126. In Grassmann's profound philosophical introduction to his Ausdehnungslehre of 1844, he suggested that Geometry, though improperly regarded as pure, was really a branch of applied mathematics, since it dealt with a subject-matter not created, like number, by the intellect, but given to it, and therefore not wholly subject to its laws alone. But it must be possible—so he contended—to construct a branch of pure mathematics, a science, that is, in which our object should be wholly a creature of the intellect, which should yet deal, as Geometry does, with extension—extension as conceived, however, not as empirically perceived in sensation or intuition.
From this point of view, the controversy between Kantians and anti-Kantians becomes wholly irrelevant, since the distinction between pure and mixed mathematics does not lie in the distinction between the subjective and the objective, but between the purely intellectual on the one hand, and everything else on the other. Now Kant had contended, with great emphasis, that space was not an intellectual construction, but a subjective intuition. Geometry, therefore, with Grassmann's distinction, belongs to mixed mathematics as much on Kant's view as on that of his opponents. And Grassmann's distinction, I contend, is the more important for Epistemology, and the one to be adopted in distinguishing the à priori from the empirical. For what is merely intuitional can change, without upsetting the laws of thought, without making knowledge formally impossible: but what is purely intellectual cannot change, unless the laws of thought should change, and all our knowledge simultaneously collapse. I shall therefore follow Grassmann's distinction in constructing an à priori and purely conceptual form of externality.
127. The pure doctrine of extension, as constructed by Grassmann, need not be discussed—it included much empirical material, and was philosophically a failure. But his principles, I think, will enable us to prove that projective Geometry, abstractly interpreted, is the science which he foresaw, and deals with a matter which can be constructed by the pure intellect alone. If this be so, however, it must be observed that projective Geometry, for the moment, is rendered purely hypothetical[132]. All necessary truth, as Bradley has shown, is hypothetical[133], and asserts, primâ facie, only the ground on which rests the necessary connection of premisses and conclusion. If we construct a mere conception of externality, and thus abandon our actually given space, the result of our construction, until we return to something actually given, remains without existential import—if there be experienced externality, it asserts, then there must be a form of externality with such and such properties. That there must be experienced externality, Kant's first argument about space proves, I think, to those who admit experience of a world of diverse but interrelated things. But this is a question which belongs to the next Chapter.
What we have to do here is, not to discuss whether there is a form of externality, but whether, if there be such a form, it must possess the properties embodied in the axioms of projective Geometry. Now first of all, what do we mean by such a form?
128. In any world in which perception presents us with various things, with discriminated and differentiated contents, there must be, in perception, at least one "principle of differentiation[134]," an element, that is, by which the things presented are distinguished as various. This element, taken in isolation, and abstracted from the content which it differentiates, we may call a form of externality. That it must, when taken in isolation, appear as a form, and not as a mere diversity of material content, is, I think, fairly obvious. For a diversity of material content cannot be studied apart from that material content; what we wish to study here, on the contrary, is the bare possibility of such diversity, which forms the residuum, as I shall try to prove hereafter[135], when we abstract from any sense-perception all that is distinctive of its particular matter. This possibility, then, this principle of bare diversity, is our form of externality. How far it is necessary to assume such a form, as distinct from interrelated things, I shall consider later on[136]. For the present, since space, as dealt with by Geometry, is certainly a form of this kind, we have only to ask: What properties must such a form, when studied in abstraction, necessarily possess?
129. In the first place, externality is an essentially relative conception—nothing can be external to itself. To be external to something is to be another with some relation to that thing. Hence, when we abstract a form of externality from all material content, and study it in isolation, position will appear, of necessity, as purely relative—a position can have no intrinsic quality, for our form consists of pure externality, and externality contains no shadow or trace of an intrinsic quality. Thus we obtain our fundamental postulate, the relativity of position, or, as we may put it, the complete absence, on the part of our form, of any vestige of thinghood.
The same argument may also be stated as follows: If we abstract the conception of externality, and endeavour to deal with it per se, it is evident that we must obtain an object alike destitute of elements and of totality. For we have abstracted from the diverse matter which filled our form, while any element, or any whole, would retain some of the qualities of a matter. Either an element or a whole, in fact, would have to be a thing not external to itself, and would thus contain something not pure externality. Hence arise infinite divisibility, with the self-contradictory notion of the point, in the search for elements, and unbounded extension, with the contradiction of an infinite regress or a vicious circle, in the search for a completed whole. Thus again, our form contains neither elements nor totality, but only endless relations—the terms of these relations being excluded by our abstraction from the matter which fills our form.
130. In like manner we can deduce the homogeneity of our form. The diversity of content, which was possible only within the form of externality, has been abstracted from, leaving nothing but the bare possibility of diversity, the bare principle of differentiation, itself uniform and undifferentiated. For if diversity presupposes such a form, the form cannot, unless it were contained in a fresh form, be itself diverse or differentiated.
Or we may deduce the same property from the relativity of position. For any quality in one position, by which it was marked out from another, would be necessarily more or less intrinsic, and would contradict the pure relativity. Hence all positions are qualitatively alike, i.e. the form is homogeneous throughout.
131. From what has been said of homogeneity and relativity, follows one of the strangest properties of a form of externality. This property is, that the relation of externality between any two things is infinitely divisible, and may be regarded, consequently, as made up of an infinite number of the would-be elements of our form, or again as the sum of two relations of externality[137]. To speak of dividing or adding relations may well sound absurd—indeed it reveals the impropriety of the word relation in this connexion. It is difficult, however, to find an expression which shall be less improper. The fact seems to be, that externality is not so much a relation as bare relativity, or the bare possibility of a relation. On this subject, I shall enlarge in Chapter IV.[138] At this point it is only important to realize, what the subsequent argument will assume, that the relation—if we may so call it—of externality between two or more things must, since our form is homogeneous, be capable of continuous alteration, and must, since our infinitely divisible form is constituted by such relations, be capable of infinite division. But the result of infinite division is defined as the element of our form. (Our form has no elements, but we have to imagine elements in order to reason about it, as will be shown more fully in Chapter IV.) Hence it follows, that every relation of externality may be regarded, for scientific purposes, as an infinite congeries of elements, though philosophically, the relations alone are valid, and the elements are a self-contradictory result of hypostatizing the form of externality. This way of regarding relations of externality is important in understanding the meaning of such ideas as three or four collinear points.