I have approached the question of why population genetics is a probabilistic theory by considering the advantages of the classical approach over an alternative maximally deterministic one. The advantages I detect for the standard treatment provide a partial vindication of two of the main accounts of the motivation for introducing probability into population genetics: Probabilities are introduced into classical population genetics because of epistemic barriers that prevent the use of a deterministic alternative, and also because they give generality to the equations that makes possible their recursive deployment.
Appendix
To see how NINPICs are modeled differently in the classical and maximally deterministic formalisms, that is, to set up a translation scheme between the formalisms, I consider how an arbitrary NINPIC is modeled in each to show how to move back and forth from one
formalism to the other. I consider a simple population whose effective population size is completely determined by its census size (Nc) and its
variance in offspring number (Vk). That is, the population lacks hierarchical structure, inter-generational variations in census size, inbreeding, biased sex ratio, and other demographic features that alter
Nev (for the impact on Nev of these demographic variables, see Nunney (1999). Suppose that in generation z, the population is beset by lethal nest predators, a NINPIC that both doubles Vk and halves Nc. Quantifying the influence of the nest predators in (3) is a matter of setting appropriate values in the first term. Suppose in a population of two competing haplotypes, the nest predators destroy 1/4 of the
haplotype i = 1 and 1/3 of i = 2 in generation z. In (3), the nest
predators are quantified by means of a cause g = w with two values, x
for predated nestlings and y for ignored ones, such that these terms are introduced into equation (3):
α1wx z =0.25, α1wy z =0. 75, α2wx z =0.3˙, α2wy z =0.6˙, w1wxz =0, w1zwy=1, w2zwx=0, w2zwy=1
Of course, the nest predators will not predate exactly the same
fractions of each haplotype in each generation. The values for α1z+1wxand
α2z+1wxwill almost certainly be different from the values for α1wxz and α2wxz . Modeling the dynamics of the population across these two generations would require knowing the new fractions of each haplotype taking each value for nest predation in generation z + 1, and so on for further generations.
Quantifying the influence of the nest predators by means of (2) is a matter of appropriately altering the values of the arguments
determining Nev. In the population at hand, Nev=(4Nc−2) /(Vk+2) (Gildenhuys 2009). If, absent the nest predators, Vk = A and Nc = B, then quantifying the influence of the nest predators in (2) is a matter of letting Vk = 2A and letting Nc = ½B. Note that this value for Nev remains unchanged across generations.
Works Cited
Abrams, M. (2007). "Fitness and Propensity's Annulment." Biology and Philosophy
22: 115-130.
Abrams, M. (2007). "How do Natural Selection and Random Drift Interact?" Philosophy of Science 74: 666-679.
Ariew, A. (1998). "Are Probabilities Necessary for Evolutionary Explanations?" Biology and Philosophy 13: 245-253.
Beatty, J. (1984). "Chance and Natural Selection." Philosophy of Science 51: 183- 211.
Bouchard, F. and A. Rosenberg (2004). "Fitness, Probability and the Principles of Natural Selection." British Journal for the Philosophy of Science 55: 693-712. Brandon, R. (2005). "The Difference between Drift and Selection: A Reply to Millstein." Biology and Philosophy 20(1): 153-170.
Brandon, R. N. and S. Carson (1996). "The Indeterministic Character of Evolutionary Theory: No "No Hidden Variables Proof" but No Room for Determinism Either." Philosophy of Science 63(3): 315-337.
Crow, J. F. and N. E. Morton (1955). "Measurement of Gene Frequency Drift in Small Populations." Evolution 9: 202-214.
Frankham, R. (1995). "Effective population size/adult population size ratios in wildlife: a review." Genetical Research 66: 95-107.
Giere, R. N. (1973). Objective Single-Case Probabilities and the Foundations of Statistics. Logic, Methodology, and the Philosophy of Science. P. Suppes, L. Henkin, G. Soisil and A. Joja. North-Holland, Elsevier. IV.
Gildenhuys, P. (2009). "An Explication of the Causal Dimension of Drift." British Journal for the Philosophy of Science 60(3): 521-555.
Gildenhuys, P. (2011). "Righteous modeling: the competence of classical population genetics." Biology and Philosophy 26: 813-835.
Glymour, B. (2001). "Selection, Indeterminism, and Evolutionary Theory." Philosophy of Science 68: 518-535.
Glymour, B. (2006). "Wayward Modeling: Population Genetics and Natural Selection." Philosophy of Science 73: 369-389.
