RICHARD T. W. ARTHUR RUSSELL’S CONUNDRUM: ON THE RELATION OF LEIBNIZ’S MONADS TO THE CONTINUUM INTRODUCTION In his influential book on Leibniz (1900), Bertrand Russell often let his new-found anti-Hegelianism come between him and his subject, attacking doctrines held by the likes of Bradley and MacTaggart in the belief that he was attacking one of their main sources. This is particularly true of his chapter on Leibniz’s treatment of the continuum, where after rather wildly accusing Leibniz of courting “the essentially Hegelian view that abstraction is falsification”,1 he proceeds to the charge that “his whole deduction of Monadism from the difficulties of the continuum, seems to bear a close analogy to a dialectical argument” (110). This is intended as a hostile criticism, as Russell makes clear, explaining that in calling the argument “dialectical” he means that Leibniz infers the reality of monads “from premisses admittedly false, and inconsistent with each other”. He charges that Leibniz’s argument, “in obtaining many reals, assumes that these are parts of matter — a premiss which it is compelled to deny in order to show that the reals are not material”.2 Russell presents Leibniz’s argument as follows: “The general premiss is: Since matter has parts, there are many reals. Now the parts of matter are extended, and owing to infinite divisibility, the parts of the extended are always extended. But since extension means repetition, what is repeated is ultimately not extended. Hence the parts of matter are ultimately not extended. Therefore it is self-contradictory to suppose that matter has parts. Hence the many reals are not parts of matter. (The argument is stated almost exactly in this form in G.VII.552.)” I doubt whether many commentators today would be prepared to accept such a “dialectical” reading of Leibniz’s reasoning. But putting aside for now the this mooted Hegelian logic, it is clear that Russell has uncovered an important problem for the interpretation of one of Leibniz’s most central concerns. For it is hard to see how monads, which Leibniz describes as mind-like units containing force and perception, could be derived from difficulties with the notion of continuous matter; or how their existence contributes towards a resolution of these difficulties. Yet Leibniz is adamant that his doctrine of substance is a solution to these difficulties, the only exit he can find from the “labyrinth of the continuum”.3 This problem has tended to dog Leibniz’s interpreters from Euler through McGuire, and few have been able to avoid finally falling into the same kind of difficulty as Russell. For no matter how subtle their attempts to explain the relation between monads and the continuum, the commentators I shall consider here all end up subscribing to some version of the view that monads are actual parts of extended material bodies. This, then, is Russell’s conundrum: if monads are not parts of continuous bodies, it is difficult to see how Leibniz could have supposed that their introduction would solve the problems of the continuum; whereas if they are, his position seems to lapse into inconsistency. I cannot, within the confines of this paper, present a full exposition of Leibniz’s solution to the problem of continuum, which would involve tracing both its origin in the words of Suárez, Fromond and others, and the respective influences of Hobbes, Descartes, Galileo, Aristotle and Plato; and also showing how his views on substance, mind and soul developed in harmony with his progress in the mathematics of the infinite, and issued in his mature dynamics.4 What I want to tackle here is altogether more modest: namely, to identify the extent to which expositions of Leibniz’s solution have been vitiated by a misidentification of monads as matter’s actual parts, a misreading which, far from leading out of Leibniz’s labyrinth, takes us straight into the dead-end of Russell’s conundrum. 1. EULER: ATOMS OF SUBSTANCE The most straightforward misinterpretation of how monads are supposed to ground the continuum is the naive atomist one, which reads Leibniz’s “atoms of substance” as indivisible parts of matter: as the analogue in the continuum of (material) substance to indivisibles in the mathematical continuum. Although it is not a position any self- respecting commentator could hold today, its consideration will set the stage for an appreciation of the difficulties to come. This was how Leonhardt Euler understood the monadology, although in fairness to him what he had in mind likely has more to do with the philosophy popularized by Christian Wolff than Leibniz. Consequently, in one of the more ironic episodes in the history of philosophy, Euler set about attacking this “solution” to the problems of the continuum in his Letters to a German Princess of 1761 — with arguments very reminiscent of