Arguing with Numbers: The Intersection of Rhetoric and Mathematics

Abstract

Part of the RSA series in transdisciplinary rhetoric, this volume brings together the insights of a diverse group of rhetorical scholars exploring the rhetorical dimensions of mathematics. There is no single perspective or approach on display as the reader is presented with studies of the rhetoric of mathematics as well as the use of rhetoric in mathematics and the rhetorical nature of mathematical language. These three prongs structure Edward Schiappa's foundational paper that explicitly informs the work of several contributors to the volume. In addition to these essentially theoretical explorations, the volume is rounded out by prescient applications that reinforce the topicality and importance of the subject matter. But any full review of the collection must begin with Schiappa's analyses.To the casual reader, no subjects could be more disconnected than rhetoric and mathematics. The language of demonstration and proofs measures an attitude of mind that values the apodictic and axiomatic while marginalizing, if not ignoring, the efforts of rhetoric. Chaim Perelman drew attention to this divide in his critique of the Cartesian ideal that detached the self-evident from the human sphere, wherein questions arise that mathematicians would consider foreign to their discipline.1 To consider numbers themselves as a source of evidence is part of what is at stake when mathematics is exposed as a human activity. Schiappa takes what Perelman abandoned and claims it as rhetorical territory. “In What Ways Shall We Describe Mathematics as Rhetorical?” answers the question in fertile ways (as subsequent papers show). The rhetorical turn of recent decades involves the rhetorical nature of mathematics on different fronts: “(1) the rhetoric of mathematics, understood as the persuasive argumentative use of mathematics; (2) rhetoric in mathematics, understood as the argumentative modes of persuasion found in written proofs and arguments throughout the history of mathematics; and (3) mathematical language as rhetorical, a sociolinguistic approach to the language of mathematics,” an approach supported by recent writings of Thomas Kuhn (33). In the first case, mathematics serves as evidence in an argument, increasing the persuasiveness of a claim. The second case refers to the argumentative and stylistic modes of persuasion found in proofs, a feature of the history of mathematics. The final case finds its motivation in the work of rhetoricians like Richard Weaver and Kenneth Burke,2 for whom all symbol use is rhetorical including that of mathematics. Mathematics is a language like others and with its own reasoning patterns operating in the discourse community of mathematicians. Schiappa illustrates each of these rhetorical aspects of mathematics with examples and bolsters their importance with argument, including a detailed discussion of the work of Kuhn. This, before taking a particularly interesting turn into ethnomathematics and the differences in how mathematics is conceived and used across cultures.Four of the papers in the collection make explicit reference to Schiappa's account and draw part of their stimulus from his distinctions; and the other analyses can be read through the lens of one or more of his distinctions, whether the papers are historical in nature or deal with contemporary questions. In the opening paper, and beyond their Introduction, the book's editors, James Wynn and G. Mitchell Reyes, open some of the relevant discussions by exploring relationships between rhetoric and mathematics. They reinforce their belief that the volume offers a timely and coordinated effort to explore the intersections of these two fields. In Schiappa's distinctions they find the appropriate routes into the subject matter. They trace the historical division between the fields, beginning with Plato and Aristotle, whose system of argument offered little overlap between rhetoric and mathematics, through to the uneven attention directed by Perelman and Olbrechts-Tyteca (not so much, although the authors’ reading of quasi-logical arguments suggests something) and Burke (quite a bit, with the explicit inclusion of mathematics as a symbolic means of communication). This reinforces the importance of rhetoric in mathematics, and much of Wynn and Reyes’ closing analyses confirm this.Two papers pursue the themes of the volume into the field of economics. Catherine Chaput and Crystal Broch Colombini explore the persuasive role of mathematics at work in the metaphor of the invisible hand. And G. Mitchell Reyes provides a detailed investigation of the 2008 financial crisis through a case study of the mathematical formula known as the Li Gaussian copula. As Reyes writes: “Unraveling this copula reveals the constitutive rhetorical force of mathematical discourse—its capacity to invent, accelerate, and concentrate economic networks” (83). The story is long and far too complex to be detailed here. But the study rewards the reader with an understanding of just how traditional rhetorical modalities (like analogy and argument) connect to the rhetorical modalities of numeracy (like abstraction and commensurability) to generate something new (114).Likewise, Chaput and Colombini draw from the traditions of rhetoric in exploring the metaphor of the invisible hand. Their concept of particular focus is energeia, the power or force that activates potential. One of the theses of the analysis is that “the metaphor of the invisible hand regulates the energetic force of economic arguments” (62), and they track the metaphor accordingly, from the work of Adam Smith to that of John Maynard Keynes, where mathematics gains a more central place in economic discussion, and on to Milton Friedman's “positivist mathematical economics” (66). Through these and further analyses, the paper successfully supports the argument that capitalism's force (energeia) emerges in part from the historical developments of the mathematization of the invisible hand.The last paper of Part 2, by Andrew C. Jones and Nathan Crick, weaves together the mathematical reasoning of Charles Sanders Peirce and the detective fiction of