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1 Introduction

I assert . . . that in any special doctrine of nature there can be only as much proper science as there is mathematics therein. For . . . proper science, and above all proper natural science, requires a pure part lying at the basis of the empirical part.

—Immanuel Kant

In the twentieth and twenty-first centuries, there has been a substantial expansion in the number of fields applying mathematics to their investigations of natural and social phenomena. Areas of social research such as psychology and sociology, which had traditionally been qualitative, have developed robust quantitative components, such as standardized intelligence tests and statistical surveys to assess the habits, beliefs, and practices of populations. In addition, fields of natural investigation. particularly genetics and evolutionary biology, have expanded their methods as described in the Forward from observation and experiment to include mathematical descriptions of the genome and the change and distribution of variation within organic populations. As a result of this expansion, mathematics has become a ubiquitous aspect of what makes a discipline “scientific,” making Kant’s invocation that “there can only be as much proper science as there is mathematics therein” seem even more relevant today as it was when he wrote it in 1786.1

Although this mathematical expansion has helped us better understand social, psychological, and biological phenomena, the path to developing and adopting mathematics in science has not always straight or easy. One widely-cited example of a mathematical science with a turbulent beginning is population genetics. In the twenty- first century, this vital field of genetics research has its own textbooks, specialists, and places in the academy. Its recent success, however, obscures a less flourishing past. Although the basic scientific and mathematical foundations for population genetics were largely in place by the first decade of the twentieth century, almost thirty years elapsed before it garnered sufficient attention and support to establish itself as a field of research worthy of an individuated identity.

Historians interested in the intellectual foundations of population genetics and its transition into an important field of inquiry have reached back to Darwin and traced its development forward, hoping to understand the reasons why such a productive, modern field of study had such a difficult maturation. Perhaps the most well-known investigation into this mystery was undertaken by historian William Provine, whose groundbreaking book, The Origins of Theoretical Population Genetics, suggests that the turmoil associated with the rise of the field was largely the consequence of an ideological conflict between Darwinians and Mendelians (ix-x).

According to Provine, Darwin and his later followers, the biometricians, believed in continuous variation in which differences between members of a species arose by the slow accretion of small variations over long periods of time. Mendelians and the supporters of mutation theory, on the other hand, believed that variation was discontinuous: varieties appeared suddenly and could introduce dramatic changes into individuals and populations of organisms. By tracing these notions about variation from Darwin through the debates between the biometricians and the Mendelians, Provine concludes that it was only when the differences between these ideological positions were resolved in the work of R.A. Fisher, J.B.S. Haldane, and Sewall Wright, that a research field of population genetics emerged (Provine 131).

Although Provine and other historians rightfully devote attention to how ideological conflicts over variation complicated the development of population genetics, their focus excludes other elements that could have contributed significantly to population genetics’ “torturous” development (Provine ix). One important element that has not been considered is whether and to what degree the beliefs about the acceptability of using mathematics to make arguments about biological phenomena might have contributed to the difficulties in establishing the field.

Although it is difficult from a twenty-first century perspective to image mathematics not being a legitimate means of researching variation, evolution, and heredity, scrutiny of the work of early researchers such Charles Darwin, Gregor Mendel, Francis Galton, Karl Pearson, and R.A Fisher suggests that this has not always been the case. Attention to their work and its reception reveals that mathematical approaches to these phenomena were caught up in a cycle of development, conflict, and persuasion that lasted almost one hundred years, a cycle that has all but been forgotten as science looks to the future and eviscerates from memory the useless, blind allies and conflicts that led to its current position. In hoping to understand the development of scientific knowledge, however, we need to look at the process of making knowledge, not just the results. Investigating these long-forgotten conflicts can help us understand not only what people believed, but also how they were moved to change their beliefs, what they perceived were good reasons for accepting or rejecting a particular position, and what lines of argument dominated the scientific landscape.

