Page:The World and the Individual, First Series (1899).djvu/234

Rh cesses involved in such ontological or existential solutions are, however, very instructive as to the nature of the ideal world; and every student of metaphysics ought to have at least an elementary acquaintance with a few concrete instances of just such investigations in mathematics.

If one hears children disputing over a fairyland of their own invention, and if the question arises whether or no there exists in that fairyland a particular being, say a fairy with six wings, a listener to the dispute easily grows impatient. “Why talk of reality or of unreality?” he says. “The six-winged fairy exists in your fairyland if you make him, and this is true because you are not talking of any real being at all, but only of make believe.” Yet in the mathematical realm it is not altogether so. Within limits, you create as you will, but the limits once found, are absolute. Unsubstantial, in one way, as fairyland, the creations of the pure mathematician’s ideality still may require of their maker as rigid, and often as baffling a search for a given kind or case of mathematical existence, as if he were an astronomer testing the existence of the fifth satellite of Jupiter, or of the variables of a telescopic star-cluster.

An equation of the wth degree, for instance, is such an ideal mathematical creation. I remember a teacher of mathematics in a far western American town, who used to scoff at the troubles of his historically more famous colleagues regarding the noted theorem as to the existence of a root of such an equation. The equation, as my friend in substance said, was a mathematician’s arbitrary creation. There was no use in calling it an equa-