Part 4

I. The Indefinite Regress.

II. Infinite series.

III. Dimensions in space and time.

Before entering upon them we must repeat that the question of number and series in mathematics is independent of the assumptions of space and time. As a science, mathematics could exist outside them: order is not necessarily spatial or temporal. Our conclusions, therefore, cannot be attacked on the ground that they are based on Euclidean conceptions of space: they are based on the laws of logic.

I. THE INDEFINITE REGRESS

Hume and, later, Kant argued that by the principle of association when we think of one quality of a thing the others are naturally brought before our minds, and thus that we get into the habit of attributing to the notion of the thing a certain group of qualities. And it is true that we do attend to a thing all at once, including in the notion of it all the qualities which we know belong to it.

Now experience, according to Leibniz, gives us an example of a unity which embraces a multiplicity of detail. Thus a thing is one substance as embodying an individual experience, and its qualities belong to it in the same sense as the constituents of experience belong to the single experience. These qualities are in relation. (The Pragmatist denies the existence of relations as part of a higher unity.) But they are not only relation, since relation always implies something more than itself. Let us take the example of number. Numbers could never have been counted if there had not been things to count. Now suppose each quality could be analysed into a new relation, we should still not get rid of the quality. At each stage there remains a quality in relation, and this goes on to Infinity. Such a constant subdivision perhaps results from our finite experience seizing facts in a disjointed way. When we analyse a law in its working, we always do seem to come to this Indefinite Regress. Now it has been the reproach against metaphysics, as uttered by the Pragmatist, that there is no correspondence in scientific fact to this road into Infinity.

W. James asserts: “But in point of fact, nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, &c., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come, that Zeno’s paradox gives trouble. And it gives trouble then only if the successive steps of change be infinitely divisible.”

The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself.

In the Boyle Lecture, delivered in Oxford in 1908, on the properties of radium, two facts emerged which show that the Indefinite Regress is now recognised in science.

First, that in the region of experiment we become aware of groups of elements allied to radium, which seem, in the number of individuals in their groups, to follow a simple arithmetical progression.

Secondly, that radio-active elements lose in activity at a certain rate, which always represents an exact proportion of the mass which remains. The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity.

II. INFINITE SERIES

There are other aspects of Infinity which we can get at by studying series, and which in the conception of series of series give strength and point to the philosophic conception of an Absolute.

Prof. C. Keyser develops this thought, and shows (in two recent articles, January and April 1909, in the Hibbert Journal) that certain theological dogmas, such as the doctrine of the Trinity, and certain attributes of the Divine Being, such as Omniscience and Omnipresence, are entirely conceivable by the human mind if regarded without the paralysing limitations of the Finite. He shows that in our mathematical formulæ which have to do with infinite series we have the exact replica of what to the lay, non-mathematical mind seem to be the paradoxes of the Athanasian Creed. He first shows that in a mathematical analogy points of view about an Infinite Being, even if partially discordant, may all be true if regard is had to His Infinity.

Further, he shows that certain assumptions, such as the whole is greater than its part, are inapplicable to Infinite Being. The conception of a Trinity in Unity in which “none is afore or after other, none is greater or less than another, but the whole three persons are co-eternal together and co-equal” is rationally conceivable by the mathematician who is familiar with the theory of manifolds.

We have, he shows, three infinite manifolds:--

E of the even integers.

O of the odd ones.

F of the fractions having integers for their terms.

No two of these have a single element in common, yet the three together constitute one manifold M, that is exactly equal in wealth of elements to each of its infinite components.

Again, there is the apparent opposition between the Omniscience of God and the freedom of man. The antithesis disappears if we realise that from the point of view of Infinites the dignity and power of Omniscience remain the same, even if some part of experience is not yet drawn into the sphere of Omniscience.

Here we have the present conceived of as a moving plane separating the unknown from the known. The “past” can be said to be known, though its content changes every instant. This is the real answer to W. James’s cry that he could accept an Absolute if it had even the fragment of an “other.” There can be this “other,” and yet the Absolute still remains an Absolute.

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.”

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?