Despite the importance of mathematics to scientific argument and epistemology, there have been few historical-philosophical, sociological, or rhetorical investigations of how scientists argue with or about the use of mathematics.2 In the history of science, the intersection between argument, science, and mathematics has been investigated by historian Peter Dear, whose book, Discipline and Experience: The Mathematical Way in the Scientific Revolution, examines sixteenth, seventeenth, and eighteenth century disputes among natural philosophers about whether mathematics could serve legitimately and authoritatively as a source for arguments about nature. In the book, Dear attempts to make sense—in the context of eighteenth century natural philosophy—of both the novelty of Newton’s physico-mathematical argument strategy and its importance to the development of a new paradigm for scientific research (248).

Although very few historians have examined the relationships between mathematics, science, and argument, there is evidence of a trend towards increasing attention to the subject. In a recent discussion, for example, in Isis—a top journal in the study of history and philosophy of science—titled, “Ten Problems in the History and Philosophy of Science,” historian and philosopher Peter Galison lists a lack of understanding of the “Technologies of Argumentation” as problem number three for philosophers and historians of science. He raises the following questions for them to pursue:

When the focus is on scientific practices (rather than discipline-specific scientific results per se), what are the concepts, tools, and procedures needed at a given time to construct an acceptable scientific argument? . . . Cutting across subdisciplines and even disciplines, what is the toolkit of argumentation and demonstration—and what is its historical trajectory? (116)

For rhetoricians of science, whose interests lay predominantly in the study of scientific argument and communicative practices, answering these sorts of questions about the relationship between mathematics, science, and argument would seem to be an important and fruitful undertaking. Despite the natural fit between scholarly interest and subject matter, however, very few rhetoricians have made efforts to examine the intersection between these three subjects. One notable exception is the work of Alan Gross, Joseph Harmon, and Michael Reidy in, Communicating Science: The Scientific Article from the 17th Century to the Present. In this book, the authors examine the developing conventions of argument and style in the scientific article, including brief descriptions of the use of mathematics.

Though the works of Dear, Gross, Harmon and Reidy begin a conversation about the role of mathematics in scientific argument, there are many important avenues currently unexplored. Questions—such as, “Do new mathematical methods have a different status of reliability as a source for arguments in science than existing ones?”; “If mathematical methods are not assumed a priori to be reliable, how do scientists make a case for their use in science?”; and “Can the reliability of mathematical methods and their use be debated and secured using methods outside of a framework of analytical argument?”— still remain and represent substantial lacunae in or understanding of the subject. The primary goal of this book is to explore these questions.

Just as scientists rely on rare diseases amd aphasias to understand the functioning of genes and language, this investigation turns its attention to a special case of the mathematization of a scientific field to find answers to these questions. Specifically, it examines the development of the mathematical study of variation, evolution, and heredity from the middle of nineteenth to the beginning of the twentieth century which eventually culminates in the emergence of important mathematical subfields of biology, including population genetics, quantitative genetics, and biostatistics. This development provides a unique opportunity to observe the nuances and difficulties in the relationships between mathematics, science, and argument.

By examining the conventions for arguing mathematically about natural phenomena and the successes and failures of advocates for a mathematical approach, I intend to advance four conclusions about the relationship of mathematics to scientific/biological argument:

 that novel mathematical arguments used to make claims about natural phenomena do not necessarily compel acceptance,

 that scientists arguing for novel mathematical warrants rely on a range of resources for generating good reasons to support their use,

 that arguments about and with mathematics in science can have non-analytical, rhetorical dimensions, and

 that conflicts over the appropriateness of using mathematics have complicated the development and acceptance of biomathematical fields such as population genetics.

A Rhetorical Approach to Scientific Epistemology

Any effort to discuss scientific argument and knowledge-making requires some explanation of one’s philosophy of scientific epistemology. The epistemological perspective guiding this rhetorical investigation can be understood by contrasting it with positions on the subject that have been previously taken up by historians, philosophers, and sociologists of science.

