4
Spatial Quantity and Qualified Space
It has already been made clear that extension is not purely and simply a mode of quantity; in other words, while it is undoubtedly legitimate to speak of quantity as extended or spatial, this does not necessarily imply that extension can be treated as quantity and nothing more. This must be insisted on again, because it is particularly important in that it reveals the insufficiency of Cartesian ‘mechanism’ and of the other physical theories derived more or less directly from it in modern times. The first thing to be noticed in this connection is that if space were purely quantitative it would have to be entirely homogeneous, and its parts would have to be indistinguishable one from another by any characteristic other than their respective sizes; this would amount to conceiving it as no more than a container without content, that is to say as something which cannot have an independent existence in manifestation, for the relation of container to content necessarily presupposes, by its very nature as a correlation, the simultaneous presence of both of its terms. The question may be put, at least with some appearance of reason, as to whether geometrical space can be conceived as endowed with some such homogeneity, but whatever may be the answer to that question no such conception of homogeneity is compatible with physical space, with the space that contains bodies, for the presence of those bodies suffices to determine qualitative differences between the parts of space they occupy — and Descartes was undoubtedly thinking of physical space, for otherwise his theory would not mean anything, since it would then not be applicable in any real sense to the world of which it claims to provide the explanation.[17] It would be useless to object that ‘empty space’ is only the starting-point of his theory because, in the first place, this would lead back to the conception of a container without content, implying an emptiness that can have no place in the manifested world, emptiness as such not being a possibility of manifestation;[18] and, in the second place, since Descartes reduces the whole nature of bodies to extension, he is compelled thenceforth to suppose that their presence adds nothing to what space itself already is. Indeed the diverse properties of bodies are no more in his eyes than mere modifications of extension; but if that be so, whence can these properties come, unless they are in some way inherent in extension itself, and how can they be so inherent if the nature of extension is lacking in any qualitative elements? Here there is something very like contradiction; indeed it would be difficult to maintain that this contradiction, and a good many others like it, is not implicit in the work of Descartes; for he, like the more recent materialists who surely have ample reason to proclaim themselves his followers, seem really to be trying to extract the ‘greater’ from the ‘lesser’. To say that a body is nothing but extension in a purely quantitative sense, is really the same as to say that its surface and its volume, which measure the portion of extension actually occupied by it, are the body itself with all its properties, which is manifestly absurd; therefore some other interpretation must be sought, and it becomes impossible to avoid the admission that extension itself is in some way qualitative, but if it is so, it cannot serve as the basis of an exclusively ‘mechanistic’ theory.
Now although these considerations show that Cartesian physics cannot be valid, they are still not sufficient to establish firmly the qualitative character of extension; indeed it might well be argued that, although it is true that the nature of bodies cannot be reduced to extension alone, yet this is just because they derive nothing from extension other than their quantitative elements. But at this point the following observation becomes pertinent: among the corporeal determinations which are undeniably of a purely spatial order, and which can therefore rightly be regarded as modifications of extension, there is not only the size of bodies, but also their situation; is situation itself therefore also purely quantitative? The partisans of a reduction to quantity will doubtless reply that the situation of a plurality of bodies is defined by their distances, and that distance is certainly a quantity — the quantity of extension that lies between them, just as their size is the quantity of extension that they occupy; but is distance sufficient by itself to define the situation of bodies in space? There is something else that cannot possibly be left out of account, and that is the direction along which distance must be measured; but, from a quantitative point of view, direction cannot but be a matter of indifference, because space cannot be considered as other than homogeneous in this respect, and this implies that particular directions in space are in no way distinguished one from another; so if direction is an effective element in situation, and if it is a purely spatial element, as it evidently is, and no less so than distance, then there must be something qualitative in the very nature of space.
In order to leave no room for doubt, physical space and bodies can be left out of the picture, nothing then remaining to be considered but a space that is in the strict sense purely geometrical, and this surely does represent what may be called space reduced to itself alone; in studying such a space, does geometry really take nothing into account but strictly quantitative conceptions? Let it be clearly understood that only the profane geometry of the moderns is now under consideration; and the question may at once be asked whether, if there proves to be anything in profane geometry that cannot be reduced to quantity, does it not immediately follow that it is even less possible and less legitimate to claim to reduce everything in the domain of the physical sciences to quantity? Even the question of situation can be left out here, because it only plays a really conspicuous part in certain special branches of geometry, which might perhaps be regarded as not constituting a strictly integral part of pure geometry:[19] but in the most elementary geometry, not only has the size of figures to be taken into account, but also their shape; and would any geometrician, however deeply imbued with modern conceptions, dare to maintain for example that a triangle and a square of equal area are one and the same thing? He would only say that they are ‘equivalent’, but he would clearly be leaving out as being understood the words ‘in respect of size’, and he would have to recognize that in another respect, namely that of shape, there is something that differentiates them; and the reason for which equivalence in size does not carry with it similitude of shape is that there is something in shape that precludes its being reduced to quantity. But this is not alclass="underline" for there is a whole section of elementary geometry to which quantitative considerations are strange, namely the theory of similar figures; similarity is in fact defined exclusively by shape and is wholly independent of the size of figures, and this amounts to saying that it is of a purely qualitative order.[20] If we now care to enquire into the essential nature of spatial shape, it will be found to be definable as an assemblage of directional tendencies: at every point in a line its directional tendency is specified by a tangent, and the assemblage of all the tangents defines the shape of the line. In three-dimensional geometry the same is true of surfaces, straight line tangents being replaced by plane tangents; it is moreover evident that the shape of all bodies, as well as that of simple geometrical figures, can be similarly defined, for the shape of a body is the shape of the surface by which its volume is delimited. The conclusion toward which all this leads could be foreseen when the situation of bodies was being discussed, namely, that it is the notion of direction that without doubt represents the real qualitative element inherent in the very nature of space, just as the notion of size represents its quantitative element; and so space that is not homogeneous, but is determined and differentiated by its directions, may be called ‘qualified’ space.
17
It is true that Descartes, at the beginning of his physics, only claims to construct a hypothetical world on the basis of certain assumptions, which can be reduced to extension and movement; but, since he is at pains to demonstrate later that the phenomena that would be produced in such a world are precisely those of which we are aware in our own, it is clear that, in spite of his purely verbal precaution, he intends to conclude that our world is in fact constituted like the world he began by imagining.
18
This argument is equally applicable against atomism, which by definition admits no positive existence other than that of atoms and their combinations, and is thus necessarily led to posit a void between the atoms for them to move about in.
19
Such are, for instance, descriptive geometry, and the geometry to which certain mathematicians have given the name of
20
This is just what Leibnitz expressed by the formula: