Leibniz on the Foundations of the Differential Calculus

This monograph consists in an interpretive essay on the foundations Leibniz provided for his calculus, together with a comprehensive collection of texts on that topic in English translation. The essay consists in five substantial chapters, preceded by an introduction and followed by a conclusion: Chapter 2: On the Metaphysics of the Continuum (1669-1676); Chapter 3: Mathematical Fictions; Chapter 4: De Quadratura Arithmetica (DQA); Chapter 5: Infinitesimals and Existence after 1676; and Chapter 6: Leibniz’s Mature Justifications of the Calculus. In the essay we argue that Leibniz first proposed the fictional nature of infinite number and infinite wholes in 1672, and infinitesimals in 1676. This was in accord with mathematical practice in the seventeenth century, where new items (e.g. negative numbers, irrational roots) could be treated as fictions in calculations, independently of questions of their existence. This formed the context in which Leibniz developed his calculus in 1675-76, and also for his treatise on quadrature (the DQA) penned in the same period. Leibniz continued to assert that bounded infinities and infinitesimals could be used in mathematics without implications for their existence for the rest of his career. Nonetheless, he had his own definite views on that question, abandoning his earlier position that they could be quantities in a loose sense, analogous to angles of contact, already in 1676. His denial of their existence dated from then, although he continued to develop arguments in favour of this position through into the 1690s. Finally, we examine the detailed accounts Leibniz provided in defence of the consistency of his calculus in the early eighteenth century, showing how they are a natural outgrowth of the views he had developed earlier, both publicly and in private. Scholars of Leibniz, as well as scholars and students in the history of science will find this to be a reference work on the subject.