VARIOUS ORDERS OF INDEFINITUDE
The logical difficulties, and even contradictions which mathematicians run up against when they consider 'infinitely great' or 'infinitely small' quantities that differ with respect to one another, and even belong to different orders altogether, arise solely from the fact that they regard as infinite that which is simply indefinite. It is true that in general they do not seem very concerned with these difficulties, but they exist nonetheless, and are no less serious for all that, as they cause the science of mathematics to appear as if full of illogicalities, or, if one prefer, of 'para-logicalities', and such a science loses all real value and significance in the eyes of those who do not allow themselves to be deluded by words. Here are some examples of the contradictions introduced by those who would allow the existence of infinite magnitudes, when they apply this notion to geometric magnitudes: if a straight line is considered to be infinite, its infinitude must be less, and even infinitely less, than the infinitude constituted by a surface such as a plane, in which both that line and an infinite number of others are also contained, and the infinitude of the plane will in turn be infinitely less than that of three-dimensional space. The very possibility of the coexistence of all of these would-be infinities, some of which are supposed to be infinite to the same degree, others to different degrees, suffices to prove that none of them can be truly infinite, even apart from any consideration of a more properly metaphysical order; indeed, as these are truths which we cannot emphasize enough, let it be said again: it is obvious that if one supposes a plurality of distinct infinites, each will have to be limited by the others, which amounts to
saying that they will exclude one another. Moreover, to tell the truth, the 'infinitists', for whom this purely verbal accumulation of an 'infinity of infinities' seems to produce a kind of 'mental intoxication', if such an expression be permissible, do not retreat in face of such contradictions, since, as has already been said, they see no difficulty in asserting that various infinite numbers exist, and that consequently one infinity can be greater or smaller than another; but the absurdity of such utterances is only too obvious, and the fact that they are commonly used in contemporary mathematics changes nothing, but only shows to what extent the sense of the most elementary logic has been lost in our day. Yet another contradiction, no less blatant than the last, is to be found in the case of a closed, hence obviously and visibly finite, surface, which nevertheless contains an infinite number of lines, as, for example, a sphere, which contains an infinite number of circles; here we have a finite container, of which the contents would be infinite, which is likewise the case, moreover, when one maintains, as did Leibnitz, the 'actual infinity' of the elements of a continuous set.
On the contrary, there is no contradiction in allowing the coexistence of a multiplicity of indefinite magnitudes of various orders. Thus a line indefinite in a single dimension can in this regard be considered to constitute a simple indefinitude of the first order; a surface, indefinite in two dimensions, and embracing an indefinite number of indefinite lines, will then be an indefinitude of the second order; and three-dimensional space, which embraces an indefinite number of indefinite surfaces, will similarly be an indefinitude of the third order. Here it is essential to point out once again that we said the surface embraces an indefinite number of lines, not that it is constituted by an indefinite number of lines, just as a line is not composed of points, but rather embraces an indefinite multitude of them; and it is again the same in the case of a volume with respect to its surfaces, three-dimensional space being itself none other than an indefinite volume. This, moreover, is basically what we said above on the subject of 'indivisibles' and the 'composition of the continuous'; it is questions of this kind that, precisely by reason of their complexity, most make one aware of the necessity of rigorous language. Let us also add in this regard that if from a certain point
of view one can legitimately consider a line to be generated by a point, a surface by a line, and a volume by a surface, this essentially presupposes that the point, the line, or the surface be displaced through a continuous motion, embracing an indefinitude of successive positions; and this is altogether different from considering each of these positions in isolation, that is, regarding the points, lines, and surfaces as fixed and determined, and as constituting the parts or elements of the line, the surface, or the volume, respectively. Likewise, but inversely, when one considers a surface to be the intersection of two volumes, a line the intersection of two surfaces, and a point the intersection of two lines, these intersections must not, of course, by any means be conceived of as parts common to the volumes, surfaces, or lines; they are only limits or extremities of the latter, as Leibnitz has said.
