realm of ideal objects, there may then arise the further question whether, within that realm, an object that meets certain new requirements, the special requirements, let us say, of a given problem, can be found or not. And this question is one whose answer, for the mathematician, is indeed hereupon not at all a matter of his arbitrary choice. He has, to be sure, created his world of mathematical objects. This world is there, as it were, by his decree, or is real, as ordinary realists would say, only in his head. It is so far like a child’s fairyland. But once created, this world, in its own eternal and dignified way, is as stubborn as the rebellious spirits that a magician might have called out of the deep.
Even the poets have told us how their heroes, once created, have often become, as it were, alive after their own fashion, so that the poet could no longer voluntarily control how they should behave. Much more, and for a far more exact reason, are the mathematician’s objects, when once created, independent of his private will. Thus then there may indeed arise the question whether, as one may now express the matter, there exists any object, within a given mathematical realm, possessed of certain properties. The what of this now sought object is defined, in advance, in terms of these mentioned properties, — properties which, as just said, usually result from the conditions of a special problem. The that of the object, its presence as a member of the ideal realm which the mathematician has before defined, is a problem such as may cause almost endless trouble before it is solved. The pro-
