Page:The World and the Individual, First Series (1899).djvu/273

254 cept, the actual process of construction belonging to, or creative of, the ideal object of which your proposition undertakes to give an account. Construction actually shown is, then, the test. This actual construction must be also not only shown, but carefully surveyed in present experience, before your proof can be estimated. The object of which you speak may be, like π, or like the total collection of all possible rational numbers, or like the quantitative infinite in any form, an object that nobody amongst us men directly observes. But, nevertheless, the fashion of its construction, the type to which it conforms, the law of its nature, the receipt for manufacturing this object, must be capable of adequate presentation in the inner experience of the mathematician, if any exact result is to be obtained. And as thus presented, the basis of the mathematician’s reasoning becomes so far the study of inner experience. The object with which he directly deals is a thing present, seen, given, tested. As our American logician, Mr. Charles Peirce has well said, exact reasoning is a process of experiment performed upon an artificial object, an object made indeed by the mathematician, but observed by him just as truly as a star or as a physiological process is observed by the student of another science, experimented upon just as truly as one experiments in a laboratory. But the marvel is that the present experience of the mathematician with his ideal object somehow warrants him in making assertions about an infinity