Leibniz on Density and Sequential or Cauchy Completeness

Abstract

The objective of this paper is to see what the basic characteristics of continuity are from a modern point of view, and whether we can find traces of them in Leibniz’s philosophical writings. It will be argued that not only density but also what we would today call sequential or Cauchy completeness is in one sense present in Leibniz’s treatment of the continuous. We will use what mathematicians call the “real line” as the paradigm for our investigation.

The responsibility for all translations is mine, though I have made use of existing translations along the way. The following abbreviations are used throughout the text A: The Leibniz-Clarke Correspondence, ed. H. G. Alexander (Manchester: Manchester University Press, 1970); BC: G.W. Leibniz philosophische Werke, ed. E. Cassirer, trans. A. Buchenau, 2 vols. (Leipzig: F. Meiner, 1924); G: Die philosophischen Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, 7 vols. (Berlin: Weidman, 1875–90); L: Gottfried Wilhelm Leibniz: Philosophical Papers and Letters, ed. and trans. Leroy E. Loemker, 2d ed. (Dordrecht: Reidel, 1969); NE: G.W. Leibniz: New Essays on Human Understanding, ed. and trans. Peter Remnant and Jonathan Bennet (Cambridge: Cambridge University Press, 1981); W: Leibniz Selections, ed. P. P. Wiener (New York: Charles Scribner’s Sons, 1951).