And yet in the case of space, itself a sensory continuum, men have found no difficulty in agreeing on a common system of measurements. As we shall see in the following chapters, the definition of the equality of different stretches of space to which men were unavoidably led was imposed upon them by the behaviour of certain bodies located in space, bodies which were deemed to remain rigid, hence to occupy equal volumes and equal lengths of space wherever they were displaced. For the present, however, we may leave these metrical considerations aside and confine our attention to the general concept of mathematical or geometrical space, which is the subject of study of the pure mathematician.
The concept of a sensory continuum, hence of perceptual space, as presented to us by crude experience, contains certain contradictions and peculiarities which it was necessary to eliminate before it could be subjected to rigorous mathematical treatment. In the first place, this perceptual space is not homogeneous, and the principle of sufficient reason demands that pure empty conceptual space be homogeneous and isotropic, the same everywhere and the same in all directions.
This homogeneity of space permits us to foresee that it must be unbounded, since a boundary would suggest a discontinuity of structure, defining an inside and an outside, hence a lack of homogeneity. Prior to Riemann’s discoveries it was thought that the absence of a boundary would necessitate the infiniteness of space. To-day we know that this belief is unjustified, for a space can be finite and yet unbounded; and two major varieties of such spaces have been discovered by mathematicians.
But the inherent inconsistencies which endure in all sensory continua constituted a still more important reason for compelling mathematicians to idealise perceptual space. In a sensory continuum, as we have seen, a sensation
cannot be distinguished from its immediate successor, the sensation
; neither can
be differentiated from