Leibniz’s Analysis of Change: Vague States, Physical Continuity, and the Calculus

One of the most puzzling features of Leibniz’s deep metaphysics is the apparent contradiction between his claims (1) that the law of continuity holds everywhere, so that in particular, change is continuous in every monad, and (2) that “changes are not really continuous,” since successive states contradict one another. In this paper I try to show in what sense these claims can be understood as compatible. My analysis depends crucially on Leibniz’s idea that enduring states are “vague,” and abstract away from further changes occurring within them at a higher resolution—consistently with his famous doctrine of "petites perceptions." As Leibniz explains further in a recently transcribed unpublished manuscript, these changes are dense within any actual duration, which is conceived as actually divided by them into states that are syncategorematically infinite in number and unassignably small. The correspondence between these unassignably small intervals between changes and the differentials of his calculus allows processes to be conceived as continuous, despite the discontinuity of the changes that occur in actuality.