The doctrine of Omnipresence follows from the argument of the Continuum (which is the aggregate of all real numbers). Thus the number of points in space of infinite dimension is no greater than the number of points in any part of space as known to us. The whole is incarnate in every part, because to each part, in however small an atom, corresponds a point in the universal whole, and the number of points in a space of infinite dimensions is equal to the number of points in a straight line however small.
And this is true not merely of points but also of forces. “The Universe is dynamic, charged throughout with innumerable modes of motion. Each point, however, of any moving thing—an ion of gas, a vibrating fibre of brain—is represented by a corresponding point in S (a small typical atom), and so within the tiny sphere—indeed, in every room, however small—the whole dynamics of the universe is depicted completely and co-enacted by motion of points and transformation of point configurations. There in miniature proceed at once the countless play and interplay of every kind of motion, small and large, simple and complex, the quivering dance of the molecule, the wave and swing of universal æther.”[29]
III. Dimensions in Space
There is another argument, one relating to the theories of time and space, which greatly affects the conception of omnipresence. This is the argument of the many dimensions, called by Keyser the “radiant concept of hyper-space, which only a generation ago was regarded, even by mathematicians—most adventurous of men—as being purposeless and vain, but which meanwhile has advanced so rapidly to commanding position that even the following statement by Poincaré, in his recent address before the International Mathematical Congress at Rome on ‘L’Avenir des Mathématiques,’ is well within the limits of conservatism: ‘Nous sommes aujourd’hui tellement familiarisés avec cette notion que nous pouvons en parler, même dans un cours d’université, sans provoquer trop d’étonnement.’ The fact is that the doctrine already exists in a vast and rapidly growing literature, flourishes in all the scientific languages of the world, and in its essential principles has become for mathematics as orthodox as the multiplication table.”
The present position of the theory is briefly this: If there did not exist a fourth dimension, we could not be aware of a third as such, and so on. Are we then looking out upon a third dimensional world, and realising it as such because we are mentally capable of conceiving dimensions beyond it? Our world sensibly contains one dimensional and two dimensional facts—the first such as a time series, for which one number is sufficient to fix a point, and the second such as a plane where position can be fixed by two numbers. Does our world contain facts of other dimensions?
“All particles of air are four-dimensional in magnitude when, in addition to their position in space, we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring-stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For in the first place one of the extremities of the measuring-stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of position, and, secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions, we shall obtain all the conceivable positions which the second extremity can assume, and consequently the dimensions must be 3 + 2 or 5 …” &c., &c.[30]
Mathematicians have for long done problems in the seventh and eighth dimensions. They have told us that you cannot tie a knot in the second dimension, because there is no up or down, and the threads would not cross—nor in the fourth, because the knot would pull out in a new direction and would not hold. But it has only lately been realised that fourth and other dimensions may be actual fact in the world round us. Of course, from the point of view of a point there are only three dimensions to be known, but to a line in the same space there are five, to the surface probably six. Our intelligence at present does not go beyond the point; but if we could think of space from the point of view of a solid, worlds upon worlds would rise before our view.
Of the fourth dimension we can discover some facts by analogy. We can count the edges of its typical figure, and apply thought to determining some of its conditions. But a more interesting subject of research is the inquiry into the light thrown by the theory of four dimensions on the determination of certain atoms in chemistry, that are known to be distinct elements, but could only be determined actually in another dimension.[31]
“In chemistry, the molecules of a compound body are said to consist of the atoms of the elements which are contained in the body, and these are supposed to be situated at certain distances from one another and to be held in their relative positions by certain forces. At first the centres of the atoms were conceived to lie in one and the same plane. But Wislicenus was led by researches in paralactic acid to explain the differences of isomeric molecules of the same structural formulæ by the different positions of the atoms in space. In fact, four points can always be so arranged in space that every two of them may have any distance from each other; and the change of one of the six distances does not necessarily involve the alteration of any other.
“But suppose our molecule consists of five atoms? Four of these may be so placed that the distance between any two of them can be made what we please. But it is no longer possible to give the fifth atom a position such that each of the four distances by which it is separated from the other atoms may be what we please. On the contrary, the fourth distance is dependent on the three remaining distances, for the space of experience has only three dimensions. If, therefore, I have a molecule which consists of five atoms, I cannot alter the distance between two of them without at least altering some second distance. But if we imagine the centres of the atoms placed in a four-dimensioned space, this can be done; all the ten distances which may be conceived to exist between the five points will then be independent of one another. To reach the same result in the case of six atoms we must assume a five-dimensional space, and so on.”[32]