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 [Pg 28] 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 . Yet no difficulty is experienced in differentiating from . Expressed mathematically, these facts yield the inconsistent series of relations ; ; . Now, an inconsistency of this sort precludes all mathematical treatment. In mathematics magnitudes cannot be both equal and unequal; they must either be one or the other. The mathematician is therefore compelled to idealise the sensory continuum of experience by assuming that were it not for the crudeness of our senses, the points or sensations , and would all be distinguishable, and that in place of ; and , we should have ; ; . But it is obvious that a continuum idealised in this way becomes atomic or discrete, since between and , as between and , no intermediary points have been mentioned. In order to re-establish continuity the mathematician is forced to postulate that between any two points and there exist an indefinite number of intermediary points, such that no one of these points has an immediate neighbour. In other words, the continuum is infinitely divisible. Thus the magnitudes 1 and 2 are not neighbours, since a number of rational fractions separate them. And no two of these fractional numbers are immediate neighbours, since whichever two such numbers we choose to select, we can always discover an indefinite number of other fractional numbers existing between them. Between any two points on a line in our continuum, however close together they may be, we have thus interposed an indefinite number of rational fractions defining points; yet, despite this fact, we have by no means eliminated gaps between the various points along our line.