Cohesion, Division and Harmony: Physical Aspects of Leibniz's Continuum Problem (1671-1686).(17th-century German physicist and philosopher Gottfried Wilhelm Leibniz)

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 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 way: it thus encompasses the laws that govern the series of states of the thing, as well as an endeavor or appetition, taking it into the next state of the series. Finally, this (immaterial) principle of unity of any given substance is the complement of its primitive passive power, or (material) principle of diversity, and substance is constituted by these primitive active and passive powers together.

I would not presume such a highly condensed account to be readily intelligible, and indeed there is much that is missing (such as the whole question of the status of infinitesimals and the calculus, Leibniz's philosophy of the infinite, and the relation of mathematics to reality), but I hope it is enough at least to set the scene, and to give some of the flavor and richness of Leibniz's assault on the continuum problem. Certainly I have no intention of giving a complete account of the latter here. But I hope to say enough to illuminate some of the main turns Leibniz took en route to his own distinctive solution.

Leibniz did once give his own account of his exit from the labyrinth, as a kind of proem to the discussion of dynamics in the second dialogue of the Phoranomus: Or, on Power and the Laws of Nature (July 1689), written just prior to his Dynamica.(2) This is interesting not least for the insight it gives into his changing views on the status of atoms, absolute space, and motion (on which I shall quote him later). Notably too, though, Leibniz there identifies a single clew that led him out of his labyrinth, in the words: "Accordingly I discovered no other Ariadnean thread that would finally extricate me from that labyrinth than the calculation of powers, assuming this metaphysical principle, that the total effect must always be equal to its full cause" (Leibniz [1689] 1991, p. 811).(3)

Of course, it may be that the labyrinth Leibniz is referring to here is not that of the continuum pure and simple, despite his references to many features that are constitutive of his solution to the latter.(4) But if it is the latter he means, I think we would do well not to take this description of the "Ariadne's thread" too literally; at any rate, I have found no convincing evidence that Leibniz's exit from the labyrinth of the continuum was immediately illuminated by this metaphysical principle, which he first announced in the summer of 1676, whereas he persisted in his labors in that labyrinth (even on a conservative estimate) until around 1683-86, when the mature solution outlined above emerged in a series of papers written at that time. On studying his papers from 1676 to 1686 that are of immediate relevance to his solution, one finds not so much one thread that delivers Leibniz from his labyrinth, as several converging strands of thought, some predominantly mathematical, others deriving from metaphysical, epistemological or physical concerns, all tugging on each other in an extremely complex web of mutual influence. Nevertheless, whatever Leibniz's rhetorical purposes in so promoting the role of the principle of equality of cause and effect, it is unlikely that the account he presents in the Phoranomus of the emergence of his views is wholly misleading. It gives a point of view, at any rate, and to the extent that it highlights the role of physical considerations in Leibniz's solution, it is very congenial to my purposes here. Accordingly, I shall structure my essay around the narrative link provided by this account, although I shall not center it on the principle of the equality of cause and effect, which I shall only treat in passing. Nor will I have the space here to examine the emergence of Leibniz's doctrine of the conservation of vis viva, or its connection with the Principle of Continuity, despite the obvious relevance of these matters.

The Problem of Cohesion

According to his testimony in the Phoranomus, Leibniz first began to wrestle with the problems of the continuum as an adolescent, when, emerging from "the prickly thornbrakes of the scholars into the pleasanter pastures of more recent philosophy," he found himself "wonderfully taken in by that flattering easiness of understanding, by which it seemed that everything which had previously been shrouded in murky notions could be comprehended by a lucid imagination. Thus after deliberating on this long and hard, I finally came to condemn forms and qualities in material things, and reduced everything to purely mathematical principles; but since I was not yet versed in geometry, I persuaded myself that the continuum consists of points, and that a slower motion is one interrupted by small intervals of rest" (Leibniz [1689] 1991, p. 803).

But Leibniz's attachment to these doctrines did not long survive his university years. In his first major attack on the continuum problem,(5) the Theoria Motus Abstracti of 1671, Leibniz disowns the thesis that motion is interrupted by rests, although the composition of the continuum from points of a certain kind is upheld. The linchpin of this account is the notion of conatus or endeavor, a notion Leibniz informs us he inherits under the name tendentia (tendency) from Weigel, his teacher at Jena, as well as (more obviously) from Hobbes.(6) The latter had attempted to assimilate Cavalieri's "indivisibles" to Euclid's points by redefining a point as "that whose quantity is not considered" (Hobbes 1668, pp. 98-99). Building on these insights, but at the same time heavily amending them, Leibniz defines a point as "that which has no extension, that is, that whose parts are indistant, whose magnitude is inconsiderable, unassignable, is smaller than can be expressed by a ratio to another sensible magnitude unless the ratio is infinite, smaller than any ratio that can be given.(7) Unlike a minimum, a "thing which has no magnitude or part,"(8) whose existence Leibniz denied, a point would have a magnitude: only it would not be assignable. Endeavor is now defined by analogy: "Endeavor is to motion as a point is to space, or as one to infinity, for it is the beginning and end of motion."(9) Thus whereas Hobbes's endeavors are true motions, being motions through a line whose quantity is not considered, Leibniz's are not true motions, but what might be called non-Archimedean infinitesimals: they-have a quantity smaller than any that can be assigned (i.e., than can be expressed by a finite ratio to a finite motion).

So far this seems exclusively metaphysical. Indeed, one of the main motivations for his work in this period is to prove that "bodies left to themselves"--i.e, bodies with no minds--would dissolve to nothing,(10) But for Leibniz metaphysics was never exclusive of physics, since it provided the latter's foundation, and he had long had his eye on a problem in physics that was grist for his mill about the inadequacy of bodies by themselves. This was the problem of cohesion, how it is that bodies hold …