Whereas Chapter 3 investigates the use of common mathematical warrants as a means of enhancing the credibility of an argument, the fourth chapter, “Hidden Value,” explores Gregor Mendel’s reliance on special mathematical warrants—mathematical principles and formulae previously unsanctioned for argument about a particular subject—in his attempts to persuade his contemporaries to accept his theory of uniform particulate inheritance. Careful scrutiny of Mendel’s arguments in “Experiments in Plant Hybridization” (1865) suggest that the mathematical principles of probability and combinatorics—mathematics “relating to the arrangement of, operation on, and selection of discrete mathematical elements belonging to finite sets or making up geometric configurations”—rather than being resources for description or verification of his experimental results, act as sources of invention for his experiments (“Combinatorial” Def. 2.). As a consequence, the mathematics function rhetorically as a creative analogy for imagining and arguing about nature before reasonable certainty had been established about its applicability to the case. I will argue that Mendel’s confidence that other members of his audience would embrace the analogy as more than creative conjecture plays a central role in his failure to persuade them of the validity of his hereditary theory.
In addition to examining the analogical characteristics of mathematics in Mendel’s argument, the chapter also investigates the complex network of ontological commitments which may have affected the reception of his mathematical arguments. A comparison of Mendel’s ontological commitments with those of his contemporaries suggests that the principles on which Mendel founded his theory were, in many cases, directly at odds with those of mainstream studies of hybridization. As a consequence, the mathematical arguments, despite their analytical rigor, could be challenged on a number of grounds unrelated to their technical execution or their verification by experiment. This vulnerability suggests that Mendel’s mathematical warrants existed in a competitive hierarchy of truths and values, some of which were either singly or collectively more compelling to his audience than the mathematical proofs which Mendel held in high esteem.
While Chapters 2, 3, and 4 probe the general divide between what was considered conventional or unconventional in mathematical arguments for mid-nineteenth century Continental and Victorian biological researchers, Chapters 5, 6, and 7 examine cases wherein explicit efforts are made to argue for the acceptance of special mathematical warrants as reliable descriptors of nature. These efforts reveal not only the reasons for the success or failure of particular attempts, but also further illuminate the scope of issues thought pertinent to, and reasons believed legitimate for, accepting or rejecting mathematical argument in science.
Chapter 5, “Probable Cause,” investigates the rhetorical success of Charles Darwin’s cousin, Francis Galton, in his endeavors to promote an analogy between the probabilistic law of error (as embodied in the bell curve) and the distribution of hereditary outcomes in human populations. Evidence from Galton’s campaign to promote this analogy in his groundbreaking book, Natural Inheritance (1889), suggests that analogies between mathematics and nature had to be argued for, and that non-analytic rhetorical strategies played a substantive role in securing their acceptance.
To make this case, the fifth chapter investigates the rhetorical strategies in Natural Inheritance, the context of its publication, and its reception. An examination of the context of publication reveals that, like Mendel. the mathematical arguments Galton relied on to establish his conclusions about variation, evolution, and heredity were considered special warrants by his audience. A close textual analysis of Natural Inheritance reveals that Galton, unlike Mendel, was aware of the lack of common acceptance of his warrants and looked for argument strategies and good reasons outside of the confines of the specialist discourse community of biological researchers to defend his conclusions. The use of general argument strategies such as narrative, visual argument, synonymia, and appeals to the values of his English Victorian readers are identified in his text, and evidence of their efficacy is presented from reviews of Galton’s work.
Whereas Chapter 5 examines a successful effort to promote a novel mathematical approach amongst conventional biological audiences, Chapter 6 investigates the failure to expand it. “Behind the Curve” explores the mathematician Karl Pearson’s efforts to develop—based on the success of Galton’s arguments in Natural Inheritance—a purely mathematical model of inheritance based on the principle of probability. This exploration reveals that Pearson’s dogged insistence that mathematics, and mathematics alone, was the key to understanding variation and evolution in natural populations, alienated his audience despite their general sympathy for Galton’s analogy between the mathematical law of error and heredity. This failure suggests that even though the suitability of the analogy had been accepted, when mathematics was advanced as a value for doing science it could be challenged rhetorically.
The final chapter, “Weightless Elephants on Frictionless Surfaces,” explores the early, twentieth century statistician R.A. Fisher’s efforts to revitalize mathematical biology in the wake of Pearson and biometricians’ failed attempts to persuade conventional biologists to side with their quantitative vision of variation, evolution, and heredity over Mendel’s. This chapter concludes that Fisher believed he needed to construct and maintain a credible ethos for his own work as well as for the general program of mathematical argument in science to reestablish biometry as a viable approach to generating new knowledge about natural selection and evolution. An investigation of his early papers and seminal book, The Genetical Theory of Natural Selection (1929), suggests that by making his complex mathematical arguments accessible to scientists with limited mathematical training, and by arguing that mathematical arguments had the virtues of practicality and inductivity, Fisher made important strides in overcoming some of the final obstacles in the long and difficult road towards the synthesis of Mendelian genetics and Darwinian natural selection that were required for the emergence of population genetics.5
A Rhetorical Approach to Mathematics, Argument, and Science
By exploring the complexity of arguing mathematically in the study of variation, evolution, and heredity from the middle of the nineteenth to the beginning of the twentieth century, this book hopes to contribute to the understanding of mathematical argument in science and its rhetorical dimensions. What it reveals about mathematics in science is that its status as a warrant for making scientific arguments is not always secure, and in some cases, requires conventional and unconventional support to be accepted as legitimate. It also advances the possibility that mathematical descriptions and arguments in-and-of-themselves may not be sufficient reasons to accept a particular scientific conclusion. Instead, mathematics exists as one node in a complex hierarchy of good reasons in competition with other values, beliefs, and truths. These conclusions illustrate the rhetorical dimensions of mathematical argument, and should thereby further encourage rhetorical investigation into mathematics not only in science, but also in other areas, such as public policy, politics, and even theoretical mathematics.
Finally, this book is dedicated to showing how a rhetorical approach to argument analysis might contribute to the efforts of historians, philosophers, and sociologists of science in their quest to understand scientific knowledge. By carefully attending to the language, organization, and argument of specific texts, and the interrelations between texts, arguers, audiences, and contexts, rhetoricians offer methods for providing concrete textual evidence to support robust characterizations of the process of argument and knowledge-making in science that are contextually sensitive and empirically grounded.
2 A Proper Science: Mathematics, Experience,