<i>The Evolution of Mathematics: A Rhetorical Approach</i>, by G. Mitchell Reyes

Could the opposition of mathematical knowledge to rhetorical invention and persuasion be a rhetorical conceit of mathematicians? Mathematicians rarely work with complex machines, do experiments, or test materials, as might be expected of STEM researchers; like rhetoricians, they scribble, talk to each other, mull over a problem while going for a walk, and distribute texts in several genres to win their jobs and share their ideas. What, then, can a rhetorical perspective—one attuned to the practices and social-material relations concomitant with texts—reveal about mathematics and a mathematician’s function in complex ecologies of discourses, relations, and things? The exciting aim of G. Mitchell Reyes’s The Evolution of Mathematics is to approach mathematics as symbolic action, drawing on new materialist approaches to embodiment and what Reyes calls translative actions and vincula—fundamental social-material relations. It offers a provocative theoretical approach and a strong demonstration of how rhetoricians can read mathematical texts, though it is less successful as a mathematical history or as an account of contemporary mathematical practice.The study of mathematical rhetoric is still a neglected field, relative to the rhetoric of engineering and the sciences, but The Evolution of Mathematics does much to address the problem by entering into conversation with a broader set of recent studies of mathematical narrative, sociology, and textuality. Much of this research comes from practitioner-researchers who, like Reyes, have some training as mathematicians: for example, the collection Circles Disturbed: The Interplay of Mathematics and Narrative edited by Apostolos Doxiadis and Barry Mazur (2012), which offers a parallel investigation, or Brian Rotman’s (1988) semiotics of math and mathematical embodiment, a core influence for Reyes, which shows the ways in which formal mathematical texts carry crucial paratextual content for their writers and readers. It is this “meta-Code” (41) accompanying proofs, rather than the formal structure of mathematical texts, that Reyes’s book works to reveal. The introduction emphasizes cases from gerrymandering to gene editing that illustrate the pervasiveness and force with which math acts socially and materially, yet the book’s aim is really to read—and read against the grain—“first moments of emergence” (3)—in work by Plato, Isaac Newton, Gottfried Wilhelm Leibniz, Gerolamo Cardano, Leonhard Euler, and Carl Friedrich Gauss—of the regimes involved in contemporary mathematical practice. Reyes analyzes the relations and paradigm shifts in each of these moments, with an overarching critique of the tacit Platonic realism that he argues still surrounds mathematics. Platonic assumptions about the objectivity and stability of mathematics are worthy of attention in part for how they are allied to ways of exercising mastery over human and nonhuman worlds through abstraction, categorization, and simplification. Reyes argues instead that math involves acts of imagination that establish translational channels between formerly separated modes of inquiry and things. The text is shadowed by two questions: Does mathematics even exist as a timeless global field? Is there today a unitary field of mathematics with shared underlying values and a common practice we could call mathematization? In some universities, there is a single department of mathematics, while, in others, “pure” and “applied” mathematics are treated separately, perhaps with statistics also given its own space and building, alongside computer science, physics, and the economics that takes the focus in the closing study. There are similarities and differences between the kinds of problems, people, methods, and intentions that inhabit each field.The focus of The Evolution of Mathematics on new materialist analysis brings a fresh perspective, yet its historical attention remains largely within the bounds of E. T. Bell’s 1937Men of Mathematics. Posthumanist and critical gestures in the book call for rhetoric “to escape the gravity of anthropocentric humanism” (58), but the elements of new materialist critique that locate dogmas of Cartesian dualism and nonhuman passivity in the insularity of a Eurocentric tradition are neglected. Mathematics, like rhetoric, is a global field whose “Western” formulation was hardly ever contained within Europe.Given Reyes’s focus on math as a means for establishing commensurability across difference, it might bear investigating what aspects of algebra’s symbolic abstraction were influenced by the discursive, cultural, and theological context of Al-Khwārizmī’s Baghdad and broader Abbasid world. In “The History of Algebra and the Development of the Form of Its Language” (2006), Ladislav Kvasz considers the linguistic practices and metaphysical assumptions involved in Arabic mathematics. Ethnomathematics proposes broadening mathematical genealogies and canons by considering assorted forms of timekeeping, inventorying, and kinship systems, as shown, for example, in the study of the Andean Quipu by Marcia Ascher and Robert Ascher (1981/1997). The works of the women Emmy Noether (see Rak 2021), Ada Lovelace (see Hammerman and Russell 2015), and Grace Hopper (1952) were particularly central in the development of mathematical and computational abstraction and suggest more complicated metanarratives about the interaction of social and mathematical contexts. If math is text, rhetoric, and practice, then should it not be open to all the problems, pedagogies, and possibilities of other rhetorical genres and histories, including cultural rhetorics? It is an accomplishment of the book to raise such problems in its reading. We start with Plato in chapters 1 and 2. A deconstructive reading surfaces the moments of aporia in which materiality and embodiment are central to the dialogue in the Meno, as Socrates uses an enslaved boy who has been given geometry problems ostensibly to illustrate the atemporal and introspectively available reality of forms (Socrates, of course, mostly achieves this by getting the boy to say yes). For Reyes, mathematical realism starts here as Plato argues for the objective existence of mathematical facts but also uses math as the example par excellence to prove the existence of such timeless forms.Our next stop (in chap. 3), roughly two millennia later, considers Newton’s and Leibniz’s use of the infinitesimal. While figured in classical mathematics (e.g., in Zeno’s paradox), Newton and Leibniz systematized the analysis of infinitesimal quantities and infinite series into the model of change, motion, and measure we know as calculus. Infinitely small and large things do not exist in the world in any perceptible way, but they are foundational for our mathematics and its usages. This “rhetorical force of the infinitesimal” (78) supported a teleological, “unifying” worldview linked with “philosophical optimism” as it organized and concentrated agency (75, 80). The application of calculus in developing the English and French Industrial Revolutions’ assemblages of machinery, economies, armies, and epistemes is not mentioned, but it also illustrates mathematics’ “translative force” rather clearly.Chapter 4 focuses further inquires into the emergence and impact of invented terms. The square root of −1, i, was first introduced by Cardano in Renaissance Italy as a trick to solve the cubic equation despite no such quantity existing in conventional arithmetic. Originally, it was seen as a “wild thought” (91), with attempts made to demonstrate the correctness of the equation without recourse to such absurd quantities. Only in the eighteenth century did Euler and Gauss bring i to the center of a new field of algebra and geometry with immense application to just about everything. In the process, a geometric and sensory metaphor for i as the vertical axis of the coordinate plane was elaborated, supporting the Lakoff and Nuñez (2000) study of the role of metaphoric extensions in abstract mathematics—Reyes expresses sympathy with their analysis of mathematical cognition while suggesting that math arises from a range of acts broader than metaphor. A supporting exploration of this material can be found in Barry Mazur’s (2003)Imagining Numbers (Particularly the Square Root of Minus Fifteen), which uses the story of i’s invention alongside John Ashbery’s Self-Portrait in a Convex Mirror (1975) to find commonality between mathematical imagination and poetic imagination.The strongest demonstration of Reyes’s method occurs in the final case study (“Algorithmic Culture and Economies of Translation” [in chap. 5]): a rhetorical reading of David Li’s default correlation formula (see Li 2000), used to price risk in the collateralized debt obligations instrumental in the 2008 subprime mortgage crash. Here, Reyes shows how the equation itself is a text whose terms have presuppositions and, in some cases, social metaphors attached to their syntactic meaning—for example, that “loans are like people and default like death” (117). His reading attends both to the text of the equation and to its embeddedness in social and material relations, as the equation circulates to win its author prominence and, more importantly, reshape the nature of mortgage markets through its relationship to scale and the distribution of (un)accountability. He follows the equation as an object and actor in its own right, drawing on Karen Barad’s (2003) agential realism. This work shows what a rhetorical perspective can bring to critical algorithm studies to complement treatments such as Cathy O’Neil’s (2017)Weapons of Math Destruction or Abeba Birhane’s research on assumptions of Cartesian dualism in machine learning systems (see, e.g., Birhane 2021).The concluding chapter reflects on the implications a rhetorical and social-material approach to math could have for its practice and teaching, with eleven itemized suggestions (134). They call to bring a rhetorical and translative perspective into the classroom, aiming to look at the paratextual content of math critically, to research and teach its “embodied inscriptive” (56) element, and to spend less time in the classroom going by rote over proofs. Paralleling Jeanne Fahnestock’s (1999) investigation of the way lines of argument are tied to scientific arguments, Reyes suggests that familiar rhetorical categories and figures operate in and shape mathematical articulations.But is mathematics necessarily allied with modernism, and is its project, as Reyes argues, necessarily to establish equivalences and translations? Some mathematicians are most eager to bring mathematical methods to other domains. Others are especially hesitant to attempt to make math act on material and social worlds. Theorizations of mathematics (much to many mathematicians’ dismay, perhaps) show up in important pieces of Continental philosophy, for example, Gilles Deleuze’s (1994) interest in the infinitesimal or Alain Badiou’s (2001) interpretation of set and category theory. In Fyodor Dostoyevsky’s The Brothers Karamazov (1879–80), non-Euclidean geometry is invoked as a metaphor for moral relativism, while for H. P. Lovecraft it indicates realms of horror and monstrosity. Rather than supporting Platonism, the writers Douglas Hofstadter (1979), Fritjov Capra (1975), James Gleick (1987), and N. Katherine Hayles (1991) have focused attention on the way mathematical theory can illustrate math’s own limits and problems of reflexivity through productions such as Kurt Gödel’s incompleteness theorems, quantum physics, chaos theory, and systems theory. Math can signify alterity, nondualism, and incomprehensibility as well as modernist reason.What, also, of projects of countermathematics, such as the heterodox schools of economics popularized after the 2008 recession? Such research shows, in mathematical terms, the contingency, empirical failures, and blinders underlying dominant supply-side theories—such as that of the default correlation algorithm—while developing alternative analytic and policymaking tools. Mathematization can be used critically to pursue alternative sociopolitical ends and inquiries. In algorithm studies, the question may turn on whether empirical and mathematical analysis of the prevalence of racial and gender bias in algorithmic systems—as demonstrated by, for example, Joy Buolamwini and Timnit Gebru (2018) and Os Keyes, Jevan Hutson, and Meredith Durbin (2019)—can convince regulators and developers to do something about it. These are offshoots and alternate paths that may not fit closely in the book, but they challenge the suggestion that mathematics everywhere and always rationalizes and abstracts rather than developing in concert and conversation with other fields of inquiry. Reyes’s focus on the specificity and entanglement of mathematical practices with the place where the theorems get scribbled ought, in fact, to open up such inquiry into these situated encounters involving mathematics.The Evolution of Mathematics may suggest a program for rhetorical research into mathematics whose full and even urgent implications it pulls back from. As in Plato’s Meno, the material and social specificities of mathematics are made secondary to its use as an exemplar for theorizing the relations of the material, discursive, and symbolic. One hopes that the methods and provocative approaches of the book will be developed further by Reyes and others producing additional rhetorical analysis of the “translative force” of particular algorithms, lectures, institutes, and calculation apparatuses. The Evolution of Mathematics is compelling in arguing that it is a fallacy to oppose rhetoric to mathematics and will be worthwhile reading for those interested in the rhetoric of math, in algorithm studies, and in new materialist approaches to technoscience.

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