Evolution by the Numbers. James Wynn
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botanists’ marked compendiums of species, Darwin searched for proof of a general pattern in the ratios of closely related species in large and small genera. He hoped to find that larger genera had a greater number of close species, while smaller genera had fewer, more distinct species. He reasoned that if this pattern did exist it would support his theory of variation and relation by descent because larger, more successful genera would be producing new species that would be recent and closely related. Conversely, unsuccessful genera would not be producing newer species, and some of the older species they had produced would have died out, leaving gaps between the existing species and making them appear less closely related (Watson to Darwin November 19, 1854 n2). Evidence of this strategy appears in the draft of the big species book from which The Origin of the Species was abstracted (Table 1).
The Origin of Species: A General Overview of the Argument
An examination of Darwin’s letters, notebooks, and reading lists reveals that he was familiar with ideas and methods of biogeography and that he adopted them in his search for evidence and arguments to support his theories of variation and evolution. In order to make the case that mathematics is an important aspect of Darwin’s argument in The Origin of the Species, however, it is imperative to show that mathematical argumentation exists in the text and to understand what role it plays in supporting Darwin’s conclusions. Towards this end, the next two sections examine the arguments in chapters two and four of text as well as the peripheral documents associated with them. This investigation reveals that Darwin employs mathematical argument in the book, and that this mathematical argumentation plays an important role in helping him invent and/or support his arguments for the existence of a process of dynamic variation and the principle of divergence of character.
To understand the significance of the arguments in chapters two and four in the overall scheme of Darwin’s argument, it is useful to review the primary conclusions he attempts to establish in the text. The basic arguments in are: (1) that organisms are highly plastic and can be made to vary to a great degree, (2) that variation accumulates over time, resulting in populations of organisms that were once related becoming physically distinct, (3) that the spread of variation in nature is the result of natural selection, and (4) that the more diversity in a species or genera the more likely it is that its members will successfully reproduce.
These primary arguments are introduced and developed in the first four chapters of the book. The other chapters of the text are concerned with presenting qualitative evidence from geology, animal behavior, comparative anatomy, and other areas of knowledge that support Darwin’s four main arguments and his efforts to address possible disputations of his position.
Chapter II: Variation under Nature
In the second chapter of The Origin of the Species, “Variation under Nature,” Darwin argues for the possibility of selection without human intervention through the process of natural selection. He achieves this goal with the help of quantified comparisons using arithmetical operations that prove that not only are taxonomic categories of species fuzzy, but also that this fuzziness can be accounted for by conceiving of diversity as the result of the dynamic process of continual variation, revealing a relationship of descent between the different levels of the taxonomic hierarchy.
In the first portion of chapter two, Darwin sets up his argument by refuting the position of special creationists who believed that each species identified in the taxonomic hierarchy marked a unique creation that was readily identifiable by the existence of an indelible set of features. He argues that if this position is correct, then there should be a definite consensus about which organisms belong in a particular category. In order to test the veracity of this assumption, Darwin quantitatively compares the categorization statistics made by experts in the field, including H.C. Watson:
Compare the several floras of Great Britain, of France, or of the United States, drawn up by different botanists, and see what a surprising number of forms have been ranked by one botanist as a good species, and by another as mere varieties. Mr. H.C. Watson, to whom I lie under deep obligation for assistance of all kinds, has marked for me 182 British plants, which are generally considered as varieties, but which have all been ranked by botanists as species. . . . Under genera, including the most polymorphic forms, Mr. Babington gives 251 species, whereas Mr. Bentham gives only 112, - a difference of 139 doubtful forms! (41)
Using their own data, Darwin reveals that even experts in plant identification and categorization come to astonishingly little agreement about which organisms should be ranked as varieties and which as separate species. By casting doubt on the fixity of taxonomic categories, he creates an opportunity to present his own theories of evolution and natural selection, which he believes more adequately account for the data.
He opens the second portion of the chapter by clearly laying out his position:
Hence I look at individual differences, though of small interest to the systematist, as of high importance for us, as being the first step towards such slight varieties. . . . And I look at varieties which are in any degree more distinct and permanent, as steps leading to more strongly marked and more permanent varieties, and at these latter, as leading to sub-species and to species. . . . I attribute the passage of a variety, from a state in which it differs very slightly from its parent to one in which it differs more, to the action of natural selection in accumulating . . . differences of structure in certain definite directions. (44).
In these lines, Darwin presents a vision of diversity in nature as a dynamic process rather than as a static condition. He argues that the small differences observed in individual organisms can spread by descent throughout successive generations, making the offspring of those individuals slightly different from the general population from which they originated. These differences can widen through the continued accumulation of variation and eventually transform distinct varieties into distinct species. This dynamic process, Darwin argues, can be attributed to natural selection, which he defines as, “the preservation of favorable variations and the rejection of injurious variations” (68).
Once he has established his position on the source and character of diversity, Darwin presents arguments to attempt to link the size and range of a group of organisms at a particular level of the taxonomic hierarchy to the number of subordinate categories of organisms associated with that group. In order to do this, he depends both on arithmetical calculations using quantitative data and on the rhetorical/logical commonplace (topos) of the more and the less with which he establishes a connection between size/range of a population and the characteristic of diversity. In Book II of the Topics, Aristotle explains this strategy of argument: 7
Moreover, argue from greater and less degrees. There are four commonplace rules. One is: see whether a greater degree of the predicate follows a greater degree of the subject. . . . For if an increase of the accident follows an increase of the subject, as we have said, clearly the accident belongs; while if it does not follow, the accident does not belong. You should establish this by induction. (The Complete Works, 114b 35–115a 6)
Using statistics from available botanical compendia, Darwin calculates the number of botanical varieties belonging to species with the greatest estimated population sizes and ranges in hopes of discovering some general pattern in, or connection between, these species: “I thought that some interesting results might be obtained in regard to the nature and relations of the species which vary most, by tabulating all the varieties in several well-worked floras” (Origin 45).
To make his calculations, Darwin first divides the species in the compendia into large and small species according to the author’s size designations. He then divides the number of species in the large and small groups by the number of varieties that are connected with them to produce an average of the number of varieties for each species, large and small. The results of these tabulations and comparisons reveal that there is a correlation between the size and range of a species’ population and the number of varieties recorded for that species (Table 2).
Table 2. Arithmetical Comparisons of the Ratios between Species and Varieties in Large and Small Genera. Reprinted from Charles Darwin, Charles Darwin’s Natural Selection, Ed. R.C. Stauffer, p. 150. © 1975. Used by permission of the publisher,