THE INFINITE AND THE CONTINUOUS
The idea of the infinite as Leibnitz most often understood it, which, let us never forget, was merely that of a multitude surpassing all number, sometimes appears under the aspect of a 'discontinuous infinite', as in the case of so-called infinite numerical sequences; but its most usual aspect, and also its most important one as far as the significance of the infinitesimal calculus is concerned, is that of the 'continuous infinite'. In this regard it is useful to recall that when Leibnitz, beginning the research that at least according to what he himself said, would lead to the discovery of his method, was working with sequences of numbers, he at first considered only differences that are 'finite' in the ordinary sense of the word; infinitesimal differences appeared to him only when there was a question of applying numerical discontinuity to the spatial continuum. The introduction of differentials was therefore justified by the observation of a certain analogy between the respective kinds of variability within these two modes of quantity; but their infinitesimal character arose from the continuity of the magnitudes to which they had to be applied, and thus, for Leibnitz, a consideration of the 'infinitely small' is closely linked to that of the 'composition of the continuous'.
Taken 'rigorously', 'infinitely small', would be partes minimae of the continuous, as Bernoulli thought; but clearly the continuous, insofar as it exists as such, is always divisible, and consequently it could not have partes minimae. 'Indivisibles' cannot even be said to be parts of that with respect to which they are indivisible, and 'minimum' can be understood here only as a limit or extremity, not as
an element: 'Not only is a line less than any surface,' Leibnitz says, 'it is not even part of a surface, but merely a minimum or an extremity'; [1] and from his point of view this assimilation between extremum and minimum can be justified by the 'law of continuity', in that according to him it permits 'passage to the limit', as we shall see later. As we have said already, the same holds for a point with respect to a line, as well as for a surface with respect to a volume; on the other hand, the infinitesimal elements must be parts of the continuous, without which they could not even be quantities; and they can be so only on condition of not truly being 'infinitely small', for then they would be nothing other than partes minimae [smallest parts] or 'final elements', of which the very existence implies a contradiction in regard to the continuous. Thus the composition of the continuous prevents infinitely small quantities from being anything more than simple fictions; but from another point of view, it is nevertheless precisely the existence of this continuity that makes them 'well-founded fictions', at least in Leibnitz's eyes: if 'within the realm of geometry they may be treated as if they were perfectly real,' this is because extension, which is the object of geometry, is continuous; and, if it is the same with nature, this is because bodies are likewise continuous, and also because there is also continuity in all phenomena such as movement, of which these bodies are the seat, and which are the objects of mechanics and physics. Moreover, if bodies are continuous, this is because they are extended and participate in the nature of extension; and similarly, the continuity of movement, as well as of the various phenomena more or less directly connected to it, derives essentially from its spatial character. Thus the continuity of extension is ultimately the true foundation of all other continuity that is observed in corporeal nature; and this, moreover, is why in introducing an essential distinction that Leibnitz did not make in this regard, we specified that in reality one must attribute
the property of 'indefinite divisibility' not to 'matter' as such, but rather to extension.
Here we need not examine the question of other possible forms of continuity, independent of its spatial form; indeed, one must always return to the latter when considering magnitudes, and its consideration thus suffices for all that pertains to infinitesimal quantities. We should, however, include together with it the continuity of time, for contrary to the strange opinion of Descartes on the subject, time really is continuous in and of itself, and not merely with respect to its spatial representation in the movement used to measure it. [2] In this regard, one could say that movement is as it were doubly continuous, for it is so in virtue both of its spatial and of its temporal condition; and this sort of combination of space and time, from which movement results, would not be possible were the one discontinuous and the other continuous. This consideration also allows the introduction of continuity into various categories of natural phenomena that pertain more directly to time than to space, although occurring in both, as, for example, with any processes of organic development. As for the composition of the temporal continuum, moreover, one could repeat everything said concerning the composition of the spatial continuum, and in virtue of this sort of symmetry which, as we have seen, exists in certain respects between space and time, one will arrive at strictly analogous conclusions; instants conceived of as indivisible are no more parts of duration than are points of extension, as Leibnitz likewise recognized, and here again we have a thesis with which the Scholastics were quite familiar; in short, it is a general characteristic of all continuity that its nature precludes the existence of 'final elements'.
All that we have said up to this point sufficiently shows in what sense one may understand that from Leibnitz's point of view, the continuous necessarily embraces the infinite; but we cannot, of course, suppose that there is any question of an 'actual infinity', as if all possible parts are effectively given whenever a whole is given; nor is there any question of a true infinity, which any determination whatsoever would exclude, and which consequently cannot be
implied by the consideration of any particular thing. Here, however, as in every case in which the idea of an alleged infinite presents itself, different from the true metaphysical Infinite, but in itself representing something other than a pure and simple absurdity, all contradiction disappears, and with it all logical difficulty, if one replaces the so-called infinite with the indefinite, and if one simply says that all continuity, when taken with respect to its elements, embraces a certain indefinitude. It is also for lack of having made this fundamental distinction between the Infinite and the indefinite that some people have mistakenly believed it impossible to escape the contradiction of a determined infinite except by rejecting the continuous altogether and replacing it with the discontinuous; thus Renouvier, who rightly denied the mathematical infinite, but to whom the idea of the metaphysical Infinite was nevertheless completely foreign, believed that the logic of his 'finitism' obliged him to go so far as to accept atomism, thus falling prey to a concept no less contradictory than the one he wished to avoid, as we saw earlier.