But what is a continuum with n dimensions; how does it differ from a continuum whose number of dimensions is larger or smaller? First, let us recall some results recently obtained by the students of Cantor. It is possible to correspond one by one the points of a line to those of a plane, or, more generally, those of a continuum with n dimensions to those of a continuum with p dimensions. This is possible, provided that one does not limit oneself to the condition that two points infinitely close to the line correspond to two points infinitely close to the plane, that is to say, to the condition of continuity.
We can thus deform the plane so as to obtain a line, provided that this deformation is not continuous
To define the continuum with n dimensions, we first have the analytic definition; a continuum with n dimensions is a set of n coordinates, that is, a set of n quantities that can vary independently of each other and to take all the real values satisfying certain inequalities. This definition, irreproachable from a mathematical point of view, can not, however, satisfy us entirely. In a continuum, the various coordinates are not, so to speak, juxtaposed to one another, they are connected with one another so as to form the various aspects of a whole. At every moment by studying space, we do what we call a change of coordinates, for example we make a change of rectangular axes; or we move to curvilinear coordinates. By studying another continuum, we also make coordinate changes, that is, we replace our n coordinates by n any continuous functions of these n coordinates. For us who derive the notion of the continuum with n dimensions, not from the above-mentioned analytical definition, but from some deeper source, this operation is quite natural; we feel that it does not alter what is essential in the continuum. For those, on the contrary, who would know the continuous only by the analytic definition, the operation would be licit, no doubt, but baroque and ill justified.
Lastly, this definition makes a good bargain of the intuitive origin of the notion of continuity, and of all the riches that this notion conceals. It fits into the type of definitions that have become so common in mathematics, since we tend to “arithmetize” this science. These definitions, irreproachable, as we have said, from the mathematical point of view, can not satisfy the philosopher. They replace the object to be defined and the intuitive notion of this object by a construction made with simpler materials; we can clearly see that we can actually do this construction with these materials, but we can see at the same time that we could do just as well many others; what it does not show is the profound reason why we have assembled these materials in this way and not from another. I do not mean to say that this “arithmetization” of mathematics is a bad thing, I say that it is not everything.
I will base the determination of the number of dimensions on the notion of cut. Let us first consider a closed curve, that is, a one-dimensional continum; if, on this curve, we mark any two points by which we will forbid ourselves to pass, the curve will be divided into two parts, and it will become impossible to pass from one to the other while remaining on the curve and without going through prohibited points. On the contrary, be a closed surface constituting a two-dimensional continuum; we can mark on this surface, one, two, any number of forbidden points; the surface will not be decomposed into two parts, it will be possible to go from one point to the other of this surface without encountering an obstacle, because we can always turn around the forbidden points.
But if we draw on the surface one or more closed curves and if we consider them as cuts that we will forbid to cross, the surface may be divided into several parts.
Let us now come to the case of space; it can not be broken down into several parts, nor by forbidding to pass by certain points, nor by forbidding to cross certain lines; we could always turn these obstacles. It will be forbidden to cross certain surfaces, that is to say certain cuts in two dimensions; and that is why we say that space has three dimensions.
We now know what a continuum n is. A continuum has n dimensions when we can break it down into several parts by making one or more cuts which are themselves continuums with n – 1 dimensions. The continuum with n dimensions is thus defined by the continuum with n – 1 dimensions; it is a definition by recurrence.
What gives me confidence in this definition, which shows me that this is how things naturally come to mind, is first of all that many authors of treaties, elementary, did something similar at the beginning of their works. They define volumes as portions of space, surfaces as the boundaries of volumes, lines like those of surfaces, points like lines; after which they stop and the analogy is obvious. Then, in the other parts of Analysis Situs, we find the important role of the cut; it is on the cut that everything rests. What, according to Riemann, distinguishes, for example, the torus of the sphere? it is because we can not draw on a sphere a closed curve without cutting this surface in two; while there are closed curves which do not cut the torus in two, and that it is necessary to practice two closed cuts having nothing in common to be sure of having divided it.
There is one more point to deal with. The continuums of which we have just spoken are mathematical continuums; each of their points is an individual absolutely distinct from the others, and, moreover, absolutely indivisible. The continuums which our senses directly reveal to us, and which I have called physical continuities, are quite different. The law of these continuities is the law of Fechner, which I will strip of the pompous mathematical apparatus which usually surrounds it to reduce it to the simple statement of the experimental data on which it rests. It is known to judge a weight of 10 grams of a weight of 12 grams; one can not distinguish a weight of 11 grams, neither that of 10 grams nor that of 12 grams. More generally there may be two sets of sensations that we distinguish from each other, without us being able to distinguish either one from the same third. Once this is done, we can imagine a continuous chain of sets of sensations so that each of them is indistinguishable from the next, although the two ends of the chain are easily discerned; this will be a one dimensional physical continuum. We can also imagine more complex physical processes. The elements of these physical continuums will still be sets of sensations (but I prefer to use the word element which is simpler). When shall we say that a system S of similar elements is a physical continuum? It is when one can consider any two of its elements as the ends of a continuous chain analogous to the one of which I have just spoken and of which all the elements belong to S. Thus a surface is continuous, if any two of its points can be joined by a continuous line which does not leave the surface.
Can we extend the notion of interruption to the physical continuums and determine the number of their dimensions? Obviously yes. Suppose we forbid certain elements of S, and all those that can not be discerned. These prohibited elements may also be in finite number, or form by their meeting one or more continuous. All of these prohibited elements will constitute a cut; and it will be possible that after having made this cut, we have shared the continuum S in several others, so that it is not possible any more to pass of any element of S to any other element by a continuous chain, no element of this chain being indistinguishable from any element of the cut.
Then a physical continuum that can be cut in this way by prohibiting a finite number of elements will have one dimension; a physical continuum will have n dimensions, if we can cut it by making cuts there which are themselves physical continuums with n – 1 dimensions.