An Essay on the Foundations of Geometry

for our present purpose, it is sufficient to observe that projective Geometry, from the first, regards the straight line as determined by two points, and the plane as determined by three, from which it follows, if we take points as possible terms for spatial relations, that the straight line and the plane may be regarded as relations between two and three points respectively. If we agree on these definitions, we can proceed to discuss the fundamental principle of projective Geometry, and to analyse the axioms implicated in its truth. 118. Projective Geometry, we have seen, does not deal with quantity, and therefore recognizes no difference where the difference is purely quantitative. Now quantitative comparison depends on a recognized identity of quality; the recognition of qualitative identity, therefore, is logically prior to quantity, and presupposed by every judgment of quantity. Hence all figures, whose differences can be exhaustively described by quantity, i.e. by pure measurement, must have an identity of quality, and this must be recognizable without appeal to quantity. It follows that, by defining the word quality in geometrical matters, we shall discover what sets of figures are projectively indiscernible. If our definition is correct, it ought to yield the general projective principle with which we set out. 119. We agreed to regard points as the terms of spatial relations, and we agreed that different points could be distinguished. But we postponed the discussion of the conditions under which this distinction could be effected. This discussion will yield us the definition of quality and the proof of our general projective principle. Points, to begin with, have been defined as nothing but the terms for spatial relations. They have, therefore, no intrinsic properties; but are distinguished solely by means of their relations. Now the relation between two points, we said, is the straight line on which they lie. This gives that identity of quality for all pairs of points on the same straight line, which is required both by our projective principle and by metrical Geometry. (For only where there is identity of quality can quantity be properly applied.) If only two points are given, they cannot, without the use of quantity, be distinguished from any two other points on the same straight line; for the qualitative relation between any two such points is the same as for the original pair, and only by a difference of relation can points be distinguished from one another. But conversely, one straight line is nothing but the relation between two of its points, and all points are qualitatively alike. Hence there can be nothing to distinguish one straight line from another except the points through which it passes, and these are distinguished from other points only by the fact that it passes through them. Thus we get the reciprocal transformation: if we are given only one point, any pair of straight lines through that point is qualitatively indistinguishable from any other. This again is, on the one hand, the basis of the second part of our general projective principle, and on the other hand the condition of applying quantity, in the measurement of angles, to the departure of two intersecting straight lines. 120. We can now see the reason for what may have hitherto seemed a somewhat arbitrary fact, namely, the necessity of four collinear points for anharmonic ratio. Recurring to the quadrilateral construction and the consequent introduction of number, we see that anharmonic ratio is an intrinsic projective relation of four collinear points or concurrent straight lines, such that given three terms and the relation, the fourth term can be uniquely determined by projective methods. Now consider first a pair of points. Since all straight lines are projectively equivalent, the relation between one pair of points is precisely equivalent to that between another pair. Given one point only, therefore, no projective relation, to any second point, can be assigned, which shall in any way limit our choice of the second point. Given two points, however, there is such a relation—the third point may be given collinear with the first two. This limits its position to one straight line, but since two points determine nothing but one straight line, the third point cannot be further limited. Thus we see why no intrinsic projective relation can be found between three points, which shall enable us, from two, uniquely to determine the third. With three given collinear points, however, we have more given than a mere straight line, and the quadrilateral construction enables us uniquely to determine any number of fresh collinear points. This shows why anharmonic ratio must be a relation between four points, rather than between three. 121. We can now prove, I think, that two figures, which are projectively related, are qualitatively similar. Let us begin with a collection of points on a straight line. So long as these are considered without reference to other points or figures, they are all qualitatively similar. They can be distinguished by immediate intuition, but when we endeavour, without quantity, to distinguish them conceptually, we find the task impossible, since the only qualitative relation of any two of them, the straight line, is the same for any other two. But now let us choose, at hap-hazard, some point outside the straight line. The points of our line now acquire new adjectives, namely their relations to the new point, i.e. the straight lines joining them to this new point. But these straight lines, reciprocally, alone define our external point, and all straight lines are qualitatively similar. If we take some other external point, therefore, and join it to the same points of our original straight line, we obtain a figure in which, so long as quantity is excluded, there is no conceptual difference from the former figure. Immediate intuition can distinguish the two figures, but qualitative discrimination cannot do so. Thus we obtain a projective transformation of four lines into four other lines, as giving a figure qualitatively indistinguishable from the original figure. A similar argument applies to the other projective transformations. Thus the only reason, within projective Geometry, for not regarding projective figures as actually identical, is the intuitive perception of difference of position. This is fundamental, and must be accepted as a datum. It is presupposed in the distinction of various points, and forms the very life of Geometry. It is, in fact, the essence of the notion of a form of externality, which notion forms the subject-matter of projective Geometry. 