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Scientific Understanding and Mathematical Abstraction

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Abstract

This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue for these claims by tracing attempts to understand the basis of natural selection from its early formulation as a statistical theory to its later development by R.A. Fisher, one of the founders of modern population genetics. Not only did these developments put natural selection of a firm theoretical foundation but its mathematization changed the way it was understood as a biological process. Instead of simply clarifying its status, mathematical techniques were responsible for redefining or reconceptualising selection. As a corollary I show how a highly idealised mathematical law that seemingly fails to describe any concrete system can nevertheless contain a great deal of accurate information that can enhance our understanding far beyond simply predictive capabilities.

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Notes

  1. The law of segregation refers to the fact that the characters that differentiate hybrid forms can be analysed in terms of independent pairs; that is, each Anlagen acts separately – they do not fuse. We can also understand this as stating that any hybrid for a given character produces an offspring distributed according to definite proportions. If the pure parental forms are A and a and the hybrid Aa then the offspring of the hybrid will be distributed according to the ration 1A:2Aa:1a. Pearson (1904) was probably the first to show the relation between the law of segregation and the stability of a population in the absence of selection.

  2. As quoted in Norton and Pearson (1976).

  3. As I have argued elsewhere (Morrison, 2000) the synthesis of Mendelism and Darwinain selection was accomplished through the employment of mathematical techniques that allowed its founders (Fisher, Sewall Wright and J.B.S. Haldane) to establish the operation of selection in Mendelian populations. To that extent the synthesis produced an enhanced understanding of selection as something compatible rather than at odds with Mendelism. But, the interesting aspect of the synthesis was that while its authors, particularly Fisher and Wright, agreed on the general conclusion each had a very different account of how selection functioned and the conditions under which is would be most effective. I have discussed this at length elsewhere (Morrison, 2006) so I won’t rehearse those arguments here, nor do I want to go into detail about the different mathematical approaches of Fisher and Wright and the extent to which those differences influenced their interpretation of selection. Instead my concern here is with the way that the mathematical approach used initially by Fisher was responsible for restructuring our ideas about selection in ways that have informed contemporary population genetics.

  4. It is perhaps important to point out here that while Darwin thought in terms of species defined as collections of individuals he didn’t have a well worked out notion of a population in the statistical sense of the term.

  5. The law of ancestral heredity stated that “the share a man retains in the constitution of his remote descendants is inconceivably small. The father transmits, on average, one-half of his nature, the grandfather one-fourth, the great-grandfather one-eight; the share decreasing step-by-step in a geometrical ratio with great rapidity.”

  6. This new law took the form of a multiple regression equation of offspring on mid-midparental ancestry:

    $$ P_{0} = \raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}{\left( {{\sigma _{0} } \mathord{\left/ {\vphantom {{\sigma _{0} } {\sigma _{1} }}} \right. \kern-\nulldelimiterspace} {\sigma _{1} }P_{1} } \right)} + \raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}{\left( {{\sigma _{0} } \mathord{\left/ {\vphantom {{\sigma _{0} } {\sigma _{2} }}} \right. \kern-\nulldelimiterspace} {\sigma _{2} }P_{2} } \right)} + 1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8{\left( {{\sigma _{0} } \mathord{\left/ {\vphantom {{\sigma _{0} } {\sigma _{3} }}} \right. \kern-\nulldelimiterspace} {\sigma _{3} }P_{3} } \right)} \ldots $$

    P 0 is the predicted deviation of an offspring from the generation mean, P 1 is a linear function of the deviation of the mid-parent from that generation mean, P 2 similarly for the mid-grandparent and σ 0, σ 1... the standard deviation of the appropriate generations of the offspring. From this formula Pearson derived theoretical values for various regression and correlation coefficients between relatives. He also generalized the geometric series of partial regression coefficients which raised the parental correlations.

  7. Pearson himself argued in his work on the χ 2 test that the sample and population constants differed on average by terms of the order 1 / √n, where n is the sample size, with the difference tending to zero as n became large. Small samples simply did not provide the proper basis for statistical work.

  8. The problem with this definition of course is that the correlation coefficients are extremely variable depending on both the character studied and the species.

  9. Here Pearson’s eugenics becomes important because natural selection was thought to no longer guarantee progress in civilised man because there was no correlation in the upper classes between viability and fertility.

  10. By population ‘structure’ I mean not just how the population was constituted but also the conditions necessary for describing that population.

  11. This also has important implications for the differences in interpreting probability in Pearson’s and Fisher’s work but that is a much longer story and one that I won’t go into here.

  12. For more on these techniques see Edwards (1994) and Ewens (2004).

  13. This debate is represented in the literature by Sober (1984) who advocates the force view and Walsh (2004) who argues for the “bookkeeping” account. For other interesting aspects of the debate see Walsh, Lewens and Airew (2002) as well as Matthen and Ariew (2002).

  14. A version of this paper was presented at the international conference on scientific understanding held at the Free University in Amsterdam in August 2005.

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Acknowledgment

I would like to thank the Social Sciences and Humanities Research Council of Canada for research support.

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Correspondence to Margaret Catherine Morrison.

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Morrison, M.C. Scientific Understanding and Mathematical Abstraction. Philosophia 34, 337–353 (2006). https://doi.org/10.1007/s11406-006-9035-7

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