of social behaviour systems (Birch 2014).

  14. Frank remembers this learning period as follows: “I took up the empirical study of fig wasp sex ratios in 1981. At that time, I also began to study Hamilton’s notes and to learn how to extend Price’s hierarchical multilevel selection analysis to apply to my empirical work” (Frank 2013, p 1174).

  15. Day and Bonduriansky use a first-order approximation for fitness as \(w(g,h;\bar{g},\bar{h})/\bar{w} \approx 1 + \beta_{g} (\bar{g},\bar{h})(g - \bar{g}) + \beta_{h} (\bar{g},\bar{h})(h - \bar{h})\), where \(\beta_{j} (\bar{g},\bar{h}) = (\partial w(\bar{g},\bar{h};\bar{g},h)/\partial j)/\bar{w}\) is the selection gradient on j.

  16. However, as \(\bar{w}\) is a random variable correlated with w, ω does not scale like typical relative fitness.

  17. Oddly, van Veelen et al (2012, p. 73) claim that any theorem is a tautology because they are analytical. However, that is not correct. Mathematical theorems are not tautologies, and not every analytical statement is a theorem or a tautology. Actually, Putnam (1975, chap. 4) argued that not all truths in mathematics are analytical as a result of Gödel’s incompleteness theorems, so there must be synthetic truths in mathematics.

  18. Andy Gardner (personal communication), Samir Okasha (personal communication).

  19. “I’ve indicated how F = ma acquires meaning trough interpretation—that is, additional assumptions about—F” (Wilczek 2005, p. 10).

  20. Actually, equation (p + q)2 = p 2 + 2pq + q 2 is simply a special product, the square of a sum. Why mathematical truths, such as the square of a sum or Price’s theorem, can represent a biological process? This is a metaphysical question. However, it is beyond the scope of this paper to solve this issue. An interesting proposal is French (2014).

  21. We can also derive mathematically, in a Price’s equation way, Newton’s second law as follows: Let the change of quantity of a body be ∆b. This change is equal to an impulse I, where I is equal to a force F multiplied by the change of time ∆t. Therefore, ∆b = I = F · ∆t. We define the change of motion of body b as the product of its mass m and its velocity v, so b = m · v. Now we can substitute, switching the order of the terms, and derive:

    \(F \cdot \Delta t = \Delta b\)

    \(F = \frac{\Delta b}{\Delta t}\)

    \(F = \frac{\Delta (mv)}{\Delta t}\)

    This is actually Newton’s original formulation. It states that the change in motion is proportional to the motive force impressed. If the mass is constant, then ∆v/∆t = a, where a is the acceleration. Therefore we obtain the familiar form, due to Euler, F = ma. Physics textbooks (Corben and Stehle 1994, p. 28; Goldstein et al 2000, pp. 1–2) introduce the second law in a very similar way (except they read it as the rate of the change in motion).

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Acknowledgments

Thanks to Valeriano Iranzo, Silvia Martínez, Jesús Alcolea, Andrés Moya, Manuel Serra, Jun Otsuka, Andy Gardner and an anonymous referee for useful comments on an earlier version of this paper. Special thanks to Andy Gardner and Samir Okasha for clarify me their position on the Price equation via personal communication. I also wish to thank Sean H. Rice for his insightful work and for clarify me his view on the Price equation and axiomatic theories via personal communication. Thanks to Bruce Walsh for sending me his great draft on the Price equation. I am grateful to Vicent Picó for providing me with insightful feedback on Newtonian mechanics and key concepts on physics, and also on previous drafts of this paper. I am also grateful to Jesús Alcolea for enlightening discussions on logic and mathematics.

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  1. Evolutionary Genetics Group, Cavanilles Institute of Biodiversity and Evolutionary Biology, C/Catedrático José Beltrán 2, 46980, Paterna, Valencia, Spain

    Victor J. Luque

  2. Departamento de Lógica y Filosofía de la Ciencia, Facultad de Filosofía y Ciencias de la Educación, Universitat de València, Avda. Blasco Ibáñez, 30, 46010, Valencia, Spain

    Victor J. Luque

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Luque, V.J. One equation to rule them all: a philosophical analysis of the Price equation. Biol Philos 32, 97–125 (2017). https://doi.org/10.1007/s10539-016-9538-y

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