IN this chapter we shall be concerned solely with manifolds and their dimensionality; and not with their metrics, which pertains to a different problem entirely. We all possess a certain instinctive understanding of what is meant by continuity. We notice, for example, that sounds, colours or tactual sensations merge by insensible gradations into other sounds, colours or tactual sensations, without any abrupt transitions. An aggregate of such continuous sensations constitutes what is called a sensory continuum or continuous manifold. That continuity is a concept which springs from experience can scarcely be doubted, and it can be accounted for by the inability of our crude senses to differentiate between impressions which are almost alike. Consider, for example, the succession of musical notes exhibited in the chromatic scale on the piano. Here we are not in the presence of a sensory continuum, for the successive sounds do not merge into one another by insensible degrees. Even an untrained ear can differentiate between a and the -sharp immediately following it. But we can conceive of a piano in which a sufficient number of semitones and intermediary notes have been interposed so that every note would be indistinguishable from its immediate successor and immediate predecessor, although we should still be able to differentiate between non-contiguous notes. It would thus be possible for us to pass through a continuous chain of sounds from any one musical sound to any other without our ear’s ever being able to detect a sudden jump; and this is what we mean by calling our aggregate of sounds a sensory continuum. Suppose now that we were to remove any one of these notes from our piano (excluding the two extreme ones). The continuity of our chain of sounds would be broken, for when we reached the missing note we should detect a sudden variation in pitch as we passed from the sound immediately preceding the removed note to the one immediately following it. In short, the removal of one of the notes would cut our continuous chain of sounds in two. [Pg 24] The dimensionality of our continuum of sounds is obviously unity, for we can assign successive numbers to the successive notes, starting from some standard note, and by this means determine them without ambiguity. Let us complicate matters somewhat by assuming that every individual note may be sounded with various intensities, but always in such a way that a note of given intensity can never be distinguished from the one sounded just a little louder or just a little softer. Once again it will be possible for us to pass in a continuous way from a note of feeble intensity to the same note sounded with louder intensity, without our ear’s ever being able to detect a variation in intensity between two successive sounds. More generally we shall be able to pass in a continuous way from a note of given pitch and given intensity to one of some other pitch and some other intensity.