The modern study of the Algebra of Logic, founded by Boole, and continued by Jevons, by Mr. Venn, and by still others in Great Britain, by Mr. Charles Peirce in America, and by Schroeder in Germany, has also contributed to set the whole theory of exact reasoning in a light at once clearer than that of old, and of a nature to reveal new problems. No longer can you venture, in the exact sciences, to make your appeal to dogmatically asserted “ultimate necessities” of reason. The mathematician is no longer fond of mere axioms. And despite what we have just said about the way in which the mathematician seems to transcend our present form of experience, a closer study shows that it is still our very experience itself that is the mathematician’s only guide to concrete results. Experience is made better by no mean, but experience makes that mean. For in modern mathematical study, even when you deal with irrational numbers, like π, and estimate their properties with an exactness that no physical experience of ours can hope to follow, — yes, even if you take the wings of the Calculus, or of the Theory of Functions, and fly unto the uttermost parts of the realm of the quantitative infinite, even there, in an unexpected, but not the less compelling sense, actual experience guides you, presented facts sustain you.
For, strangely enough, the logical outcome of this whole recent review of the bases of mathematical science can be expressed by saying that the modern mathematician rightly doubts every attempt to prove any proposition in his science unless, in trying to prove, you can first empirically show him, in a fashion that he can ac-
