[5] That the choice of an ordering relation is all-important for the determination of dimensionality may be gathered from the following examples. Consider a number of diapasons emitting notes of various pitch. We should have no difficulty in ranging these various diapasons in order of increasing pitch. This ordering relation would be instinctive and not further analysable, since it would issue from that mysterious human capacity which enables all men to assert that one sound is shriller than another. Human appreciation is thus responsible for this definition of order in music; and we may well conceive of men whose reactions to sonorous impressions might differ from ours and who in consequence would range the diapasons in some totally different linear order. As a result, a note which we should consider as lying “between” two given notes might, in the opinion of these other beings, lie outside them. In either case, however, if we should assume that the various notes differed in pitch from one another by insensible gradations, a sensory continuum of sound pitches would have been constructed, though what we should call a continuum of pitches would appear to the other beings as a discontinuity of notes, and vice versa. In either case the respective sensory continua would be one-dimensional, since by suppressing any given note, continuity of passage would be broken. And now let us suppose that these strange beings were still more unlike us humans. Let us assume that not only would every note of their continuum appear to them as indiscernible from the one immediately preceding it and the one immediately following it, but that it would also be impossible for them to differentiate each given note from two additional notes. This assumption is by no means as arbitrary as it might appear, for we know full well that even normal human beings experience considerable difficulty in differentiating extreme heat from extreme cold and we know, too, that people afflicted with colour blindness are unable to distinguish red from green. But then, to return to our illustration, we should realise that the sound continuum, when ordered with this novel understanding of nextness or contiguity, would no longer remain one-dimensional.
In other words, in virtue of this new ordering relation, imposed by the idiosyncrasies of their perceptions, the same aggregate of notes would range itself automatically into a two-dimensional sensory continuum. This is what is meant when we claim that an aggregate of elements has of itself no particular dimensionality and that some ordering relation must be imposed from without. We must realise, therefore, that when we speak of the space of our experience as being three-dimensional in points, no intrinsic property of space can be implied by this statement. It is only when account is taken of the complex of our experiences interpreted in the light of that sensory order which appears to be imposed upon out co-ordinative faculties that the statement can acquire meaning. These conclusions are by no means vitiated by such facts as our inability to get out of a closed room or to tie a knot in a space of an even number of dimensions. Unfortunately, owing to lack of time, we cannot dwell further on these difficult questions.
[6] Here again we are neglecting, from motives of simplicity, our awareness of the focussing effort and of that of convergence. If we take these into account, we are in all truth considering not merely one private perspective, but a considerable number. Even so, our data would be incomplete.
[7] This difficulty of counting points might be obviated to a certain degree were space to be considered discrete or atomic; for in that case we might count the atoms of space separating points and thereby establish absolute comparisons between distances. But here again the procedure would be artificial, for it would be nullified unless we were to assume that the spatial voids separating the successive atoms were always the same in magnitude; furthermore (as Dr. Silberstein points out in his book, “The Theory of Relativity”), we should be in a quandary to know how a succession of atoms would have to be defined, since this definition would depend on a definition of order. At any rate, we need not concern ourselves with atomic or discrete manifolds, for Riemann assumed that mathematical space was a continuous manifold. In view of the quantum phenomena we may eventually be led to modify these views and to attribute a discrete nature to space, but this is a vague possibility which there is no advantage in discussing in the present state of our knowledge.
[8] Were it not for this restriction, comparisons of distance in space situated in different places could never be obtained by transporting rods from one place to another; for since the measurements yielded by our standard rods when they had reached their point of destination would depend essentially on the route they had followed, they could scarcely be called rigid. When the restriction is adhered to we obtain the most general type of Riemannian spaces or geometries exemplified by the three major types, the Euclidean type, the Riemannian type and the Lobatchewskian type. Weyl, however, dispenses with Riemann’s postulate and thereby obtains a more generalised type of space or geometry.
The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discussed a posteriori and not a priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.
So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.
[9] The association of a straight line with the shortest distance between two points only holds, however, provided the several dimensions of the space are of an identical nature. When we consider continua in which the several dimensions differ in nature, as in the space-time of relativity, the straight line may turn out to be the longest distance between two points.
[10] In the chapter on Weyl’s theory we shall mention Weyl’s method briefly.
[11] We are referring solely to those bodies which would yield the numerical results of Riemann’s or Lobatchewski’s geometry—both these types of geometry being compatible with the homogeneity of space.