content with determining the order and position of given elements in the sense of an analysis situs). But that is also all that can be said a priori; whether two observers will coincide in their judgment as to the simultaneity of two elements of Experience must for the time being remain an open question. We must not forget, too, that we have hitherto paid no attention to the metrical properties of a physical continuum and we cannot a priori preclude the possibility of difference in this direction without offending against the methodological principle of economy in hypotheses, and unduly circumscribing possibilities which are in no way at variance with the laws of our thought.
49. But even here we must not go too much to extremes; the metrical part of our study, which we have deferred to a later work, and which is of the greatest importance for physics, lies in investigating the relation of “interval” (distance) between two elements of a given continuum; this relation in a multidimensional continuum is a certain mathematical function of its components, which we will call “distances,” and which enable us uniquely to denote individual elements by co-ordinates already having more than a descriptive function. It would, indeed, be meaningless to affirm that we are free to premise that the interval between two elements will be different for two observers: an interval is an interval, that is, something in the given continuum, and though different observers may call it by different names, their speaking of it in different ways does not alter it. We may state this fact in different words, or, if we wish, lay it down as a postulate, saying that the total interval between any two elements of Experience is independent of, or invariant with respect to, the observer. It is otherwise with the partial “distances”: all that is required in their case is that a certain function of them, including them all, should produce the same total interval. Here the main problem will be whether there is in a single Experience something (a structure) which directly determines these partial “distances,” or whether it is left to the observer to determine them (in our case spatially and temporally) as well as he can. In reality it appears that this freedom is left to the observer, or, properly speaking, imposed upon him; and upon this freedom depend those differences in estimating the spatial and temporal lengths which appear so paradoxical in the theory of Relativity. Even this freedom, however, I maintain, is not absolute, being limited by the principle, which I call the principle of maximum uniformity, and which requires space-time distances to be so determined as to permit, in relation to one another, of the most uniform possible description of Experience.