Leibniz’s own. For Euler supposed monads to be parts of bodies that result from a “limited division” (Euler 1843, pp. 48—49), “ultimate particles which enter into the composition of bodies” (p. 39). Naturally he had little difficulty in making hay with this straw man, drawing out the contradiction between this idea of a limited division and the fact (acknowledged by Leibniz) that matter is infinitely divis- ible, and therefore divisible without limit. Leibniz, of course, was adamant that although the reality of any phenomenal body is constituted by an infinite aggregate of monads, these monads, having no extension, are not parts out of which any extended body can be composed. But, strictly speaking, matter is not composed of constitutive unities, but results from them, since matter or extended mass is nothing but a phenomenon founded in things. like a rainbow or mock-sun, and all reality belongs only to unities ... Substantial unities, in fact, are not parts but foundations of phenomena. (To de VoIder, 30/6/1704: G.ll.268) Euler’s interpretation is also flatly contradicted by Leibniz’s repeated assertions that phenomena are not only infinitely divided, they have no smallest part (G.II.268, G.II.305, GM.II.157, etc.); and that monads, far from being such smallest parts of the bodies, are presupposed in as small a part of a body as we want to consider (G.II.135, G.II.301, G.ll.305, G.I1.436, G.IJ.329). So much for the naive atomist reading. Although it had the merit of clarity in suggesting a plausible connection between Leibniz’s monadology and his infinitesimal calculus, it is evident from the above objections that no such simple connection is to be expected, but something considerably more labyrinthine. 2. RESCHER: IDEAL AND REAL WHOLES But before proceeding to more elaborate interpretations of the relation of monads to the continuum, it is worth enquiring how in the face of this uncompromising preliminary conclusion, there could remain any justification for interpreting monads as actual parts of extended phenomena. The answer is that the tendency to interpret monads this way has been fostered largely by Leibniz’s frustratingly terse statements of his solution. These usually consist in the remark that in ideals the whole is prior to its parts, which are merely potential or indeterminate, whereas in actuals the parts are real and determinate, and prior to the whole; and that confusion about the continuum could be avoided if we would only heed this distinction between determinate and indeterminate parts, and refrain from looking for actual parts in ideal wholes, and vice versa. Here it is natural to interpret these “actual parts” of the real whole as the monads that give rise to a phenomenal body, as the following passage would suggest: It is also obvious from what I have said that in Actuals there is only discrete Quantity, namely the multiplicity of monads or simple substances, a quantity which is in fact greater than any given number for any aggregate that is sensible or corresponds to phenomena. But continuous Quantity is something ideal, which pertains to possibles and to actuals considered as possibles. The continuum, that is, involves indeterminate parts, whereas in actuals there is nothing indefinite — indeed, in them any division that can be made, is made. Actuals are composed as is a number out of unities, ideals as a number out of fractions: the parts are actual in the real whole, not in the ideal whole. In fact we are confusing ideals with real substances when we seek actual parts in the order of possibles, and indeterminate parts in the aggregate of actuals, and we entangle ourselves in the labyrinth of the continuum and inexplicable contradictions. (To de VoIder, 19/1/1706: G.II.282) Thus just as any number can be divided into parts in innumerable different ways, so that it has no parts that can be identified as the parts out of which is composed — no determinate parts — so too, for the same reason, no continuous quantity such as time or Cartesian matter can have determinate parts either. But it is axiomatic for Leibniz that anything actual must have such determinate parts, just as integers are made up out of a determinate number of units. If therefore follows that continuous quantities, such as time and Cartesian matter, must be entia rationis or ideal entities. But passages such as these only seem to push us inexorably toward a more comprehensive atomism. For if continuity is to be found only in ideal wholes, it seems that everything real must be merely discrete. But in this case how does Leibniz’s position constitute a solution to the problem of continuum? Nicholas Rescher explains things as follows. Leibniz, he contends, locates the source of the difficulties of the continuum in the “failure to distinguish between the ideal or phenomenal and the real or actual” (1967, p.