Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal by G. Mitchell Reyes

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. 164

Newton 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. 168

A 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. 169

With 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. 172

One 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. 175

As 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. 177

In 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

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