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Leibniz's Law of Continuity 1 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity 1 “A COMPLETE DENIAL OF THE CONTINUOUS”? LEIBNIZ'S LAW OF CONTINUITY “A Complete Denial of the Continuous”? Leibniz's Law of Continuity 1. Russell’s and Cantor’s charges of inconsistency As has often been noted, Leibniz’s philosophy was a virtual hymn to the continuous. His axiom that nullam transitionem fieri per saltum, “no transition occurs by a leap,” was central to his critique of the Cartesian natural philosophy, as he made clear in his correspondence with the Cartesian de Volder (G II168). And in the New Essays he writes: Nothing takes place suddenly, and it is one of my great and best confirmed axioms that nature never makes leaps. I call this the Law of Continuity… (1704/1981, 56) In nature everything happens by degrees, and nothing by leaps, and this rule regarding change is part of my law of continuity. But the beauty of nature, which insists upon perceptions which are distinct from one another, requires the appearance of leaps and musical cadences (so to speak) amongst phenomena … (1704/1981, 473; Russell 1900, 222) For this, however, as in many other instances, Bertrand Russell takes Leibniz to task for want of consistency, charging that “In spite of the law of continuity, Leibniz’s philosophy may be described as a complete denial of the continuous” (Russell 1900, 111). Russell has a point. He documents Leibniz’s denial that anything actual is continuous, and also his oft-repeated doctrine that “the continuum is ideal because it has indeterminate parts, whereas in the actual everything is determinate” (111). He also gives the following passage from Leibniz’s letter to de Volder in the appendix of leading passages for this section: Matter is not continuous but discrete, and actually infinitely divided, though no assignable part of space is void of matter. But space, like time, is something not substantial, but ideal, and consists in possibilities, or in an order of coexistents that is in 2 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity some way possible. And thus there are no divisions in it but such as are made by the mind, and the part is posterior to the whole. In real things, on the other hand, unities are prior to the multiplicity, and multiplicities exist only through unities. (The same holds of changes, which are not really continuous.) (To De Volder, 11 Oct, 1704; G II 278-79; Russell 1900, 245; Loemker 1976, 536) So actuals—simple substances and their changes—are not really continuous. Nor are bodies or their changes, since as composites they are simply infinite aggregates or multiplicities of monads: In actuals there is only discrete quantity, namely the multiplicity of monads or simple substances, although this 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 insofar as they are possible. (To De Volder, Jan 19, 1706; G II 282; Russell 1900, 245; Loemker 1976, 539) Yet this denial of continuity for substances and their changes seems to be incompatible with what Leibniz explicitly says elsewhere: Not only is everything that acts an individual substance, but also every individual substance acts continuously… (On Nature Itself, FW 215) I also take it for granted that every created thing is subject to change, and therefore the created monad as well; and indeed that such change is continuous in every one. (Monadology, 1714, §10, FW 269) 3 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity But continuity is found in time, extension, qualities and motion, and in fact in all natural changes, for these never take place by leaps (Initia Rerum Mathematicarum Metaphysica, 1715; GM VII 17-29) In what follows I hope to show that the incompatibility between these various statements is only apparent. In so doing I shall try to shed some light on Leibniz’s philosophy of the continuous. I shall not be attending to subtle changes in Leibniz’s views in his later years; in any case, I do not believe that his basic position on these matters underwent any profound change after the early 1680s. This is especially so of Leibniz’s interpretation of infinitesimals as fictions, which I believe he had already formulated in 1676 as a consequence of his syncategorematic interpretation of the infinite. To be sure, Leibniz did not succeed in explaining this interpretation clearly to his contemporaries, but I do not read his later tactical hesitations as evidence for changes of heart on the status of infinitesimals.1 Thus I shall not be trying to trace changes in his description of his position, nor how they were occasioned by external factors. In defence of such an unhistorical approach, I plead economy; its justification must rest with whatever