This would be the formula of the physical continuum, as crude experience gives it to us, whence arises an intolerable contradiction that has been obviated by the introduction of the mathematical continuum. This is a scale of which the steps (commensurable or incommensurable numbers) are infinite in number but are exterior to one another, instead of encroaching on one another as do the elements of the physical continuum, in conformity with the preceding formula.
The physical continuum is, so to speak, a nebula not resolved; the most perfect instruments could not attain to its resolution. Doubtless if we measured the weights with a good balance instead of judging them by the hand, we could distinguish the weight of 11 grams from those of 10 and 12 grams, and our formula would become:
A < B, B < C, A < C.
But we should always find between A and B and between B and C new elements D and E, such that
A = D, D = B, A < B; B = E, E = C, B < C,
and the difficulty would only have receded and the nebula would always remain unresolved; the mind alone can resolve it and the mathematical continuum it is which is the nebula resolved into stars.
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 E1, E2, ..., En, such: (1) that they all belong to C; (2) that each of them is indistinguishable from the following, E1 indistinguishable from A and En 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 E1, E2, ..., En satisfying the first two conditions, there will be an element E indistinguishable from one of the elements of the cut. In the first case we can go from A to B by a continuous route without quitting C and without meeting the cuts; in the second case that is impossible.