new life to this old story; and the realm of that which undertakes to be real only in so far as it is true, is a realm of very distinctly present interest for the philosophy of recent natural science.
As for the purely mathematical instances, in general, however, they are not at all limited to the geometrical ones. Modern Analysis, and the Theory of Functions, contain very many propositions of the class that are sometimes called “Existence-Theorems.” That there exists a root for any algebraic equation of the nth degree; that there exists a differential coefficient for a given function; that, on the other hand, there exist functions continuous throughout given intervals which still have within those intervals no differential coefficients; that the limits of this or that variable quantity (for instance, of convergent infinite series), exist: — such are examples that may be more or less familiar even to students who have, like myself, to confine themselves to decidedly elementary mathematics. Avoiding, however, the mathematical form of expression, one may here try to make clear the metaphysically important nature of theorems of this sort very much as follows: In pure mathematics, the student deals with certain objects that, upon their face, are the products of purely arbitrary definitions. The mathematician builds up these his objects, as, for instance, the objects of pure Analysis, very much as he pleases. His ideas are in so far his facts. So far one would suppose, then, that no questions about existence would trouble the mathematician. But when one looks closer, one sees that when the mathematician has once built up such a notion of some
