Chapter 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.[26]

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.[27]

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.[28]

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.”[29]