- Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes

Among his achievements in all areas of learning, Leibniz’s contributions to the development of European mathematics stand out as especially influential. His idiosyncratic metaphysics may have won few adherents, but his place in the history of mathematics is sufficiently secure that historians of mathematics speak of the “Leibnizian school” of analysis and delineate a “Leibnizian tradition” in mathematics that extends well past the death of its founder. This great reputation rests almost entirely on Leibniz’s contributions to the calculus. Whether he is granted the status of inventor or co-inventor, there is no question that Leibniz was instrumental in instituting a new method, and his contributions opened up a vast new field of mathematical research.

The foundations of this new method were a matter of some controversy, however. The Leibnizian calculus at least appears to violate traditional strictures against the use of infinitary concepts, and critics charged Leibniz with abandoning classical standards of rigor and lapsing into incoherence and error. In response, Leibniz argued that his methods were rigorous and that they did not suppose the reality of infinitesimal quantities. But the critics of the calculus were not the only ones who engaged Leibniz in a discussion of the nature of the infinitesimal. Even those sympathetic to the new method gave differing interpretations of the doctrine of infinitesimal differences, and Leibniz’s own doctrine evolved out of dissatisfaction with alternative foundations for the calculus, as well as a desire to satisfy the demands of his “traditionalist” critics.

The fundamental thesis in Leibniz’s mature account of the foundations of the calculus is that infinitesimals are well-grounded fictions. Although it is not without its difficulties, the doctrine seems to solve or avoid problems inherent in two alternative treatments of the calculus, one (due **[End Page 6]** to Johann Bernoulli) that regards infinitesimals as real positive quantities and another (from John Wallis) that takes them as nothing, or “non-quanta.” At the same time the fictional infinitesimal allows a response to the traditionalists, whose criteria of rigor permit only finite magnitudes in mathematics. I do not wish to say that Leibniz’s theory arises out of a Hegelian synthesis of Wallis’s thesis and its Bernoullian antithesis. Nevertheless, the fictional treatment of the infinitesimal clearly appears designed in response to them and to the critics of the calculus. If I am right, we can see this doctrine take shape through the 1690s as Leibniz tries to settle on an interpretation of the calculus that can preserve the power of the new method while placing it upon a satisfactory foundation. ^{1}

This essay is divided into six parts in roughly chronological order. The first is a brief overview of Leibniz’s formulation of the calculus, including its background in Hobbes’s doctrine of conatus. The second outlines objections that Bernard Nieuwentijt and Michel Rolle raised to the calculus. The third considers Leibniz’s response to Nieuwentijt, and particularly the proposal that the calculus can be based on “incomparably small” magnitudes. Section four examines Leibniz’s correspondence with John Wallis and his rejection of Wallis’s claim that infintesimals lack quantity. The fifth section considers Leibniz’s correspondence with Bernoulli and its connection to the project of replying to Rolle. The final section then tries to make some sense of the Leibnizian theory of the fictional infinitesimal.

# 1. Leibniz, Hobbes, and the Problem of Quadrature

The story of Leibniz’s invention (or, if you prefer, discovery) of the calculus has been told many times, both by Leibniz himself and by numerous commentators. ^{2} I do not propose to recount it in any detail, but it is important that we consider it at least in outline. Leibniz’s first mathematical investigations were apparently in the field of combinatorics; he reports in the essay *Historia et origo calculus differentialis* that he “took great delight in the properties and combinations of numbers” (GM V, p. 395), and these studies led him to publish his *Dissertatio de Arte Combinatoria* in 1666 (GM V, pp. 10–79). Leibniz became particularly interested in the **[End Page 7]** properties...