Glymour, B. (2013). "The wrong equations: a reply to Gildenhuys." Biology and Philosophy 28: 675-681.
Graves, L., B. L. Horan and A. Rosenberg (1999). "Is Indeterminism the Source of the Statistical Character of Evolutionary Theory?" Philosophy of Science 66(1): 140- 157.
Haldane, J. B. S. and S. D. Jayakar (1963). "Polymorphism due to selection of varying direction." Journal of Genetics 58: 237-242.
Hedgecock, D. (1994). Does variance in reproductive success limit effective
population sizes of marine organisms? Genetics and Evolution of Aquatic Organisms. A. R. Beaumont. London, Chapman & Hall: 122-134.
Hedrick, P. W. (2005). Genetics of Populations. Boston, Jones and Bartlett.
Hedrick, P. W. (2005). "Large Variance in Reproductive Success and the Ne/N Ratio." Evolution 59(7): 1596-1599.
Hedrick, P. W. (2010). Genetics of Populations. Sudbury, MA, Jones and Bartlett. Horan, B. L. (1994). "The Statistical Character of Evolutionary Theory." Philosophy of Science 61(1): 76-95.
Kerr, B. and P. Godfrey-Smith (2002). "Individualist and Multi-level Perspectives on Selection in Structured Populations." Biology and Philosophy 17(4): 477-517. Kirkpatrick, M., T. Johnson and N. H. Barton (2002). "General Models of Multilocus Evolution." Genetics 161: 1727-1750.
Laland, K. N., F. J. Odling-Smee and M. W. Feldman (1999). "Evolutionary Consequences of Niche Construction and Their Implications for Ecology."
Proceedings of the National Academy of Sciences of the United States of America
96(18): 10242-10247.
Laplace, P. S. (1814[1951]). A Philosophical ESsay on Probabilities. New York, Dover Publications.
Lyon, A. (2011). "Deterministic Probability: neither chance nor credence." Synthese
Millstein, R. (1996). "Random Drift and the Omniscient Viewpoint." Philosophy of Science 63: S10-S18.
Millstein, R. (2003). "Interpretations of Probability in Evolutionary Theory." Philosophy of Science 70: 1317-1328.
Millstein, R. L. (2002). "Are Random Drift and Natural Selection Conceptually Distinct?" Biology and Philosophy 17(1): 33-53.
Millstein, R. L. (2003). How not to argue for the indeterminism of evolution: A look at two recent attempts to settle the issue. Determinism in Physics and Biology. A. Hutterman. Paderborn, Germany, Mentis: 91-107.
Nunney, L. (1996). "The Influence of Variation in Female Fecundity on Effective Population Size." Biological Journal of the Linnean Society 59: 411-425.
Nunney, L. (1999). "The Effective Size of a Hierarchically Structured Population." Evolution 53: 1-10.
Ramsey, G. (2013). "Driftability." Synthese.
Rice, S. (2004). Evolutionary Theory: Mathematical and Conceptual Foundations. Sunderland, MA, Sinauer and Associates.
Rosenberg, A. (1994). Instrumental Biology, or The Disunity of Science. Chicago, University of Chicago Press.
Rosenberg, A. (2001). "Discussion Note: Indeterminism, Probability, and Randomness in Evolutionary Theory." Philosophy of Science 68: 536-544.
Rosenberg, A. (2001). "Indeterminism, Probability and Randomness in Evolutionary Theory." Philosophy of Science 63: 536-544.
Sansom, R. (2003). "Why Evolution is Really Indeterministic." Synthese 136: 263- 279.
Sober, E. (1984). The Nature Of Selection: Evolutionary Theory In Philosophical Focus. Cambridge, Mass., MIT Press.
Sober, E. (2010). Evolutionary Theory and the Reality of Macro Probabilities. Probability in Science. E. Eells and J. Fetzer. Boston, Springer: 133-162.
Stamos, D. (2001). "Quantum Indeterminism and Evolutionary Biology." Philosophy of Science 68: 164-184.
Walsh, D. M., T. Lewens and A. Ariew (2002). "The Trials of Life: Natural Selection and Random Drift." Philosophy of Science 69(3): 452-473.
Weber, M. (2001). "Determinism, Realism, and Probability in Evolutionary Theory." Philosophy of Science 68: S213-S224.
Weber, M. (2005). Darwinism as a theory for finite beings. Darwinism and
Philosophy. V. Hosle and C. Illies. Notre Dame, University of Notre Dame Press: 275- 297.
Wright, S. (1938). "Size of population and breeding structure in relation to evolution." Science 87: 430-431.