sense I am adjudged to have made of Leibniz’s apparently contradictory remarks by adopting it. As starting point, I am going to examine a similar allegation of inconsistency concerning Leibniz’s doctrine of the actual infinite. Indeed, one can see a strong parallel between Russell’s remark already quoted above, that “In spite of the law of continuity, Leibniz’s philosophy may 1 Thus I must reluctantly dissent from the analysis given by Douglas Jesseph in his (1998), who takes it as “clear that the most natural formulation of the Leibnizian calculus makes it straightforwardly committed to the reality of infinitesimals”, and then sees Leibniz as developing the fictional treatment of infinitesimals “through the 1690s” in response to criticisms. Of course, I agree with Jesseph that in this period “Leibniz tries to settle on an interpretation of the calculus that can preserve the power of the new method while placing it on a satisfactory foundation”; I just do not see the fictional interpretation as new. I have similar disagreements with the way the Henk Bos frames the issue in his classic article (1974-74) on the Leibnizian calculus, as I shall explain below. Contra Bos (54), Leibniz did not introduce infinitesimals as “true quantities” that we was then obliged to treat as fictions; rather, he had already developed the syncategorematic interpretation, whose great advantage was that it did not commit him to actually existing infinitesimals, and yet provided an adequate foundation for the calculus. 4 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity be described as a complete denial of the continuous”, and a similar remark of Cantor’s about Leibniz’s philosophy of the infinite:2 Even though I have ... quoted many places in Leibniz’s works where he comes out against infinite numbers, ... I am still, on the other hand, in the happy position of being able to cite pronouncements by the same thinker in which, to some extent in contradiction with himself, he expresses himself unequivocally for the actual infinite (as distinct from the Absolute).3 Again, there is good textual evidence for the incompatibility, as Cantor documents. First there are the ecstatic pronouncements in favour, as in Leibniz’s letter to Foucher: I am so much for the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that it affects it everywhere, the better to indicate the perfections of its Author. Thus I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and that consequently the least particle ought to be considered as a world full of an infinity of different creatures (Leibniz 1693, G.I.416.) Then there are the denials that there can be such a thing as an infinite number, even juxtaposed with the embrace of the actual infinite, as in the New Essays: It is perfectly correct to say that there is an infinity of things, i.e. that there are always more of them than can be specified. But it is easy to demonstrate that there is no infinite 2 I should stress that Russell himself did not misinterpret Leibniz on the infinite. On the contrary, I believe he was the first to point out that there is no inconsistency between Leibniz’s upholding of the actual infinite and his denial of infinite number: “[the principle] that infinite aggregates have no number … is perhaps one of the best ways of escaping from the antinomy of infinite number” (Russell 1900, 117). He does, however, misconstrue Leibniz on infinite aggregates, and on his “deduction of monadism”: see (Arthur, 1989). 3 Georg Cantor, “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (1883), in Cantor (1932, 179). 5 “A Complete Denial of the Continuous”? Leibniz's Law of Continuity number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes. (Leibniz 1704/1981, 157, Russell 1900, 244) Cantor, of course, believed these views inconsistent. On the one hand he wished to call on Leibniz as a historical precedent for upholding the actual infinite; on the other, Leibniz’s denial of infinite number seemed not only to contradict this, but to be belied by Cantor’s own theory of the transfinite. Others have agreed with him: Rescher says that Cantor’s theory of transfinite numbers, point-set topology and measure theory “have shown that Leibniz’s method of attack was poor. Indeed, Galileo had already handled the problem more satisfactorily ...” (Rescher 1967, 111); Gregory Brown concurs (Brown 1998, 122-123; 2000, 23-24): Leibniz suffered a failure of nerve, and would have done better to have embraced infinite number and anticipated Cantor in the process. 2. The syncategorematic infinite Against this view, however, I have argued elsewhere (1999, 2001a, b, c) that there is no inconsistency in Leibniz’s account of the actual infinite.
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