The Physical Continuum of Several Dimensions

We have discussed above the physical continuum as it is derived from the immediate evidence of our senses, from the rough results of Fechner’s experiments.

These results are summed up in the contradictory formulæ:

A = B,
B = C,
A < C.

How is this notion generalised?

How from it may be derived the concept of continuums of several dimensions?

Consider any 2 aggregates of sensations. We can either distinguish between them, or we cannot, just as in Fechner’s experiments the weight of 10 grams could be distinguished from the weight of 12 grams, but not from the weight of 11 grams.

This is all that is required to construct the continuum of several dimensions.

Let us call one of these aggregates of sensations an element.

It will be in a measure analogous to the point of the mathematicians, but will not be, however, the same thing. We cannot say that our element has no size, for we cannot distinguish it from its immediate neighbours, and it is thus surrounded by a kind of fog.

If the astronomical comparison may be allowed, our “elements” would be like nebulæ, whereas the mathematical points would be like stars.

If this be granted, a system of elements will form a continuum, if we can pass from any one of them to any other by a series of consecutive elements such that each cannot be distinguished from its predecessor.

This linear series is to the line of the mathematician what the isolated element was to the point.

Before going further, I must explain what is meant by a cut. Let us consider a continuum C, and remove from it certain of its elements, which for a moment we shall regard as no longer belonging to the continuum. We shall call the aggregate of elements thus removed a cut.

By means of this cut, the continuum C will be subdivided into several distinct continuums; the aggregate of elements which remain will cease to form a single continuum. There will then be on C two elements, A and B,mathematical magnitude.

which we must look upon as belonging to two distinct continuums. We see that this must be so, because it will be impossible to find a linear series of consecutive elements of C (each of the elements indistinguishable from the preceding, the first being A and the last B), unless one of the elements of this series is indistinguishable from one of the elements of the cut.

It may happen, on the contrary, that the cut may not be sufficient to subdivide the continuum C. To classify the physical continuums, we must first of all ascertain the nature of the cuts which must be made in order to subdivide them.

If a physical continuum, C, may be subdivided by a cut reducing to a finite number of elements, all distinguishable the one from the other (and therefore forming neither one continuum nor several continuums), we shall call C a continuum of one dimension.

If, on the contrary, C can only be subdivided by cuts which are themselves continuums, we shall say that C is of several dimensions; if the cuts are continuums of one dimension, then we shall say that C has two dimensions; if cuts of two dimensions are sufficient, we shall say that C is of three dimensions, and so on.

Thus the notion of the physical continuum of several dimensions is defined, thanks to the very simple fact, that two aggregates of sensations may

be distinguishable or indistinguishable.

The Mathematical Continuum of Several Dimensions

The conception of the mathematical continuum of n dimensions may be led up to quite naturally by a process similar to that which we discussed at the beginning of this chapter. A point of such a continuum is defined by a system of n distinct magnitudes which we call its co-ordinates.

The magnitudes need not always be measurable; there is, for instance, one branch of geometry independent of the measure of magnitudes, in which we are only concerned with knowing, for example, if, on a curve ABC, the point B is between the points A and C, and in which it is immaterial whether the arc AB is equal to or twice the arc BC.

This branch is called Analysis Situs. It contains quite a large body of doctrine which has attracted the attention of the greatest geometers, and from which are derived, one from another, a whole series of remarkable theorems. What distinguishes these theorems from those of ordinary geometry is that they are purely qualitative.

They are still true if the figures are copied by an unskilful draughtsman, with the result that the proportions are distorted and the straight lines replaced by lines which are more or less curved.mathematical magnitude.

As soon as measurement is introduced into the continuum we have just defined, the continuum becomes space, and geometry is born. This is discussed in Part 2.