Page:Mind-a quarterly review of psychology and philosophy, vol33, no129 (1924).djvu/14

 in works on the foundations of mathematics, the whole theory of order can be developed. The exact logical formulation of the properties of these order-relations offers considerable difficulties with which we shall not concern ourselves here, referring the reader to standard works on the matter; in these works we can also satisfy ourselves that the order-relations can be defined without any reference to space and time, and are intrinsically independent of these concepts; the concept of order is there fore purely formal and its use in the definition of space and time cannot involve us in a vicious circle.

14.1. We say that a given aggregate of any elements (or mutually exclusive parts) is ordered, if of any two of its elements we can say that one is before the other, or, what is the same thing, that one is after the other; or if of any three elements of the aggregate we can say that one is between the other two. A given aggregate of elements is said to be properly ordered, when the case “X is between A and B” excludes the cases “B is between X and A” and “A is between X and B”'; that is, if we say that A is before B, we cannot also say that B is before A.

Boundary.—15. With the aid of the concept of dissection it is possible to define the concept of boundary, which is of the greatest importance in the theory of ordered aggregates; the concept is one of the most difficult to define in a purely formal way, and, as in the case of order, for its exact definition we shall refer the reader to standard works on the foundations of mathematics. We shall here content ourselves with an incomplete definition which we think sufficient for our purpose, and define the boundary of the entity R in the entity X (of which R is a part), as that characteristic of R, which R has in common with co-R, and by virtue of which R is also distinct from co-R.

The definition is sufficient to enable us to see that the boundaries of two entities can intersect (that two entities can have a part of their boundary in common). Of two non-intersecting entities, which have a part of their boundary in common, we shall say that they are joined; two non-intersecting entities which are not joined we shall call separate. Further, we shall say that an entity R is enclosed in the entity X, if every part of R is a part of X, and if for every part of R (say r) there can be found two parts of co-R, (say a and b) such that r is between a and b. It can be readily seen that if R is an entity enclosed in X, the whole of its boundary will be a part of the boundary of co-R, the rest of the boundary of which will be defined when the