and phoros, then empirical laws in which the phoros and theme are conflated seems to be something different altogether.
Once a mathematical formula has made the transition from an analogy to a law or principle of nature, it can be used as a warrant for making further arguments about phenomena both related and unrelated to the original subject of the induction. In “Of the Application of Inductive Truths,” in Philosophy of the Inductive Sciences, Whewell offers astronomical tables as an example of how quantitative laws, once established inductively, are extended deductively to draw conclusions about subjects considered in the original induction, but not specifically used in the calculation of the laws: “Tables of great extent have been calculated, with immense labor, from each theory, showing the place which the theory assigned to the heavenly body at successive times; and thus, as it were, challenging nature to deny the truth of discovery” (2: 426). In Whewell’s example, he cites the laws of planetary motion, arrived at by observing a few heavenly bodies, and then extrapolated in tables to describe the motion of other like objects, as instances where deduction is applied to the same class of subjects that were considered in the original induction.
In other cases, an empirical law can be used to predict phenomena which were not the original subjects of the induction by which the law was established. Herschel cites Newton’s and others’ applications of the theory of gravitational attraction to deduce the anomalies in the motions of the planets as an example where inductively established laws lead, via deductive extrapolation, to arguments about phenomena not considered under the original laws:
We must set out by assuming this law [of gravitational attraction] . . . we then, for the first time, perceive a train of modifying circumstances which had not occurred to us when reasoning upwards from particulars to obtain the fundamental law; we perceive that all the planets attract each other . . . and as this was never contemplated in the inductive process. (Discourse 201)
By developing further mathematical calculations from the law of gravity to describe the amount of influence planets have on one another, and then using the results to predict eccentricities in their orbital paths, Newton and others seeking to verify or extend his theory of gravitation proved that the law of gravity accounted for anomalies in planetary motion that had previously puzzled researchers. If the theory of gravity had not been able to suitably account for these effects, for which it had been deduced to be the cause, then the credibility of the law would have been in jeopardy (202).
In the final stage of induction, mathematical analogies make an important transition from tentative conclusions to generally accepted warrants for further arguments. Although the laws established from these analogies are still open to emendation and clarification, they have passed an important threshold after which they are generally considered accepted principles of nature. As a result of their new status, they can serve as axioms from which extrapolations can be made about subjects that fall under their jurisdiction, or about phenomena not originally considered. In this capacity, mathematical warrants serve as engines of invention, suggesting new pathways for expanding natural investigation.
Conclusion
By examining in tandem the works of two of the most influential, nineteenth-century philosophers/methodologists of science, this chapter has endeavored to provide the background for assessing what constitutes the usual or commonly accepted criteria for making mathematical argument in science in the nineteenth century, and the appropriate stages by which mathematical warrants were thought to develop. Though fundamental disagreement existed between Herschel and Whewell on the ultimate source of natural knowledge, they both agreed that without quantitative laws, nature’s intricate and sometimes impossible-to-observe operations could never be brought to light. They also believed that the strength of mathematical arguments resided in their capacity to illuminate these operations in a precise and rigorous manner, which spared natural researchers from the weakness of memory and the illusions of experience. These obvious benefits of mathematics helped it to persist in biological investigations of variation, evolution, and heredity despite general disagreements over the applicability of mathematical laws to biological phenomena. Conclusions supported by quantified data or mathematical operations could be considered more precise and rigorous than those that did not.
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,”