122. We may now sum up the results of our analysis of projective Geometry, and state the axioms on which its reasoning is based. We shall then have to prove that these axioms are necessary to any form of externality, with which we shall pass, from mere analysis, to a transcendental argument. The axioms which have been assumed in the above analysis, and which, it would seem, suffice to found projective Geometry, may be roughly stated as follows: I. We can distinguish different parts of space, but all parts are qualitatively similar, and are distinguished only by the immediate fact that they lie outside one another. II. Space is continuous and infinitely divisible; the result of infinite division, the zero of extension, is called a point[129]. III. Any two points determine a unique figure, called a straight line, any three in general determine a unique figure, the plane. Any four determine a corresponding figure of three dimensions, and for aught that appears to the contrary, the same may be true of any number of points. But this process comes to an end, sooner or later, with some number of points which determine the whole of space. For if this were not the case, no number of relations of a point to a collection of given points could ever determine its relation to fresh points, and Geometry would become impossible[130]. This statement of the axioms is not intended to have any exclusive precision: other statements equally valid could easily be made. For all these axioms, as we shall see hereafter, are philosophically interdependent, and may, therefore, be enunciated in many ways. The above statement, however, includes, if I am not mistaken, everything essential to projective Geometry, and everything required to prove the principle of projective transformation. Before discussing the apriority of these axioms, let us once more briefly recapitulate the ends which they are intended to attain. 123. From the exclusively mathematical standpoint, as we have seen, projective Geometry discusses only what figures can be obtained from each other by projective transformations, i.e. by the operations of projection and section. These operations, in all their forms, presuppose the point, straight line, and plane[131], whose necessity for projective Geometry, from the purely mathematical point of view, is thus self-evident from the start. But philosophically, projective Geometry has, as we saw, a wider aim. This wider aim, which gives, to the investigation of projectively equivalent figures, its chief importance, consists in the determination of qualitative spatial similarity, in the determination, that is, of all the figures which, when any one figure is given, can be distinguished from the given figure, so long as quantity is excluded, only by the mere fact that they are external to it. 124. Now when we consider what is involved in such absolute qualitative equivalence, we find at once, as its most obvious prerequisite, the perfect homogeneity of space. For it is assumed that a figure can be completely defined by its internal relations, and that the external relations, which constitute its position, though they suffice to distinguish it from other figures, in no way affect its internal properties, which are regarded as qualitatively identical with those of figures with quite different external relations. If this were not the case, anything analogous to projective transformation would be impossible. For such transformation always alters the position, i.e. the external relations, of a figure, and could not, therefore, if figures were dependent on their relations to other figures or to empty space, be studied without reference to other figures, or to the absolute position of the original figure. We require for our principle, in short, what may be called the mutual passivity and reciprocal independence of two parts or figures of space. This passivity and this independence involve the homogeneity of space, or its equivalent, the relativity of position. For if the internal properties of a figure are the same, whatever its external relations may be, it follows that all parts of space are qualitatively similar, since a change of external relation is a change in the part of space occupied. It follows, also, that all position is relative and extrinsic, i.e., that the position of a point, or the part of space occupied by a figure, is not, and has no effect upon, any intrinsic property of the point or figure, but is exclusively a relation to other points or figures in space, and remains without effect except where such relations are considered. 125. The homogeneity of space and the relativity of position, therefore, are presupposed in the qualitative spatial comparison with which projective Geometry deals. The latter, as we saw, is also the basis of the principle of duality. But these properties, as I shall now endeavour to prove, belong of necessity to any form of externality, and are thus à priori properties of all possible spaces. To prove this, however, we must first define the notion of a form of externality in general. Let us observe, to begin with, that the distinction between Euclidean and non-Euclidean Geometries, so important in metrical investigations, disappears in projective Geometry proper. This suggests that projective Geometry, though originally invented as the science of Euclidean space, and subsequently of non-Euclidean spaces also, deals really with a wider conception, a conception which includes both, and neglects the attributes in which they differ. This conception I shall speak of as a form of externality. 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