In the last century, notions of scientific epistemology have tended either towards logical positivist/empiricist models of science or towards social constructivist models. For the logical positivist/empiricist, scientific propositions and theories are thought to be systematically verified or falsified by appealing to physical evidence in conjunction with logical-linguistic constructs and deductions. For the social constructionist, rationality is located in the commitments of a scientific community to seeing nature in a particular way. Although logical positivist/empiricist approaches to scientific epistemology were extremely influential in the late decades of the nineteenth and the early decades of the twentieth century, they met a series of challenges from philosophers such as Karl Popper, W.V. Quine, and Ludwig Wittgenstein, culminating in Thomas Kuhn’s The Structure of Scientific Revolutions, which substantially decreased their appeal. Kuhn’s investigation offered a fairly comprehensive vision of scientific knowledge-making that rejected the possibility that fixed, rational principles could be appealed to in times of epistemic crisis. From Kuhn’s paradigmatic perspective, major changes in the conceptual framework of a scientific community could only occur when an existing paradigm had become so troublesome that its adherents began the process of developing alternative paradigms to replace it. This feature of paradigmatic change precludes falsification or verification by rejecting the possibility of a rational, external position from which a paradigm could be judged (145).

One of the consequences of Kuhn’s model of scientific epistemology is that it not only eliminates the possibility for “objective” logical-linguistic constructs and deductions to guide argument and decision-making in science, but also the prospects for any reasonable common ground to exist between members of old and new paradigms. The absence of a third position, or alternative reasonable perspective, from which arguments supporting or challenging a paradigm over its alternative can be made or judged, raises important questions such as: “How is it that researchers working in a particular field during a time of revolution choose one paradigm over another?” and “How do communities of scholars with different points of view decide that one school of thought’s natural metaphysic is sufficiently better than its competitors’ and should be embraced as a paradigm?”

In response to the first question, Kuhn argues that the choice of a paradigm is made on the basis of a personal rather than a communal calculus. Novices entering a field with conflicting paradigms, for example, choose a paradigm to apprentice under according to their own individual sensibilities about which one they more closely identify with. Similarly, established participants in an existing paradigm either remain steadfast in their support for it, or experience a sudden, personal conversion to the alternative. In both cases, the transformation cannot be compelled by any commonly held good reasons for preferring one position over the other (Kuhn155). In response to the second question, Kuhn offers no criteria at all, stating only that “to be accepted as a paradigm, a theory must seem better than its competitors” (17).

As a challenge to logical positivist/empiricist approaches to scientific epistemology, Kuhn’s concept of paradigm adoption swings away from what the rhetorical theorist Kenneth Burke, in the Rhetoric of Motives, calls a grammatical stance: a search for a set of universal propositions and procedures which would account for its epistemological robustness (21–23). In correcting the perceived errors of the grammatical position, however, Kuhn moves towards a radically opposed symbolic stance in which rationality is bound to idiosyncratic, personal reasons for choice rather than some loci of rationality shared by the larger community. The perspective on knowledge and argument employed in this investigation takes an epistemological middle ground between the universal and the idiosyncratic. This middle path is uniquely fitted to a rhetorical perspective because it rejects, on the one hand, analytical self-evidence by embracing the centrality of audience in argumentation, and in so doing, the necessarily communal and probabilistic nature of argument (Perelman and Olbrechts-Tyteca 1–10). On the other hand, it takes up the position that discourse communities overlap and interconnect and, as a consequence, good reasons can exist outside of a single discourse community and affect persuasion within it (Aristotle, Rhetoric I, ii 15–25). This allows for alternative avenues of rationality and common ground to exist, even in cases where different paradigms or schools of thought compete, and dispenses with the necessity of reverting to personal calculi to make decisions in these types of crises.

Rhetorical Method

From a rhetorical perspective, foci for analysis can include, but are certainly not limited to: (1) the good reasons and forms of evidence and argument that discourse communities find acceptable, (2) the effects of these dimensions of argument on the choices that speakers and writers make in constructing arguments, and (3) the ways that audiences judge their choices. Analyses centered on these foci address questions like, “Who is the audience for a scientific argument?”; “What conventions govern the way researchers participating in a particular scientific discourse community are expected to argue?”; “What facts, beliefs, and values do participants within a particular research community use to judge the validity and reliability of methods and conclusions?”; and “What broader sets of facts, beliefs, and values might influence the making or judging of arguments?”

