Some Perspectives on Leibniz’s Nominalism and Its Sources

The chapter considers the presence of nominalist motives in the development of Leibniz’s logical and ontological thought. The discussion begins with Leibniz’s Preface to his reedition of the work of the Renaissance nominalist Nizolius, and emphasizes Leibniz’s acceptance of antirealistic assumptions, his balancing of them with Platonic elements, and his rejection of Hobbes’s conventionalist implications. There is also a consideration of the deflationary treatment of abstract terms that Leibniz offers as part of his program of providing ontological clarification by way of semantic analysis. Leibniz applies an analysis of abstract talk stemming from Hobbes, with the aim of availing himself of the resources of certain expressive devices in logical and scientific language, while avoiding realistic commitments. Leibniz’s final profession of a “precautionary nominalism” confirms his preference for ontological economy, but also leaves unresolved some problems in his “austere” account of the ontology of predication and change.

Set—Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be a key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later.