'INFINITE DIVISION' OR INDEFINITE DIVISIBILITY
For Leibnitz, not only is matter divisible, but all its parts are 'actually sub-divided without end, ... each part into parts, each having some movement of its own';[1] and he emphasizes this point above all in order to offer theoretical support to the concept we last explained: 'It follows from the actual division that in every part of matter, however small it might be, there is as it were a world consisting of innumerable creatures. [2] Bernoulli likewise supposes this actual division of matter in partes numero infinitas [into infinitely many parts], but he draws from it conclusions Leibnitz did not accept: 'If a finite body,' he says, 'has parts infinite in number, I have always believed, and still do, that the smallest of these parts must have an unassignable, or infinitely small, ratio to the whole';[3] to which Leibnitz responds: 'Even if one agrees that there is no portion of matter that is not actually divided, one does not, however, arrive at indivisible elements, or at parts smaller than all others, or infinitely small, but only at ever smaller parts, which, however, are ordinary quantities, just as in augmentation one arrives at ever greater quantities. [4] Thus it is the existence of minimae portiones [smallest parts], or of 'final elements', that Leibnitz contests;
for Bernoulli, on the contrary, it seems clear that actual division implies the simultaneous existence of all the elements in question, just as, if an 'infinite' sequence be given, all of its constituent terms must be given simultaneously, which implies the existence of a terminus infinitesimus [infinitesimal limit]. But for Leibnitz the existence of this limit is no less contradictory than that of an 'infinite number', and the notion of a smallest of numbers, or a fractio omnium infima [a part smaller than all others], no less absurd than that of a greatest of numbers. What he considers to be the 'infinity' of a sequence is characterized by the impossibility of arriving at a final term, and matter would likewise not be 'infinitely' divided if this division could ever be completed and end at 'final elements'; and it is not only that we could not in fact ever arrive at these final elements, as Bernoulli concedes, but that they should not exist in nature at all. There are no indivisible corporeal elements, or 'atoms' in the proper sense of the word, any more than there are indivisible fractions that cannot yield ever smaller fractions in the numerical order, or, in the geometric order, linear elements that cannot be divided into ever smaller elements.
In all of this Leibnitz basically takes the word 'infinite' in exactly the same sense as he does when speaking of an 'infinite multitude'; for him, to say of any sequence, including that of the whole numbers, that it is infinite is not to say that it must come to a terminus infinitesimus or an 'infinite number', but on the contrary that it must have no final term, since its terms are plus quam numero designari possint [more than can be numbered], that is, they constitute a multitude that surpasses all number. Similarly, if one can say that matter is infinitely divided, this is because any one of its portions, however small, always encloses such a multitude; in other words, matter does not have partes minimae [smallest parts] or simple elements, it is essentially a composite: 'It is true that simple substances, that is, those that do not exist by aggregation, really are indivisible, but they are immaterial, and are only principles of action. [5] It is in the sense of an innumerable multitude-which, moreover, is the sense Leibnitz most commonly employs-that the idea of the
so-calledinfinite can be applied to matter, to geometric extension, and in general to the continuous, taken in relation to its composition; besides, this sense is not exclusive to the infinitum continuum [continuous infinite] but extends to the infinitum discretum [discrete infinite] as well, as we have seen both in the example of the multitude of all the numbers and in that of the 'infinite sequence'. This is why Leibnitz was able to say that a magnitude is infinite insofar as it is 'inexhaustible', which means that 'one can always take a magnitude as small as one likes', and, 'it remains true, for example, that 2 is as much as 1/1+1/2+1/4+1/8+1/16+1/32+..., which is an infinite series, comprised at once of all fractions with a numerator of 1 and denominators in double geometric progression, although only ordinary numbers are ever used, that is, one never introduces any infinitely small fraction, or one with an infinite number for its denominator. [6] Moreover, what was just said allows us to understand how Leibnitz, while affirming that the infinite, as he understands it, is not a whole, nevertheless could apply this idea to the continuous: a continuous set, as any given body, indeed constitutes a whole, even what we above called a true whole, logically anterior to its parts and independent of them, but it is obviously always finite as such; it is therefore not with respect to the whole that Leibnitz is able to call it infinite, but only with respect to its parts into which it can be divided, and only insofar as the multitude of these parts effectively surpasses every assignable number. This is what one might call an analytical conception of the infinite, since in fact, it is only analytically that the multitude in question is inexhaustible, as we shall explain later.
