[147]. It must be carefully noted that distance as thus defined is not properly a quantitative relation, and involves no notion of magnitude, but only of relative place in a series. It should also be observed that in assuming the existence of such a unique relation between every pair of points, it is tacitly taken for granted that the number of dimensions of the spatial order is finite. In a space of an infinite number of dimensions, such unique relation would be impossible. (See Russell, Foundations of Geometry, p. 161 ff.) Our justification for making this assumption, as also for taking time to be of one dimension only, seems to be that it is indispensable for all those practical purposes which depend on our ability to create a science of Geometry, and that we have no positive ground for assuming the opposite. Thus ultimately the assumption appears to be of the nature of a postulate.
[148]. The ablest detailed account of the relativity of spatial position readily accessible to the English reader, will be found in Mr. Russell’s Foundations of Geometry, chaps. iiiA, iv. Mr. Russell has since, in Mind for July 1901, attempted to prove the opposite view, that positions in space and time are inherently distinct, but without discussing his own previous arguments for relativity. Into the purely mathematical part of Mr. Russell’s later contentions I am not competent to enter. I may, however, suggest that the question of Metaphysics cannot be decided merely by urging, as Mr. Russell does, that fewer assumptions are required to construct a geometry on the hypothesis of absolute than on that of relative position. The superior convenience of an assumption for certain special purposes is no proof of its ultimate intelligibility. And when Mr. Russell goes on to admit that points in space are indistinguishable for us, he seems to me to give up his case. For is not this to admit that, after all, the space with which we deal in our geometrical science is relative from beginning to end? How differences of quality of which we, by hypothesis, can know nothing, can help or hinder our scientific constructions, it is indeed hard to see.
[149]. This may be brought home even to those who, like myself, are not mathematicians, by the perusal of such a work as Lobatchevsky’s Untersuchungen zur Theorie der Parallel-Linien, where a consistent geometry of triangles is constructed in entire independence of the postulate of parallelism. Of course, in the end it must be a mere question of nomenclature whether a form of serial order independent of these quasi-empirical restrictions is to be called “space” or not.
[150]. It must be carefully remembered that the essential defect of the indefinite regress is not its interminableness, but its monotony. We ourselves held that Reality is an individual composed of lesser individuals which repeat the structure of the whole, and that the number of these individuals need not be finite. But, in our view, the higher the order of individuality the more self-explanatory was its structure, whereas in the indefinite regress an incomprehensible construction is endlessly repeated in the same form.
[151]. Normally, that is; for brevity’s sake I omit to note the possible case of a coherent dream-life continued from night to night. In principle there would be a difference between the case of the space and time of such a dream-life and those of our waking hours.
[152]. So the events of my dreams, though not occupying any place in the temporal series of the events of waking life, are so far logically connected with that series as both sets of events stand in relation to certain identical elements of psychical temperament and disposition. Another interesting case is that of so-called “dual personality.” The experience of both the two alternating personalities can be arranged in a single temporal series only because of the way in which both sets are inwoven with the systematic interests of other men, whose personality does not alternate, or alternates with a different rhythm. If all mankind were subject to simultaneous alternations of personality, the construction of a single time-series for all our experiences would be impossible. In this discussion I have throughout followed the full and thorough treatment of the problem by Mr. Bradley, Appearance and Reality, chap. 18.
[153]. Otherwise, conceptual space and time are, as we have seen, derivatives of the number-series, and we have already learned that the number-series leads to the problem of summing an endless series, and is therefore not an adequate way of representing ultimate Reality. (Bk. II. chap. 4, § 10). Another form of the same difficulty would be that conceptual space and time are applications of the numerical series,—but application to what? To a material which is already spatial and temporal. All these puzzles are only different ways of expressing the essential relativity of space and time. But see the anti-Kantian view in, e.g., Couturat, L’Infini Mathématique, pt. 2.
[154]. Compare Prof. Royce’s remarks, The World and the Individual, Second Series, lect. 3, “The Temporal and the Eternal,” p. 134. I should certainly have had to acknowledge considerable obligation to Prof. Royce’s discussion had not the present chapter been written before I had an opportunity of studying it.
[155]. Against the plausible attempt to solve the problem by simply thinking of the whole physical order as forming a “specious present” to the Absolute Experience, we may urge that the “specious present” itself regularly consists for us of a multiplicity of detail, which we apprehend as simultaneous without insight into its inner unity as the embodiment of coherent system. Hence the direct insight of the Absolute Experience into its own internal meaning or structure cannot be adequately thought of as mere simultaneous awareness of the detail of existence. So long as a succession is merely apprehended as simultaneous, its meaning is not yet grasped.