Valid in its own measure it is, — to say that is to utter the deep commonplace of St. Augustine’s form of the ontological proof of the existence of God. For it must indeed be true that there is a Veritas. Yet mere Veritas, mere validity, still remains to us a conception as unintelligible as it is insistently present to our thought. And our difficulty at the last time came thus to light: In mathematics, you define and prove valid assertions, and deal with entities, such as roots of equations, and properties of functions, whose Being seems to mean only their validity. But how do you prove these propositions about validity? How do you test the existence of your mathematical objects? Merely by experimenting upon your present ideas. What is there before you as you thus experiment? At each step of your procedure, one moment’s narrow contents extend to the very horizon of your present finite mathematical experience. Yet if your procedure is, indeed, as it pretends to be, valid, the truth that you define embraces eternity, and predetermines the structure and the valid existence of an infinity of objects that you regard as external to the thought which defines them. Your world of objects then is here boundless; your human grasp of these objects is even pitiably limited. Validity thus implies, in the world of the mathematical entities, a twofold character. As presented, as seen by you, as here realized, the observed validity is apparently given in experience, indeed, but as a mere internal meaning, — the creature of the instant. But as objective, as genuine, the validity is a part of the endless realm of mathematical truth, a realm that is, to use Aristotle’s term, the Unmoved Mover of all your finite struggle for