According to what we have just said, each dimension introduces as it were a new degree of indeterminacy to space, that is, to the spatial continuum insofar as it is subject to indefinite increase of extension and thus yields what could be called successive powers of the indefinite; [1] and one can also say that an indefinite quantity of a certain order or power contains an indefinite multitude of indefinite quantities of a lower order or lesser power. As long as it is only a question of the indefinite in all of this, these considerations, as well as others of the same sort, remain perfectly acceptable, for there is no logical incompatibility between multiple and distinct indefinite quantities, which, despite their indefinitude, are nonetheless of an essentially finite nature, and which, like any other particular and determined possibility, are therefore perfectly capable of coexisting within total Possibility, which is alone infinite, since it is identical to the universal All. [2] These same considerations take on an impossible and absurd form only when the indefinite is confused with the infinite; thus, as with the notion of the 'infinite multitude', we once again have an instance in which the contradiction inherent in a socalled determined infinite is concealed, deforming another idea that, although in itself not at all contradictory, is nonetheless rendered virtually unrecognizable.
We have just spoken of various degrees of indeterminacy in relation to quantities taken in the increasing direction; by applying the same notion to the decreasing direction we have already justified above the consideration of various orders of infinitesimal quantity, the possibility of which is all the more understandable in the light of the correlation we noted earlier between indefinitely increasing and indefinitely decreasing quantities. Among indefinite quantities of various orders, those of orders apart from the first will always be indefinite with respect to those of the preceding order as well as to ordinary quantities; inversely, among infinitesimal quantities of various orders, it is just as legitimate to consider those of each order as infinitesimal not only with respect to ordinary quantities, but also to the infinitesimal quantities of the preceding orders. [3] There is no absolute heterogeneity between indefinite quantities and ordinary quantities, nor again between infinitesimal quantities and ordinary quantities; in short, it is only a question of a difference of degree, not of kind, since, in reality, the consideration of indefinitude, whatever the order or power in question, never takes us out of the finite; again, it is the false conception of the infinite that introduces the appearance of a radical heterogeneity between the different orders of quantity, which at bottom is completely incomprehensible. In doing away with this heterogeneity, a kind of continuity is established quite different from that which Leibnitz envisaged between variables and their limits, and much better grounded in reality, for contrary to what he believed, the distinction between variable and fixed quantities essentially implies a difference of nature.
Under these conditions, ordinary quantities themselves can in a way be regarded as infinitesimal with respect to indefinitely increasing quantities, at least when we are dealing with variables, for, if a quantity is capable of being rendered as great as one likes with respect to another, inversely the latter will by the same token become as small as one likes with respect to the former. We say that it must be a question of variables because an infinitesimal quantity must always be conceived of as essentially variable, and this restriction is inherent in its very nature; moreover, quantities belonging to two different orders of indefinitude are inevitably variable with respect to one another, and this property of relative and reciprocal variability is perfectly symmetric, for, in accordance with what was just said, to consider one quantity to be indefinitely increasing with respect to another, or this latter indefinitely decreasing with respect to the first, amounts to the same thing; without this relative variability there could be neither indefinite increase nor indefinite decrease, but only definite and determined ratios between the two quantities.
In the same way, whenever there is a change in position with respect to two bodies $A$ and $B$, to say that body $A$ is in motion with respect to body $B$, and, inversely, that body $B$ is in motion with respect to body $A$, also amounts to the same thing, at least insofar as the change is only considered in and of itself; in this regard the concept of relative motion is just as symmetric as that of relative variability, which we were just considering. This is why, according to Leibnitz, who used it to demonstrate the inadequacy of Cartesian mechanism as a physical theory claiming to furnish an explanation for all natural phenomena, one cannot distinguish between a state of motion and a state of rest when one is limited solely to the consideration of changes in position; to do so one must bring in something of another order, namely, the notion of force, which is the proximate cause of such changes, and which alone can be attributed to one body rather than to another, as it allows the true cause of change to be located in one body and in that body alone. [4]