to a given type of order, form a collection, or an aggregate; and if the order of a given aggregate is to be unambiguous, the members of this aggregate must be uniquely determined, and must be mutually exclusive: they must be what we shall later denote by the term elements of the aggregate. Externality and ‘elementarity’ are only different consequences of the same relation, which we denote by the term extension, and which thus together with the relation of order, will constitute the basis of our definition of space and time.
To formulate this definition, to find what in our perceptions constitutes space and time, we shall have to undertake an analysis of the perceptual datum; and for this purpose we must collect and define a number of concepts which will serve us as the tools and foundations of our analysis. As we wish to eliminate the danger of a possible logical circle in our definition, we must be on the alert to prevent any specially spatio-temporal concepts from creeping surreptitiously into our definition of space and time through the material on which this definition is to be based; consequently great stress will be laid upon the purely formal, non-perceptual character of our basic materials; hence also the purely logical, formal character of this section.
Extension.—9. The first fundamental and undefinable, although generally and unambiguously intelligible relation, which we shall place at the basis of our investigation, is the relation of whole to part, which we shall briefly denote as the relation of inclusiveness. This relation is purely formal: although it holds good as a special case in the world of our perceptions, it can be also predicated of non-perceptual, purely conceptual entities: of numbers, speech, sensations, etc.; it therefore does not of itself contain anything spatial or temporal.
Extension we shall define as the possibility of inclusiveness: a given entity will be said to possess extension, to be extended, if taken by itself it admits of the relation of inclusiveness, i.e., if it is divisible into parts. The relation of extension is implied in the relation of inclusiveness, and is therefore as purely formal a concept as the latter; but although, being void of all spatial and temporal content, it possesses the characteristics necessary for the purposes of our definition, it is not adequate to define space and time. To say, for instance, that space is pure extension, would be equal to maintaining that any extended pure concept, e.g., human speech, has spatial relations; the fact that all extended entities admit of spatial representation is not, as we can readily see, a proof of the spatiality of extension: the