Historical Analysis

This investigation relies on several different methods of analysis, including historical analysis, close textual analysis, and audience response analysis to draw conclusions about the relationship between scientific and mathematical arguments in the development of mathematical approaches to the study of variation, evolution, and heredity. Historical analysis is used to establish the dimensions of the scientific debates surrounding these phenomena and the perceived role of mathematics in the debate from the middle of the nineteenth to the beginning of the twentieth century. This aspect of analysis draws on a wide range of sources, including archived letters, scientific articles, philosophies of science, reader reviews of primary texts, and secondary historical accounts to establish the contours of the debate. These resources also supply evidence for characterizing the role of mathematical argument in science during the period under investigation.

Using philosophies of science to establish the conventions for scientific arguing in a particular historical period is not, to my knowledge, a method that has been previously exploited by rhetoricians of science. Current rhetorical work analyzing scientific argument relies either on a scientific figure’s knowledge of rhetoric or on second-hand accounts of the conventions of scientific argument. In Jean Dietz Moss’s work on the Copernican controversy, Novelties in the Heavens, for example, Moss offers readers historical evidence that scientific figures such as Galileo and Kepler had studied and/or taught rhetoric. This evidence proves a particular connection between rhetorical treatises and a scientist’s use of characteristically rhetorical strategies of argument—such as the employment of ornamental language to gain the attention and admiration of the reader. While this approach offers a robust connection between specific rhetorical training and argument, it limits the rhetorical analyst to cases in which scientists can be proven to have had a rhetorical education. Though these limitations are not prohibitive in investigations of Renaissance science, they become severely restrictive for science in the nineteenth century—a time when rhetoric was largely absent from standard education and during which no new substantive treaties on rhetoric were published (Houlette ix). Further, by restricting rhetorical investigations to facets of argument learned through a rhetorical education, the range of scientific argument that can be explored is unnecessarily narrowed based on distinctions between dialectic and rhetoric, which are practically very difficult to maintain.

Other rhetorical analysts of science have adopted a broader sense of the argumentative territory open to rhetorical discussion and analysis. However, in their efforts to identify the conventions of scientific argument, they depend on secondary rather than primary sources. In Lawrence Prelli’s substantive work on the rhetoric of science, A Rhetoric of Science: Inventing Scientific Discourse, for example, he relies on Thomas Kuhn’s discussions in The Structure as the source for his problem-solution topoi and evaluative topoi.3 Though Kuhn is certainly considered a reputable source for understanding scientific argument, neither he nor Prelli offer any primary source evidence to corroborate that scientists endorsed these lines of reasoning as conventional places for finding arguments in science.

The methodological contribution of this book is in its use of primary source material, specifically the writings of nineteenth century philosophers of science, as sources for constructing a robust description of the conventions for arguing with mathematics during this period. By relying on primary source material, it avoids problems of reliability and contextual sensitivity. In addition, it broadens the scope of materials available to rhetoricians for analyzing scientific argument and offers a means by which the division between common and special lines of argument can be made. These divisions are formulated positively based on examinations of the actual conventions articulated by a scientific discourse community rather than negatively as anything not existing in a particular treatise or set of treatises on rhetoric. Investigating the articulated conventions of science in conjunction with scientific arguments provides a broader and more accurate picture of what constitutes or does not constitute a common or special line of argument, and thereby what aspects of scientific argument are or are not being employed rhetorically.

Close Textual Analysis

While the historical analyses in the book are aimed primarily at establishing the conventions for mathematical and scientific argument, close textual analyses of the works of featured arguers offer insight into their specific choices of language, organization, and argument. These choices illuminate not only the persuasive goals and strategies of arguers, but also what these arguers may have believed about their audiences.

This type of analysis is conducted in this investigation using a number of pre-existing analytical categories in modern and classical rhetoric, such as ethos, stases, loci, value hierarchies, etc. as well as a detailed assessment of choices in language, organization, and argument in the text. The applications of these analytical categories are intended—in addition to their utilitarian function of illuminating the character and structure of the argument—to illustrate that categories for analyzing discourse and argument exist within rhetoric that might be profitably used to expand our understanding of the role of mathematics in scientific argument.

Audience Response Analysis

Finally, unlike the two previous analytical methods, which are primarily designed to illuminate argument conventions and strategies, the third method, audience response analysis, is designed to provide insight into persuasion. As some rhetoricians have pointed out, rhetorical analysts have made bold pronouncements about the persuasive affects of texts without supplying evidence from the audience to support their contentions.4 The analysis in this book endeavors to make claims about the reasons that late nineteenth and early twentieth century biological researchers judged mathematical concepts and formulae to be reliable or unreliable grounds for arguments about variation, evolution, and heredity. As a consequence, it is necessary not only to discuss the conventions of scientific argument, but also the specific reasons given by audiences for accepting or rejecting them.

Examining the responses of individual audience members in conjunction with the conventions of scientific argument has a number of benefits. First, by examining the two together, it is possible to know whether members of a particular audience were or were not appealing to convention to support their praise or excoriation of a scientific argument. This knowledge provides a method of checking whether scientific conventions were taken seriously, considered unreasonable ideals, or not considered applicable in particular situations. Second, by examining individual responses it is possible to understand whether sources of good reasons for accepting or rejecting mathematical argument were limited to the conventions outlined in scientific philosophies. If alternative good reasons exist, their presence suggests that there may be values, beliefs, and truths from outside the confines of a specific scientific discourse community impacting the development of scientific knowledge. Their existence would indicate that broader, rhetorical lines of argument are implicated in reasoning about the validity of mathematical warrants in making scientific arguments about biological phenomena.

To make claims about what audiences might have thought about a particular application of a mathematical argument to some aspect of variation, evolution, and heredity, each chapter includes close readings of audience responses to the texts of the featured arguers. These responses are primarily from book reviews written at or around the time a featured text was released or, in the cases of journal articles, private or public responses offering commentary on the research. All audience responses are assessed by close textual analyses, singling out the reasons given by respondents for supporting or challenging the featured arguments. In addition, the reasons are compared to the featured arguer’s theory of his audience to draw conclusions about why they may have succeeded or failed in their persuasive endeavor.

Preview of Chapters

Each chapter in the book is dedicated to investigating some aspect of the relationship between science, mathematics, and argument. In the first three chapters, the focus is on articulating the general conventions and limitations of mathematical argument in science. The remaining three chapters explore the rhetorical dimensions of making mathematical arguments in science.

Chapter 2, “A Proper Science,” explores nineteenth century epistemological and ontological commitments about the appropriate relationship between mathematics and science by examining the philosophies of two of the period’s most influential, natural philosophers. A close reading of William Whewell’s, Philosophy of the Inductive Sciences (1840), and John Herschel’s, Preliminary Discourse on Natural Philosophy (1831), suggests that quantification and mathematical reasoning were considered assets in the production of knowledge about nature because they contributed precision and rigor to scientific research and reasoning. Investigations of these texts also reveal the stages in which quantification and mathematical reasoning contributed to science and the process by which mathematical arguments might be elevated from hypothetical analogies to deductive laws of nature. The views of these two, influential philosophers about the status of mathematical arguments in science provides a framework for assessing the status of mathematical arguments, the choices of arguers as they seek to defend them, and the reactions of nineteenth century audiences as they move to accept or reject mathematical arguments as a legitimate means for making knowledge about nature.

The third chapter, “A New Species of Argument,” examines the rise of the mathematical treatment of variation and evolution in Charles Darwin’s work, The Origin of the Species (1859). It makes the case, contrary to most current scholarship on Darwin, that his work relies on common mathematical warrants—generally accepted lines of reasoning for making mathematical arguments in science—both to support and invent arguments about dynamic variation, relation by descent, and the principle of divergence of character. In addition, the chapter suggests that his use of mathematics for support and invention may have been a rhetorical move on Darwin’s part to establish an ethos of precision and rigor for what he knew would be controversial positions. An examination of popular philosophies of science, which had been read by Darwin and were revered by many nineteenth century philosophers, suggests that the naturalist had attempted to follow the conventions for developing robust scientific arguments through the use of quantification and mathematical operations.

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.