- Cohesion, Division and Harmony: Physical Aspects of Leibniz’s Continuum Problem (1671–1686)

Leibniz often claimed that his struggles with the problem of the composition of the continuum and its solution were formative for his theory of substance. As has long been recognized, mathematical considerations— especially his creation of the differential calculus and the work on the summation of infinite series—were highly relevant. But the role of physical considerations has been comparatively neglected, and it is this I want to address in this paper by discussing three topics from physics which appear to have been particularly important for Leibniz in formulating his solution: the problem of cohesion, the problem of “the solid and the liquid,” and the implications of the relational nature of motion.

Of course, if the composition of the continuum is understood as a purely mathematical problem, one may well wonder what bearing physical considerations could have on it. But for Leibniz and his contemporaries, the problem was not restricted to the composition of purely mathematical entities—such as whether a line is composed out of points or infinitesimals or neither—but was understood as applying to all existing quantities and their composition. In this wider sense, the continuum problem is: what (if any) are the first elements of things and their motions? Are there atoms or indivisible elements of substance? Is space composed of points, or time of moments? These metaphysical questions are in turn linked to some pressing problems of physics: for Descartes, the actual division of (at least some parts of) matter into indefinitely small particles is a necessary condition for motion through unequal spaces in the plenum; for Galileo, the supposition of indivisible voids between the indivisible parts of matter explains the cohesion of bodies; for Hobbes, the Galilean analysis of the continuous motion of bodies into infinite degrees requires a foundation in terms of **[End Page 110]** endeavors, infinitely small (but unequal) beginnings of motion, and these same endeavors are the cornerstones of his materialist psychology.

Leibniz inherits this wider conception of the continuum problem, and it is the whole cluster of problems concerning infinite divisibility, the actual infinite, the existence of atoms of matter or substance, and the analysis of continuous space, time, and motion to which his characteristic allusions to the “labyrinth of the continuum” refer. ^{1} Given this complexity it is not at all easy to summarize what Leibniz took his solution to the problem to be. But in broad brush strokes, it involves at least the following: insofar as anything is continuous, its parts are indiscernible from one another, and thus indefinite. The continuum is therefore not an actually existing thing, a whole composed of determinate parts, but an abstract entity. In existing things, by contrast, the parts are determinate, and are prior to any whole that they compose. Matter, for example, considered abstractly (i.e., as primary matter), is a homogeneous, continuous whole, consisting in a pure potentiality for division; but taken concretely (i.e., as secondary matter), it is at any instant not only infinitely divisible, but actually infinitely divided by the differing motions of its parts. Thus no part of matter, however small, remains the same for longer than a moment; even shape or figure is evanescent, and a body with an enduring figure is something imaginary. Similarly, there is no stretch of time, however small, in which some change does not occur. Change, on the other hand, can only be understood in bodies as an aggregate of two opposed states at two contiguous or “indistant” moments; but again nothing can remain in precisely the same state for longer than a moment, so the supposed enduring states of bodies must themselves be to some extent imaginary. Thus the perduring element in matter is not something material, i.e., explicable in terms of the extended, motion and figure. There must, however, be such a perduring element in any part of matter however small, which is the principle of all the changes occurring in it. This immaterial principle, Leibniz concludes, consists in a primitive force of acting. It bestows unity on a substance by taking that thing through all its states in a lawful...