If we now question the worth of the idea of 'infinite division', we must recognize that, as with the 'infinite multitude', it contains a certain portion of truth, though its manner of expression is anything but safe from criticism. First of all, it goes without saying that, in accordance with all that we have explained so far, there can be no question of infinite division, but only of indefinite division; and on the other hand it is necessary to apply this idea not to matter in general, which would perhaps have no meaning, but only to bodies, or
to corporeal matter if one insists on speaking of 'matter' here, in spite of the extreme obscurity of the notion, and the many equivocations to which it gives rise. [7] In fact, it is to extension that divisibility properly pertains, not to matter, in whatever sense this is understood, and the two could only be confused were one to adopt the Cartesian concept, according to which the nature of bodies consists essentially and uniquely in extension, a concept, moreover, that Leibnitz also did not admit. If, then, all bodies are necessarily divisible, this is because they possess extension, and not because they are material. Now let us again recall that extension, being something determined, cannot be infinite; hence, it obviously cannot imply any possibility more infinite than itself; but as divisibility is a quality inherent to the nature of extension, its limitations can only come from this nature itself; as long as there is extension, it is always divisible, and one can thus consider its divisibility to be truly indefinite, its indefinitude being conditioned, moreover, by that of extension. Consequently, extension as such cannot be composed of indivisible elements, for these elements would have to be extensionless to be truly indivisible, and a sum of elements with no extension can no more constitute an extension than a sum of zeros can constitute a number; this is why, as we have explained elsewhere, [8] points are not the elements or parts of a line; the true linear elements are always distances between points, which latter are only their extremities. Moreover, Leibnitz himself envisaged things thus in this regard, and according to him, this is precisely what marks the fundamental difference between his infinitesimal method and Cavalieri's 'method of indivisibles', namely, that he does not consider a line to be composed of points, or a surface of lines, or a volume of surfaces: points, lines, and surfaces are here only limits or extremities, not constituent elements. It is indeed obvious that points, multiplied by any quantity at all, can never produce length, since, rigorously speaking, they are null with respect to length; the true elements of a magnitude must always be of the same nature as the magnitude, although incomparably less: this leaves no room for
'indivisibles', and what is more, it allows us to observe in the infinitesimal calculus a certain law of homogeneity, which implies that ordinary quantities and infinitesimal quantities of various orders, although incomparable among themselves, are nonetheless magnitudes of the same species.
From this point of view one can say in addition that the part, whatever it be, must always preserve a certain 'homogeneity' or conformity of nature with the whole, at least insofar as the whole is considered able to be reconstituted by means of its parts, by a procedure comparable to that used in the formation of an arithmetical sum. Moreover, this is not to say that no simple thing exists in reality, for composites can be formed, starting from their elements, in a way completely different from this; but then, to speak truly, these elements are no longer properly 'parts', and as Leibnitz recognized, they can in no way be of a corporeal order. What is indeed certain is that one cannot arrive at simple, that is, indivisible, elements without departing from the special condition that is extension; the latter could not be resolved into such elements without ceasing to be as extension. It immediately follows that there cannot exist indivisible corporeal elements, as this notion implies a contradiction; for indeed, such elements would have to be without extension, and then they would no longer be corporeal, for by very definition the word 'corporeal' necessarily entails extension, although this is not the whole nature of bodies; thus, despite all the reservations we must make in other regards, Leibnitz is at least entirely right in his position against atomism.
But until now we have spoken only of divisibility, that is to say the possibility of division; must we go further and admit with Leibnitz an 'actual division'? This idea is also not exempt from contradiction, for it amounts to supposing an entirely realized indefinite and on that account is contrary to the very nature of indefinitude, which, as we have said, is always a possibility in the process of development, hence essentially implying something unfinished, not yet completely realized. Moreover, there is in fact no reason to make such a supposition, for when presented with a continuous set we are given the whole, not the parts into which it can be divided, and it is only we who conceive that it is possible for us to divide this whole
into parts capable of being rendered smaller and smaller so as to become less than any given magnitude, provided the division be carried far enough; in fact, it is consequently we who realize the parts, to the extent that we effectuate the division. Thus, what exempts us from having to suppose an 'actual division' is the distinction we established earlier on the subject of the different ways of envisaging a whole: a continuous set is not the result of the parts into which it is divisible but is on the contrary independent of them, and, consequently, the fact that it is given to us as a whole by no means implies the actual existence of those parts.
Likewise, from another point of view and passing on to a consideration of the discontinuous, we can say that if an indefinite numerical sequence is given, this in no way implies that all the terms it contains are given distinctly, which is impossible precisely inasmuch as it is indefinite; in reality, to give such a sequence is simply to give the law that enables one to calculate the term occupying a determined position, or, for that matter, any position whatsoever within the sequence. [9] If Leibnitz had given this answer to Bernoulli, their discussion on the existence of the terminus infinitesimus would thereby have been brought to an immediate close; but he would not have been able to do so without logically being led to renounce his idea of 'actual division', unless he were to deny all correlation between continuous and discontinuous modes of quantity.
Be that as it may, as far as the continuous is concerned at any rate, it is precisely in the 'indistinction' of its parts that we can see the
root of the idea of the infinite such as it was understood by Leibnitz, since, as we said earlier, this idea always carries with it a certain amount of confusion; but this 'indistinction', far from presupposing a realized division, tends on the contrary to exclude it, even apart from the completely decisive reasons we have just noted. Therefore, even if Leibnitz's theory is right insofar as it is opposed to atomism, it must be corrected elsewhere if it is to correspond to truth; the 'infinite division of matter' must be replaced by the 'indefinite divisibility of extension'; here, in its briefest and most precise expression, is the conclusion to which all the considerations we have just set forth ultimately lead.