that the very bases of mathematical science are certain ultimate necessities of thought, for which no empirical warrant can be given. The world of validity also often appears as a world containing an essentially eternal truth. But, as it may now be asked, does our experience, as such, ever compass eternity? Moreover, one who asserts the objective validity of an idea, even in a merely temporal sense, transcends by his very assertion the circle of his present experience. In brief, every form of Critical Rationalism involves a confidence in a reasoning process. But is reasoning identical with experience?
These considerations may serve to introduce a still further reflection upon the deeper meaning of our Third Conception. As a fact, it is far too easy to talk of validity without analyzing its foundation. But if you thus analyze, you are led to a view of the nature of ideas, and of the reasoning process, which indeed shows that our very conception of validity needs a further supplement before it can be accepted as at once consistent, and adequate to its own undertaking.
The theory of reasoning has received, in recent logical and scientific thought, an extensive reëxamination, which students of metaphysics can no longer ignore. Nowhere has this theory been more carefully revised than in the history of modern elementary mathematics. A frequent experience of inconsistencies and of apparent paradoxes, due to extremely subtle errors in exact method, has led mathematicians, within the past fifty years, to a thoroughgoing attempt at a review of the very bases of Arithmetic, of Geometry, and of Analysis.