Edgar Allen Poe, specifically the Dupin trilogy that includes “The Murders in the Rue Morgue.” The discussion identifies similarities between Poe's forensic analyst and Peirce's mathematician, offering a further case of rhetoric in mathematics. Like Burke in the earlier paper, Peirce is a thinker who understands rhetoric as the effective communication of signs—although I would not want to be taken as suggesting similarities between Burke and Peirce beyond this—and this would apply to all signs, including the mathematical. Poe's detective Dupin further illustrates Peirce's method of abduction, and Jones and Crick take us through the steps involved, from hypothesis to confirmation (while also using the wrong turn of the real case behind “The Mystery of Marie Rogêt” to show how abductive reasoning can fail).Part 3, on mathematical argument and rhetorical invention, begins with Joseph Little's adoption of Schiappa's taxonomy for his study of the Saturnian account of atomic spectra, the most technical paper in the collection. That said, the historical case study of Hantaro Nagaoka underlying the discussion is quite accessible. The investigation of atomic spectra begins with a puzzle involving different appearances under different conditions. Little addresses responses to this by looking at rhetoric in Nagaok's mathematics, specifically his use of an analogy between the behaviour of material in Saturn's rings and that of atoms in what is known as the Zeeman effect. Little then analyzes the rhetoric of Nagaoka's mathematics, showing that “a mathematical equation can function indexically, symbolically, and qualitatively in a given case without taking on a computational role (164). Finally, he completes the Schiappian analysis with an account of Nagaoka's mathematical language as rhetorical in the debate that ensued between Nagaoka and the mathematical physicist G.A. Schott.Jeanne Fahnestock's paper, “The New Mathematical Arts of Argument: Naturalists Images and Geometric Diagrams,” completes Part 3. The study takes its place among Fahnestock's meticulously wrought accounts of rhetorical thinking in the history of science.3 She plunges the reader immediately into a discussion of the depiction of scallops in Martin Lister's publications of 1695. Illustrated with original drawings from the account, the rhetorical importance of image reproduction combined with geometrical ways of seeing diagrammatically is shown to underlie arguing in sixteenth century natural philosophy to an extent “that is difficult to appreciate from a twenty-first century perspective that separates the mathematical and the verbal” (174). Fahnestock believes these features underlie arguing because, unlike today, grounding all disciplines (including mathematics) was dialectic in the form of a general art of argumentation. The dialectic in question is Philip Melanchthon's Erotemata dialectics, a work which Fahnestock has just translated into English (Fahnestock 2021). This is a dialectic in which mathematics plays a detailed role, and the paper proceeds to provide a history of this work that blends naturally into a deeper history of the argumentative use of diagrams. Her conclusions point to how, through geometrically controlled images. mathematical ways of viewing the natural world issued in today's “mathematically constructed world” (204).The final two essays comprise Part 4, and both deal with the role of mathematics in education. James Wynn's “Accommodating Young Women” explores some of the gender biases in the way mathematics is taught but more specifically provides a lengthy case study of the rhetorical devices used by TV star and math scholar Danica McKellar to turn middle school girls to the study of mathematics through her book Math Doesn't Suck. This involves an interesting application of epideictic rhetoric to a contemporary subject of concern, and the strategies used are both traditional and innovative. Essentially, McKellar strives to modify the image of mathematics, and Wynn's study of her attempts is both fascinating and instructive.The final paper in the collection, Michael Dreher's “Turning Principles of Action into Practice,” studies the National Council of Teachers of Mathematics’ (NCTM) rhetoric in reforming mathematics education. Two of Schiappa's categories come into play here: rhetoric of mathematics and in mathematics. Built on a historical account of philosophies of mathematics education, and incorporating several pertinent anecdotes, Dreher reveals the successes and failures of the NCTM's persuasive attempts to counter the idea that mathematical ability is inherent in only few and instead promote wide success in students’ mathematical achievement. It is a challenge that continues, and Dreher makes clear the difficulties still to be faced.This is, in sum, an eclectic set of papers gathered around a few common agreements and unified by a deep conviction of the importance of challenging any vestiges of the traditional belief that rhetoric and mathematics occupy different, even competing, spheres. The stand-out paper, testified to by the importance accorded it by many of the other studies in the book, is Schiappa's. One could say that it is worth the price of the book, but that would be unfair to the many other fine pieces of scholarship collected here.The observant reader will also have noted that much of the forgoing discussion refers to rhetoric and mathematics, while the title of the volume speaks of arguing. In fact, the attention to argumentation is pervasive, and this book takes its place among a recent appreciation of the role of mathematics in argumentation,4 while answering the kinds of dismissive critiques we once witnessed from skeptics like Alan Sokal and Jean Bricmont,5 who attempted to maintain the rhetoric/mathematics gap by suggesting that those who crossed it (at least from one direction) were unknowledgeable interlopers. It was one of Schiappa's opening insights that “If we replace the word “rhetoric” with “argument” . . . we find considerable recent interest in “mathematical argumentation” as a social and pedagogical practice” (43). And, as I have noted, this is repeatedly corroborated in this highly recommended book.