Despite obvious strengths, mathematical reasoning could be challenged on the grounds that it did not accurately reflect experience. As a consequence, mathematical formulae and reasoning had to be tested against evidence from repeated observations and experiments under a variety of conditions. As both modern and historical cases reveal, it is only in the presence of data that mathematical applications and arguments thrive in science. Chapter 4, for example, examines how Mendel’s work fell into obscurity because it lacked a broader data set to support its conclusions, and Chapter 5 explores how Galton’s work succeeded in part because of his herculean efforts to collect data in support of his theory of inheritance.
In addition to describing the qualities that make mathematical argument robust, this chapter has also illustrated the stages by which mathematical knowledge achieves legitimacy. Understanding where argument is perceived to be in this process provides insight into why a particular argument may or may not be considered rhetorical, and for what reasons. When I use the term “rhetorical” here, I am talking about argument which is probable rather than certain; argument which produces agreement from a variety of sources, which includes, but is not limited to, emotions, beliefs, and values; and finally, argument that relies on a number of general strategies/tools for argument, including figures, tropes, and topoi as means to secure agreement and establish understanding. Scientific arguments at the beginning stages of mathematization take on a rhetorical dimension because they rely heavily on the prestige accorded to mathematical deductive rigor and precision to make their scientific case, which initially has only a limited amount of inductive, empirical evidence to support it. In the middle stages of the process, the rhetorical dimension of mathematical arguments shifts from a reliance on the ethos of rigor and precision of mathematics to a dependence on analogy to establish understanding and secure agreement. Finally, in the last stages of quantitative induction, fused analogies are no longer rhetorical because they can be used as a common ground for further argument. However, because the possibility of a challenge always exists, they have the potential to lose their status as reliable warrants for scientific argument and fall once again into the realm of the probable, the rhetorical.
In the chapters that follow, the conventions for mathematical argument set out in Herschel’s and Whewell’s philosophies provide an epistemological framework for assessing the strategies of arguers as they attempt to advance mathematical programs for the study of variation, evolution, and heredity, and their successes or failures in making their cases. These investigations illustrate the utility of scientific philosophies and methodologies in understanding the epistemological context of mathematical argument in science, and it’s possible rhetorical dimensions.
3 Evolution by the Numbers: Mathematical Arguments in The Origin of Species
I have deeply regretted that I did not proceed far enough at least to understand something of the great and leading principles of mathematics; for men thus endowed seem to have an extra sense.
—Charles Darwin
So every idea of Darwin—variation, natural selection, sexual selection, inheritance, prepotency, reversion—seems at once to fit itself to mathematical definition and to demand statistical analysis.
—Karl Pearson
Because Darwin’s impact on the social and scientific developments in the nineteenth and twentieth century has been so large, his work, particularly The Origin of Species, has garnered a lot of attention from scholars in a variety of fields, including rhetoric, history, philosophy, and literature. In rhetoric alone, his persuasive efforts have been the focus of papers by authors such as: John Angus Campbell, Jeanne Fahnestock, Alan Gross, and Carolyn Miller (Campbell, “Perspective,” “Polemical,” “Rhetorician,” “Invisible,” “Believed;” Fahnestock, “Series Reasoning;” Gross, “Taxonomy;” and Miller and Halloran). Their works explore different aspects of his rhetorical strategy in The Origin of the Species, including the use of analogy between the human breeder and nature, to help his audience understand the operation of natural selection and the importance of the rhetorical figures incrementum and gradatio in making the argument about variation and diversity among groups of organisms.
Despite the wide range of topics and issues in argument covered by rhetorical scholars, there has as yet been no substantive discussion about the rhetorical importance of mathematics in making arguments in The Origin of the Species. The purpose of this chapter is to offer arguments and analyses that suggest that Darwin relies heavily on mathematical elements such as quantification and basic arithmetical operations for support and invention of his arguments for dynamic variation, relation by descent, and the principle of divergence of character in The Origin of Species. It will also make the case that, by following the best practices of quantitative induction, Darwin hoped to establish an ethos of precision and rigor for his work which was commensurate with the rising importance of quantification to the study of biological phenomena in the middle of the nineteenth century.
Mathematical Darwin?
Though historians and philosophers of science have expended considerable effort tracing the development of different mathematical fields and examining their political, cultural, and even rhetorical influence (e.g., Cullen; Patriarca), they have not, with rare exceptions, taken up investigations into the role of mathematics in Darwin’s arguments. A survey of eleven books and nine articles published by historians, philosophers, and rhetoricians of science, most published in the last twenty-five years, reveals that few texts associate Darwin’s arguments with mathematical reasoning (Appendix A). Those texts that do associate the two predominantly comment either on the lack of mathematical reasoning in the text, or on Darwin’s inability to use mathematics to make his case (Ghislen; Hull; Gale; Depew and Weber). Only four assign any real importance to mathematics in Darwin’s arguments (Browne; Schweber; Parshall; Bowler).
Mathematics in The Origin of Species
The previous examination of selected books and articles in the history, philosophy, and rhetoric of science suggests that many modern scholars do not believe or have not considered mathematical argument as an important facet of Darwin’s persuasive strategy in The Origin of Species. These results raise the question, “If mathematics plays such an important role in Darwin’s argument, why is it that so few scholars in rhetoric and history bothered to write about it?”
A cursory review of the text itself reveals that there are very few places where mathematical symbols, numbers, tables, equations, etc. are used. This scarcity of mathematical notation is puzzling even to those who argue in favor of the importance of mathematics in The Origin of the Species, like historian Janet Browne, who comments on the scarcity of mathematics in the text:
That Darwin’s botanical arithmetic has been neglected by historians is partly his own fault. In On the Origin of Species, he barely referred to his botanical statistics or the long sequence of calculations which he had undertaken from 1854 to 1858. He compressed and simplified these into a few meager paragraphs, giving his reader only six pages of statistical data to fill out the discussion of “variation of nature” in Chapter II. (53)
Despite its absence in the actual text, a brief review of Darwin’s notebooks, letters, the published manuscript of his “big species book,” and Variation of Plants and Animals under Domestication, reveals the extent to which mathematics influenced the development of his theories.1 In these publications, Darwin supplies his readers not only with lists of quantitative evidence and calculations, but also with occasional glimpses of the degree to which these data and calculations helped him formulate his conclusions.
The existence of precisely quantified data and calculations in these extrinsic sources, however, still does not explain why, if they were so important to Darwin’s argument, the majority of them were left out of his text. The answer to this query is provided by Darwin himself in the introduction to The Origin of the Species.
I can here give only the general conclusions at which I have arrived, with a few facts in illustration, but which, I hope, in most cases will