CHAPTER XVI Relationship between the Point and Space
The question raised by the final observation in the last chapter calls for further examination, but we do not propose to go fully into the question of space in all its implications, since this would more properly fall to be dealt with in a study of the conditions of corporeal existence. The first thing to be said is that the distance between two immediately adjacent points, which we have been led to consider as a result of the introduction of continuity into the geometrical representation of the being, may be regarded as the limit of space in the sense of indefinitely decreasing quantities; in other words, it is the smallest space possible, after which there remains no spatial condition at all, and it would not be possible to suppress it without departing from the realm of existence that is subject to that condition. Therefore, when space is divided indefinitely, [1] and when this division is carried as far as is possible, that is, to the limits of the spatial possibility by which divisibility is conditioned (and which is indefinite in the decreasing as well as the increasing sense), what is arrived at as the final result is not a point, but rather the elementary distance between two points. It follows from this that for spatial extension to exist there must be already two points, and the (one-dimensional) expanse which is realized by their simultaneous presence, and which is precisely the distance between them constitutes a third element which expresses the relationship between the two points, by at once joining and separating them. This distance, moreover, when regarded as a relation, is plainly not composed of parts, for if it were, the parts into which it could be resolved would simply be other relationships of distance of which it is logically independent, just as from the numerical point of view unity is independent of fractions. [2] This is true for any distance, when envisaged solely in respect of the two points that are its extremities, and is a fortiori true for an infinitesimal distance, which is in no way a definite quantity, but solely expresses a spatial relation between two immediately adjacent points, such as two consecutive points in any line. Again, the points themselves, considered as extremities of a distance, are not parts of the spatial continuum, although the distancerelation assumes that they are conceived as situated in space ; it is thus really distance that is the true spatial element.
Accordingly, it is not possible in all strictness to say that a line is formed of points, and the reason for this is not difficult to understand, for, since each of the points is without extension, their mere addition, even if they are in indefinite multitude, can never form an extension; in reality, the line is constituted by the elementary distances between its consecutive points. In the same way, and for a similar reason, if we consider an indefinitude of parallel straight lines in a plane, we cannot say that the plane is constituted by the combination of all these lines, or that they are true constitutive elements of the plane; the true elements are the distances between those lines, distances which make them distinct lines and not confounded, and if the lines do form the plane in a certain sense, it is not by themselves but by their distances that they do so, as in the case of the points of a line. Again, a three-dimensional expanse is not composed of an indefinitude of parallel planes, but of the distances between all those planes.
However, the primordial element, that which exists by itself, is the point, since it is presupposed by distance and distance is only a relationship; hence space itself presupposes the point. The latter may be said to contain in itself a virtuality of extension, which it can only develop by first duplicating itself, placing itself so to speak opposite to itself, and then by multiplying (or better, sub-multiplying) itself indefinitely, so that manifested space in its entirety proceeds from differentiation of the point, or, to speak more exactly, from the point in so far as it differentiates itself. This differentiation however is real only from the viewpoint of spatial manifestation; it is illusory in respect of the principial point itself, which does not thereby cease to be in itself that which it was, and whose essential unity can in no way be affected thereby. [3] The point, considered in itself, is in no wise subject to the spatial condition, for on the contrary it is the principle of that condition: it is the point that realizes space and produces extension by its act, which, in the temporal condition (but only therein), is translated by movement ; but, in order to realize space thus, it is bound, by some one of its modalities, to situate itself in space, which indeed is nothing without it, and which it will completely fill by the deployment of its own virtualities. [4] Successively in the temporal condition, or simultaneously outside that condition (which, be it observed in passing, would take us outside ordinary three-dimensional space), [5] it identifies itself, with all the potential points in space in order to realize the latter. Thus space must be regarded as no more than a mere potentiality of being, which is nothing else than the total virtuality of the point conceived in its passive aspect, the locus or container of all the manifestations of its activity, a container which has no existence except through the realization of its possible content. [6]
Being without dimensions, the primordial point is also without form; hence it does not belong to the order of individual existences. It does not individualize itself in any way except when it situates itself in space, and then not in itself, but solely by one of its modalities, so that strictly speaking it is these latter that are really individualized, and not the principial point. Besides, if there is to be form, there must already be differentiation, hence multiplicity realized in a certain measure, which is possible only when the point opposes itself, if the expression is permissible, by means of two or more of its modalities of spatial manifestation; and it is this opposition, fundamentally, that constitutes distance. The realization of distance is the first accomplishment of space, which without it, as was said, is but a mere potentiality of receptiveness. We would also observe that distance at first exists only virtually in the spherical form that was mentioned earlier, which is the form that corresponds to the minimum of differentiation; being "isotropic " in respect of the central point, with nothing to distinguish one particular direction from any other; the radius, which is here the expression of distance (taken from the centre to the periphery), is not actually drawn and does not form a component part of the spherical figure. The actual realization of distance is made explicit only in the straight line, of which it is the initial and fundamental element, as the result of the specifying of a certain given direction. Thereafter, space can no longer be regarded as "isotropic" ; from this standpoint it must be referred to two symmetrical poles (the two points between which there is distance) instead of being referred to a single centre.
The point, which realizes the whole of space, as has just been shown, makes itself the centre of space by measuring it along all its dimensions through the indefinite extension of the branches of the cross in the six directions, or towards the six cardinal points of space. It is thus "Universal Man ", of whom this cross is the symbol (and not individual man, who, as such can realize nothing outside his own state of being), that is truly the " measure of all things ", to use the expression of Protagoras which we have quoted elsewhere, [7] though it is unlikely that the Greek sophist was himself aware of this metaphysical interpretation. [8]