[1] Quoted from a lecture by the present writer on “The Law of Correspondences Mathematically Considered,” delivered before The Theological and Philosophical Society on 26th April 1912, and published in Morning Light, vol. xxxv (1912), p. 434 et seq.

The Pythagorean doctrine of the Cosmos, in its most reasonable form, however, is confronted with one great difficulty which it seems incapable of overcoming, namely, that of continuity.  Modern science, with its atomic theories of matter and electricity, does, indeed, show us that the apparent continuity of material things is spurious, that all material things consist of discrete particles, and are hence measurable in numerical terms.  But modern science is also obliged to postulate an ether behind these atoms, an ether which is wholly continuous, and hence transcends the domain of number.[1] It is true that, in quite recent times, a certain school of thought has argued that the ether is also atomic in constitution—­that all things, indeed, have a grained structure, even forces being made up of a large number of quantums or indivisible units of force.  But this view has not gained general acceptance, and it seems to necessitate the postulation of an ether beyond the ether, filling the interspaces between its atoms, to obviate the difficulty of conceiving of action at a distance.

[1] Cf. chap. iii., “On Nature as the Embodiment of Number,” of my A Mathematical Theory of Spirit, to which reference has already been made.

According to BERGSON, life—­the reality that can only be lived, not understood—­is absolutely continuous (i.e. not amenable to numerical treatment).  It is because life is absolutely continuous that we cannot, he says, understand it; for reason acts discontinuously, grasping only, so to speak, a cinematographic view of life, made up of an immense number of instantaneous glimpses.  All that passes between the glimpses is lost, and so the true whole, reason can never synthesise from that which it possesses.  On the other hand, one might also argue—­extending, in a way, the teaching of the physical sciences of the period between the postulation of DALTON’S atomic theory and the discovery of the significance of the ether of space—­that reality is essentially discontinuous, our idea that it is continuous being a mere illusion arising from the coarseness of our senses.  That might provide a complete vindication of the Pythagorean view; but a better vindication, if not of that theory, at any rate of PYTHAGORAS’ philosophical attitude, is forthcoming, I think, in the fact that modern mathematics has transcended the shackles of number, and has enlarged her kingdom, so as to include quantities other than numerical.  PYTHAGORAS, had he been born in these latter centuries, would surely have rejoiced in this, enlargement, whereby the continuous as well as the discontinuous is brought, if not under the rule of number, under the rule of mathematics indeed.