Note 63 — Introductory material

In the first place there is, if I am not mistaken, considerable obscurity in the definition of a manifold, of which an almost verbal rendering was given in Chapter I. What is meant, to begin with, by a general conception capable of various determinations? Does not this property belong to all conceptions? It affords, certainly, a basis for counting, but if continuous quantity is to arise, we must, surely, have some less discrete formulation. It might afford a basis, for example, for the distinction of points in projective Geometry, but projective Geometry has nothing to do with quantity. Something more fluid and flexible than a conception, one would think, is necessary as the basis of continua. Then, again, what is meant by a quantum of a manifold? In space, the answer is obvious: what is meant is a piece of volume. But how about Riemann's other continuous manifold, colour? Does a quantum of colour mean a single line in the spectrum, or a band of finite thickness? In either case, what are the magnitudes to be compared? And how is superposition necessary, or even possible? A colour is fixed by its position in the spectrum: two lines in the same spectrum cannot be superposed, and two lines in different spectra need not be—their positions in their respective spectra suffice, or even, roughly, their immediate sense-quality. The fact is, Riemann had space in his mind from the start, and many of the properties, which he enunciates as belonging to all manifolds, belong, as a matter of fact, only to space. It is far from clear what the magnitudes are which the various determinations make possible. Do these magnitudes measure the elements of the manifold, or the relations between elements? This is surely a very fundamental point, but it is one which Riemann never touches on. In the former case, the superposition which he speaks of becomes unnecessary, since the magnitude is inherent in the element considered. We do not require superposition to measure quantities corresponding to different tones or colours; these can be discovered by analysis of single tones or colours. With space, on the other hand, if we seek for elements, we can find none except points, and no analysis of a point will find magnitudes inherent in it—such magnitudes are a fiction of coordinate Geometry. The magnitudes which space deals with, as we shall see in Chapter III., are relations between points, and it is for this reason that superposition is essential to space-measurement. There is no inherent quality in a single point, as there is in a single colour, by which it can be quantitatively distinguished from another. Thus the conception of a manifold, as defined by Riemann, either does not include colours, or does not involve superposition as the only means of measurement. From this dilemma there is no escape.