In the general theory of relativity, space-time appears in two ways: first, as providing a four-dimensional order; secondly, as giving rise to the metrical concept of "interval." Both are relations between "points," but both are treated mathematically as differential relations. This requires us to solve a purely mathematical problem: what is the function or process which tends towards these relations as a limit? This is on the assumption that space-time is continuous, which we do not know it to be. Let us begin with this hypothesis, and proceed afterwards to the hypothesis of discreteness. In the absence of evidence, it is necessary to develop both. For the present, therefore, I assume space-time to be continuous. This involves, or at least renders natural, the assumption that there is an infinite number of events compresent with any given event; I shall make this assumption also so long as I assume continuity. "Compresence" is assumed to be a symmetrical relation, which every term in its field has to itself, and whose field is capable of being well ordered. A group of five events is capable of a relation called "co-punctuality," which means, in effect, that there is a region common to all five. A group[Pg 377] of more than five events is called "co-punctual" when every quintet chosen out of it is co-punctual. A "point" is defined as a co-punctual group of events which cannot be added to without ceasing to be co-punctual. "Events" are defined as the field of the relation of compresence. Hence, by means of not implausible axioms, we arrive at the space-time order presupposed in the assignment of co-ordinates. This part of the theory is straightforward. When we come to "interval" there is more difficulty. In the discussion of measurement we decided, following Eddington, that equality of two intervals is what has to be defined, and that this has to be defined as a limit when both intervals tend towards zero. For this purpose, we supposed a relation of five points , , , , ' which we may express in the words: " is more nearly a parallelogram than ." From this, by means of a certain apparatus of axioms, we can arrive at what seems to be metrically necessary for mathematical physics. But this procedure is somewhat artificial. It seems natural to suppose that our relation of five points arises as follows: between any two points there is a relation, which for the moment we will call "separation," and the separation of and is more like that of and than like that of and . Thus we shall have to do with degrees of resemblance between separations of point-pairs; these separations, however, cannot exist only for infinitesimal distances, but must exist for finite distances, at any rate if they are sufficiently small.