that the mathematician, because of his far more controllable subject-matter, is generally surer of finding his way erelong past these contrasts to the truth that he seeks, while in the physical sciences the ontological errors may persist longer.
As to the method of work used by mathematicians in such cases, where the existence of an object is in question, I again speak quite as a layman in this field; but, so far as I have observed, the mathematicians, in proving the sort of existence of which they speak, proceed very much like students of other types of real Being. To prove the existence of an object whose what is already stated, but whose that is in question, the mathematician may simply produce, as it were, before your eyes, an object of the desired type, and may then let you observe that it meets the requirement. In such cases he works somewhat as a naturalist might do. He shows you the object and says: “See, it exists.” Or again, he may be unable to do this; but instead he may try a sort of experiment with his already accessible ideal objects, and the result of this experiment may give you an indirect but infallible sign that a being of the precise sort here in question must exist, even if it cannot be directly produced. This more indirect method of showing that a being of a given type exists, may roughly be compared to the devices by which the spectroscope reveals the existence of an element in a star, by showing the characteristic lines of the element.
In brief, then, in talking of this his shadowland of ideal beings, the pure mathematician illustrates, in ways often very remarkable, how manifold may be the meanings that
