UC San Diego

This dissertation concerns a topic that has been unduly neglected by historians of Early Modern philosophy and philosophers of mathematics alike: the highly original conception of number advanced by Gottfried Wilhelm Leibniz in the seventeenth and early eighteenth centuries. I aim to answer several questions regarding that conception, thereby illustrating its historical and philosophical importance: (1) How does Leibniz define the concept of number?; (2) Into which ontological category does Leibniz think numbers fall?; (3) Which sorts of numbers — e.g. rational, irrational, complex — does Leibniz think are conceptually legitimate, and to what extent does he realize that his own definitions commit him to the acceptance of certain kinds of numbers as such?; and (4) How does Leibniz think we acquire knowledge about numbers?

In the course of answering these questions, I aim to show that Leibniz’s conception of number is philosophically significant insofar as it unites the most productive aspects of earlier conceptions into one that goes a long way toward allowing him to accommodate numbers that had not been previously viewed as conceptually legitimate (e.g. irrationals and complex numbers); provides an original ontology of number as a certain kind of relation; and anticipates the core views of the logicist school in the philosophy of mathematics.

The dissertation is organized in a way that reflects the four core questions: I begin by discussing the intellectual climate in seventeenth-century mathematics in Chapter 1; I move on to an analysis of Leibniz’s conceptual characterization of number in Chapter 2; I argue that this characterization is consistent with Leibniz’s ontology of number (and explain the nature of that ontology) in Chapter 3; I discuss the scope of Leibniz’s view of number in Chapter 4, arguing that he is committed to the existence of different sorts of non-rational numbers, while also delineating the conceptual and technical limitations of his views; I explain his epistemology of number in Chapter 5; and I close by arguing that his views — conceptual, ontological, and epistemological — anticipate those of the logicists in Chapter 6.