156. This converse deduction, as regards Free Mobility, is not very difficult, and follows from the argument of Section A[162], which I will briefly recapitulate. In the first place, externality is an essentially relative conception—nothing can be external to itself. To be external to something is to be an other 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—it can have no intrinsic quality, for our form consists of pure externality, and externality contains no shadow or trace of an intrinsic quality. Hence we derive our fundamental postulate, the relativity of position. From this follows the homogeneity of our form, for any quality in one position, which marked out that position from another, would be necessarily more or less intrinsic, and would contradict the pure relativity. Finally Free Mobility follows from homogeneity, for our form would not be homogeneous unless it allowed, in every part, shapes or systems of relations, which it allowed in any other part. Free Mobility, therefore, is a necessary property of every possible form of externality.
157. In summing up the argument we have just concluded, we may exhibit it, in consequence of the two preceding paragraphs, in the form of a completed circle. Starting from the conditions of spatial measurement, we found that the comparison, required for measurement, could only be effected by superposition. But we found, further, that the result of such comparison will only be unambiguous, if spatial magnitudes and shapes are unaltered by motion in space, if, in other words, shapes do not depend upon absolute position in space. But this axiom can only be true if space is homogeneous and position merely relative. Conversely, if position is assumed to be merely relative, a change of magnitude in motion—involving as it does, the assertion of absolute position—is impossible, and our test of spatial equality is therefore adequate. But position in any form of externality must be purely relative, since externality cannot be an intrinsic property of anything. Our axiom, therefore, is à priori in a double sense. It is presupposed in all spatial measurement, and it is a necessary property of any form of externality. A similar double apriority, we shall see, appears in our other necessary axioms.
II. The Axiom of Dimensions[163].
158. We have seen, in discussing the axiom of Free Mobility, that all position is relative, that is, a position exists only by virtue of relations[164]. It follows that, if positions can be defined at all, they must be uniquely and exhaustively defined by some finite number of such relations. If Geometry is to be possible, it must happen that, after enough relations have been given to determine a point uniquely, its relations to any fresh known point are deducible from the relations already given. Hence we obtain, as an à priori condition of Geometry, logically indispensable to its existence, the axiom that Space must have a finite integral number of Dimensions. For every relation required in the definition of a point constitutes a dimension, and a fraction of a relation is meaningless. The number of relations required must be finite, since an infinite number of dimensions would be practically impossible to determine. If we remember our axiom of Free Mobility, and remember also that space is a continuum, we may state our axiom, for metrical Geometry, in the form given by Helmholtz (v. [Chap. I. § 25]): "In a space of n dimensions, the position of every point is uniquely determined by the measurement of n continuous independent variables (coordinates).[165]"
159. So much, then, is à priori necessary to metrical Geometry. The restriction of the dimensions to three seems, on the contrary, to be wholly the work of experience[166]. This restriction cannot be logically necessary, for as soon as we have formulated any analytical system, it appears wholly arbitrary. Why, we are driven to ask, cannot we add a fourth coordinate to our x, y, z, or give a geometrical meaning to x4? In this more special form, we are tempted to regard the axiom of dimensions, like the number of inhabitants of a town, as a purely statistical fact, with no greater necessity than such facts have.
Geometry affords intrinsic evidence of the truth of my division of the axiom of dimensions into an à priori and empirical portion. For while the extension of the number of dimensions to four, or to n, alters nothing in plane and solid Geometry, but only adds a new branch which interferes in no way with the old, some definite number of dimensions is assumed in all Geometries, nor is it possible to conceive of a Geometry which should be free from this assumption[167].
160. Let us, since the point seems of some interest, repeat our proof of the apriority of this axiom from a slightly different point of view. We will begin, this time, from the most abstract conception of space, such as we find in Riemann's dissertation, or in Erdmann's extents. We have here, an ordered manifold, infinitely divisible and allowing of Free Mobility[168]. Free Mobility involves, as we saw, the power of passing continuously from any one point to any other, by any course which may seem pleasant to us; it involves, also, that, in such a course, no changes occur except changes of mere position, i.e., positions do not differ from one another in any qualitative way. (This absence of qualitative difference is the distinguishing mark of space as opposed to other manifolds, such as the colour- and tone-systems: in these, every element has a definite qualitative sensational value, whereas in space, the sensational value of a position depends wholly on its spatial relation to our own body, and is thus not intrinsic, but relative.) From the absence of qualitative differences among positions, it follows logically that positions exist only by virtue of other positions; one position differs from another just because they are two, not because of anything intrinsic in either. Position is thus defined simply and solely by relation to other positions. Any position, therefore, is completely defined when, and only when, enough such relations have been given to enable us to determine its relation to any new position, this new position being defined by the same number of relations. Now, in order that such definition may be at all possible, a finite number of relations must suffice. But every such relation constitutes a dimension. Therefore, if Geometry is to be possible, it is à priori necessary that space should have a finite integral number of dimensions.
161. The limitation of the dimensions to three is, as we have seen, empirical; nevertheless, it is not liable to the inaccuracy and uncertainty which usually belong to empirical knowledge. For the alternatives which logic leaves to sense are discrete—if the dimensions are not three, they must be two or four or some other number—so that small errors are out of the question[169]. Hence the final certainty of the axiom of three dimensions, though in part due to experience, is of quite a different order from that of (say) the law of Gravitation. In the latter, a small inaccuracy might exist and remain undetected; in the former, an error would have to be so considerable as to be utterly impossible to overlook. It follows that the certainty of our whole axiom, that the number of dimensions is three, is almost as great as that of the à priori element, since this element leaves to sense a definite disjunction of discrete possibilities.
III. The Axiom of Distance.
162. We have already seen, in discussing projective Geometry, that two points must determine a unique curve, the straight line. In metrical Geometry, the corresponding axiom is, that two points must determine a unique spatial quantity, distance. I propose to prove, in what follows, (1) that if distance, as a quantity completely determined by two points, did not exist, spatial magnitude would not be measurable; (2) that distance can only be determined by two points, if there is an actual curve in space determined by those two points; (3) that the existence of such a curve can be deduced from the conception of a form of externality, and (4) that the application of quantity to such a curve necessarily leads to a certain magnitude, namely distance, uniquely determined by any two points which determine the curve. The conclusion will be, if these propositions can be successfully maintained, that the axiom of distance is à priori in the same double sense as the axiom of Free Mobility, i.e. it is presupposed in the possibility of measurement, and it is necessarily true of any possible form of externality.