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