In their essay [Rhetoric and Mathematics] one

gains the double sense in which Davis and Hersh mean to discuss mathematics as

rhetorical: in the first sense they consider how mathematics is used rhetorically in

other fields (in economics for example). In the second sense Davis and Hersh

maintain that there are “rhetorical modes of argument and persuasion” at play in the

doing of mathematics.16 Here they are interested in showing the non-foundational

nature of mathematical proof. In both cases Davis and Hersh deploy a rhetoric

highly Aristotelian in flavor, one bent on disclosing the available means of persuasion

surrounding different forms of mathematical discourse. p. 162

I’ve struggled to find this, but the citation is P. Davis and R. Hersh, “Rhetoric and Mathematics,” in The Rhetoric of the Human Sciences:

Language and Argument in Scholarship and Public Affairs, ed. J. Nelson, A. Megill, and D.

McCloskey (Madison, WI: University of Wisconsin Press, 1987), 54.

Similar to Campbell, Gross, Ceccarelli, and other rhetoricians who have worked to

illuminate the rhetoric of science, the goal of this essay is to uncover the rhetorical

dimensions of mathematics that allow for certain methods to arise and become

accepted as dominant mathematical models. By focusing explicitly on the invention

of a novel mathematical concept, I hope to show the role of rhetoric in the

constitution and dissemination of the Calculus. A purely intentional, agent-centered

model of rhetoric (like the one deployed by Davis and Hersh), however, is insufficient

to the task of understanding the rhetorical force of the infinitesimal. There

is an excess to seventeenth-century mathematical and scientific intention in the

infinitesimal that raised debate in the first place. In order to analyze that excess, one

must go beyond an intentional notion of rhetoric. This is not to suggest agency and

intention play no role in my analysis (they both are fundamental to my critical

practice); rather, I want to combine an agent-centered approach with a discursive

one in order to show that the infinitesimal itself, far beyond the intentions of

Newton or Leibniz, required a rhetorical response.[..]

In the last analysis, this essay makes an effort to understand not only the rhetorical

debates between Newton, Leibniz, and their contemporaries, but also the rhetorical

force of the infinitesimal; that is, the variety of arguments that constituted the

concept of the infinitesimal and opened up a radically non-representational realm of

mathematical practice. As counterintuitive as it may sound, the infinitesimal had its

own rhetorical force independent of any author, and seventeenth-century intellectuals

found themselves forced to respond to it.p. 164Newton and Leibniz created a number of metaphors to account for their notion

of infinitesimals, and the sounds of confusion and criticism they heard were the

sounds of their scientific community slowly accepting and integrating the essential

concept of the infinitesimal into their framework. […]The constitutive rhetoric of the infinitesimal, then, generated two primary criticisms

that Newton and Leibniz would have to address rhetorically. The first

highlighted the apparent paradox of “infinitesimals in degree,” that is, the way the

Calculus deployed greater and lesser infinitesimals. [..] The second and

perhaps more powerful critique struck the language of ratios Newton and Leibniz

deployed later in their careers. Critics argued that ratios implied a magnitude that

infinitesimals lack by definition. […] The vagueness of infinitesimals, the expansion of mathematical language,

the resulting inconsistencies, the metaphysical and ontological questions—Newton

and Leibniz would have to negotiate all of these using the persuasive resources of

language. p. 166

The thrust of the paper is about strategies employed by Newton and Leibniz in discussing infinitesimals, such as:

Newton organized his language

so the emphasis was placed on the ratios of infinitesimals rather than on

infinitesimals themselves. p. 168A quick glance back to his treatment of

infinitesimals provides several examples of his ambiguous language. Consider that he

defined infinitesimals at first to be those “just nascent principles of finite magnitudes,”

but what are “just nascent principles?” Should we interpret nascent to mean

initial, or beginning, or fundamental, and in each case, does not the phrase have

different meanings? To argue that this ambiguity was strategic, however, would be

to assume that Newton had a greater understanding of infinitesimals than he

demonstrated. It is more reasonable to say that Newton used the language at his

disposal to explain a novel concept in the best ways he could while preempting

criticisms from others. p. 169With the belief in continuity, Leibniz confidently made his position on

infinitesimals clear: “I take for granted the following postulate,” he stated matter-offactly,

“in any supposed transition, ending in any terminus, it is permissible to

institute a general reasoning, in which the final terminus may also be included.”62

Imagine, then, that you have two points on a line, and one point is greater than the

other. If the lesser point approaches the greater until both numbers are equal, it is

reasonable to argue that in the instant before the two numbers become equal, a final,

ultimate quantity would be transgressed. Essentially, this was the intuitive logic

behind Leibniz’s use of infinitesimals. Unfortunately for Leibniz, although this

reasoning appeared sound, the existence of an infinitesimal quantity was untestable,

unobservable, and unrepresentable geometrically. It was built on general reasoning

and not on empirical science or Euclidean geometry. Leibniz suffered the same

criticism as Newton; he could not prove in modern terms the existence of

infinitesimals. p. 172One strategy used by Newton and Leibniz was to argue that the internal

consistency of mathematics trumped the need for empirical verification. They then

reorganized their methods so as to emphasize the ratio of infinitesimals rather than

the infinitesimals themselves. p. 175As large a role as the constitutive rhetoric of the infinitesimal (by which I mean

the apparatus of interrelated argumentative structures giving presence to the

infinitesimal) played in subsequent mathematical and scientific discourse, we should

not neglect its influence in other areas. The infinitesimal would prove fundamentally

important in several other European intellectual circles. In philosophy, for example,

thinkers began to integrate the Calculus into discussions of metaphysics and

ontology, allowing “the fantasy of a transcendental origin, an ultimate guarantor of

Truth unsituated in time, space or history” to shape the intellectual climate of

eighteenth-century Europe. p. 176

This might be useful:

The essay by Davis and Hersh begins an inquiry

into the rhetoric of mathematics, but they do not attend to the constitutive rhetoric

that emerges with the invention of novel mathematical concepts. p. 177In order to understand the rhetorical dynamism of infinitesimals, this essay has

approached mathematical rhetoric without an emphasis on the substance/appearance

dichotomy handed down by our Greek ancestry. I believe this is a problematic

binary that makes it more difficult to see rhetoric as constitutive of mathematical

and/or scientific discourse because the binary positions rhetoric as secondary to

whatever substance it conveys. Such a perspective shackles language to a logic of

representation in which language is the mediator between subject and object. […] The upshot of such critical maneuvers is the ability to

analyze the constitutive rhetoric of mathematical invention, to view mathematics not

as a language “true” to “reality,” but as a tool for helping humans cope with the

world. As Richard Rorty pointed out recently, “that Newton’s vocabulary lets us

predict the world more easily than Aristotle’s does not mean that the world speaks

Newtonian.”92 Mathematics does not seek out the truth lying dormant in nature; it

seeks out means for negotiating the world humans continuously encounter. p. 178