In order to illustrate elliptical space in two dimensions, as we have illustrated spherical space, we must assume that we limit the surface of our sphere to one of its hemispheres. It might appear that in this case the surface would no longer be unbounded since it would stop abruptly at the equator, but the connectivity of elliptical space is such that every point on the equator is identical with its antipodal point on the other side of the equator. It would be as though, in our concrete representation, there were cross-connections of zero length uniting into one sole point, each pair of points situated diametrically opposite one another on the equator.
It is exceedingly difficult to visualise how it can be that a diameter joining two antipodal points of the equator should be vanishing in length, but the difficulty arises from our endeavouring to represent the distance between two points by a line joining these two points. Such a procedure may often be successful, but in the more complex cases of connectivity it breaks down completely. As Eddington remarks, for obscure psychological reasons we find it natural to represent a distance between two points by a line joining them, believing that this line represents a something fundamental in nature. But it is probable that the line we trace does not have the fundamental importance with which we credit it, and that it cannot be regarded as having any intrinsic significance outside of the ease with which it often introduces itself into our graphical description of natural phenomena. Distance would be more akin to a difference of temperature between two points, and it would never enter our minds to represent this difference of temperature by a line joining the two points. Why in the case of distance such a graphical representation appears obligatory is more or less of a mystery. At any rate the graphical representation is not always possible, and our description of elliptical space gives us an illustration to this effect.
Incidentally we see that the problem of space, even when restricted to the particular cases studied by Riemann, is not exhausted by the sole concept of congruence. Connectivity or routes of continuous passage between points must also be taken into consideration. The study of connectivity opens up a new branch of geometry also created by Riemann and known as Analysis Situs.[21]
CHAPTER VI
TIME
OUR awareness of the passage of time constitutes one of the most fundamental facts of consciousness, and our sensations range themselves automatically in this one-dimensional irreversible temporal series. In this respect we must concede a certain difference between space and time, the former being, as we have seen, a concept due in the main to a less inevitable synthetic co-ordination of sense impressions. Yet, on the other hand, there exists a marked similarity between space and duration, in that they both manifest the characteristics of sensory continuity.
Thus, in the case of time, two sensations
and
may be so close together that it will be impossible for us to determine which of the two was sensed first; the same may happen in the case of two sensations