Note 5 — Chapter I · MANIFOLDS

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.