Abstract
Robert Batterman and others have argued that certain idealizing explanations have an asymptotic form: they account for a state of affairs or behavior by showing that it emerges “in the limit”. Asymptotic idealizations are interesting in many ways, but is there anything special about them as idealizations? To understand their role in science, must we augment our philosophical theories of idealization? This paper uses simple examples of asymptotic idealization in population genetics to argue for an affirmative answer and proposes a general schema for asymptotic idealization, drawing on insights from Batterman’s treatment and from John Norton’s subsequent critique.
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Notes
Quoting Batterman (2010, p. 22).
Weisberg (2013) provides a treatment of the question.
Some writers reserve “drift” for the effects of the third class of processes enumerated in the opening paragraph of this section, thus not for the stochastic element of natural selection. My use of the term in the main text to include variability in the effects of selection is the more standard one in population genetics.
Individual offspring are not always represented in the models of population genetics; many models keep track only of the composition of the gene pool as a whole. More comprehensive models keep track of genotypes, and thus in effect of individuals. In what follows it will be convenient to suppose that we are dealing with the more realistic genotype models.
In such a population, an organism passes on all its genes to its offspring, so there is no meiosis-like source of genetic drift. Stochasticity comes in many other ways, however, such as environmental accidents that kill one but not the other of two genetically identical individuals. There is plentiful drift to be found, then, in finite populations of the sort considered here.
I assume that the infinite population contains infinite numbers of both \(A\) and \(B\) variants. If not, then there are well-defined relative frequencies of zero and one but there is no dynamics to model.
For example, Gillespie (2004, p. 1) promises that the contents of his first chapter, which is based entirely on deterministic models, will provide “true insights”. Anyone who has looked knows, however, that scientists very rarely comment, in uncontroversial cases, on which aspects of their theories and models are explanatory. Indeed, the term “explanation” turns up only occasionally in the official scientific literature, although many scientists will be happy to tell you in their popular writing and elsewhere that explaining the world is science’s ultimate aim—as in Weinberg’s (1992, Chap. 2), with its insistent chain of whys.
There is of course something absurd about the numbering scheme. John Roberts suggested to me a less outré, though more complex, way to represent limiting frequencies in an infinite population: imagine the ecosystem as consisting of infinitely many tiles each containing a finite population. If the relative frequencies on each tile are the same, then however the tiles are ordered, the limiting frequency will be the same and equal to the relative frequency. One might feel obliged, however, to model the population dynamics of each tile separately, which would result in each tile having a stochastic dynamics and thus in different tiles exhibiting different relative frequencies at any given time. How to aggregate these relative frequencies to obtain the overall expected frequency? A determinate answer to that question requires an ordering of tiles, just as the original question required an ordering of organisms. It is a delicate matter—and one concerning which population geneticists have no interest whatsoever.
In many systems the population size does not naturally stay fixed. The derived model therefore cannot model such systems, but no matter: that’s not what it’s for.
What counts as an appropriate metric when constructing the extrapolation space? That is determined by the target phenomenon, the behavior that you ultimately wish to predict or explain. A good metric is one that picks out limiting structures such that the target phenomena predicted by the terms in an extrapolation sequence converge on the target phenomenon predicted by the sequence’s limiting structure. What metric to use, then, to evaluate the convergence of the target phenomena? That depends on your theoretical aims—for example, on what you consider to be a prediction that is close to the actual behavior—but appropriate choices are usually quite obvious.
In this regard, the realistic model is already idealized. More realistic models take into account the molecular realization of the genetic material, and replace the infinite alleles idealization with the infinite sites idealization.
References
Abrams, M. (2006). Infinite populations and counterfactual frequencies in evolutionary theory. Studies in History and Philosophy of Biological and Biomedical Sciences, 37, 256–268.
Batterman, R. W. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. Oxford: Oxford University Press.
Batterman, R. W. (2010). On the explanatory role of mathematics in empirical science. British Journal for the Philosophy of Science, 61, 1–25.
Belot, G. (2005). Whose devil? Which details? Philosophy of Science, 72, 128–153.
Edwards, A. W. F. (1977). Foundations of mathematical genetics. Cambridge: Cambridge University Press.
Elgin, C. (2007). Understanding and the facts. Philosophical Studies, 132, 33–42.
Gillespie, J. H. (2004). Population genetics: A concise guide (2nd ed.). Baltimore, MD: Johns Hopkins University Press.
McMullin, E. (1985). Galilean idealization. Studies in History and Philosophy of Science, 16, 247–273.
Norton, J. D. (2012). Approximation and idealization: Why the difference matters. Philosophy of Science, 79, 207–232.
Pincock, C. (2012). Mathematics and scientific representation. Oxford: Oxford University Press.
Potochnik, A. (2017). Idealization and the aims of science. Chicago: University of Chicago Press.
Ramsey, G. (2013). Driftability. Synthese, 190, 3909–3928.
Strevens, M. (2008). Depth: An account of scientific explanation. Cambridge, MA: Harvard University Press.
Strevens, M. (2017). How idealizations provide understanding. In S. R. Grimm, C. Baumberger, & S. Ammon (Eds.), Explaining understanding: New perspectives from epistemology and philosophy of science. New York: Routledge.
Wayne, A. (2012). Emergence and singular limits. Synthese, 184, 341–356.
Weinberg, S. (1992). Dreams of a final theory. New York: Pantheon.
Weisberg, M. (2007). Three kinds of idealization. Journal of Philosophy, 104, 639–659.
Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. Oxford: Oxford University Press.
Acknowledgements
For invaluable discussion and feedback, thanks to Bob Batterman and John Norton, Kate Vredenburgh, the Research Triangle Philosophy of Science Reading Group, the audience at ISHPSSB 2013, and the anonymous reviewers at Synthese.
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Strevens, M. The structure of asymptotic idealization. Synthese 196, 1713–1731 (2019). https://doi.org/10.1007/s11229-017-1646-y
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DOI: https://doi.org/10.1007/s11229-017-1646-y