Note 19 — Introductory material

Conceptions of magnitude, according to Riemann, are possible there only, where we have a general conception, capable of various determinations (Bestimmungsweisen). The various determinations of such a conception together form a manifold, which is continuous or discrete, according as the passage from one determination to another is continuous or discrete. Particular bits of a manifold, or quanta, can be compared by counting when discrete, and by measurement when continuous. "Measurement consists in a superposition of the magnitudes to be compared. If this be absent, magnitudes can only be compared when one is part of another, and then only the more or less, not the how much, can be decided" (p. 256). We thus reach the general conception of a manifold of several dimensions, of which space and colours are mentioned as special cases. To the absence of this conception Riemann attributes the "obscurity" which, on the subject of the axioms, "lasted from Euclid to Legendre" (p. 254). And Riemann certainly has succeeded, from an algebraic point of view, in exhibiting, far more clearly than any of his predecessors, the axioms which distinguish spatial quantity from other quantities with which mathematics is conversant. But by the assumption, from the start, that space can be regarded as a quantity, he has been led to state the problem as: What sort of magnitude is space? rather than: What must space be in order that we may be able to regard it as a magnitude at all? He does not realise, either—indeed in his day there were few who realized—that an elaborate Geometry is possible which does not deal with space as a quantity at all. His definition of space as a species of manifold, therefore, though for analytical purposes it defines, most satisfactorily, the nature of spatial magnitudes, leaves obscure the true ground for this nature, which lies in the nature of space as a system of relations, and is anterior to the possibility of regarding it as a system of magnitudes at all.