Yet up to this point we have not introduced the notion of the number of dimensions. What is meant when we say that a mathematical continuum or that a physical continuum has two or three dimensions?
First we must introduce the notion of cut, studying first physical continua. We have seen what characterizes the physical continuum. Each of the elements of this continuum consists of a manifold of impressions; and it may happen either that an element can not be discriminated from another element of the same continuum, if this new element corresponds to a manifold of impressions not sufficiently different, or, on the contrary, that the discrimination is possible; finally it may happen that two elements indistinguishable from a third may, nevertheless, be distinguished one from the other.
That postulated, if A and B are two distinguishable elements of a continuum C, a series of elements may be found, E1, E2, ..., En, all belonging to this same continuum C and such that each of them is indistinguishable from the preceding, that E1 is indistinguishable from A, and En indistinguishable from B. Therefore we can go from A to B by a continuous route and without quitting C. If this condition is fulfilled for any two elements A and B of the continuum C, we may say that this continuum C is all in one piece. Now let us distinguish certain of the elements of C which may either be all distinguishable from one another, or themselves form one or several continua. The assemblage of the elements thus chosen arbitrarily among all those of C will form what I shall call the cut or the cuts.
Take on C any two elements A and B . Either we can also find a series of elements E 1 , E 2 , ..., E n , such: (1) that they all belong to C ; (2) that each of them is indistinguishable from the following, E 1 indistinguishable from A and E n from B ; (3) and besides that none of the elements E is indistinguishable from any element of the cut . Or else, on the contrary, in each of the series