the fog away from some of the philosophical notions. The universe cannot be constructed by mere thinking, mathematics and other thinking are not the result of a universal characteristic, the intuition has been reinstated, and at the same time conditioned in its action, the real source of verity in mathematics has been exhibited. Says Brunschvicg:1 ress. The mathematical intellectualism is henceforth a positive doctrine, but one that inverts the usual doctrines of positivism: in place of originating progress in order, dynamics in statics, its goal is to make logical order the product of intellectual progThe science of the future is not enwombed, as Comte would have had it, as Kant had wished it, in the forms of the science already existing; the structure of these forms reveals an original dynamism whose onward sweep is prolonged by the synthetic generation of more and more complicated forms. No speculation on number considered as a category a priori enables one to account for the questions set by modern mathematics space affirms only the possibility of applying to a multiplicity of any elements whatever, relations whose type the intellect does not undertake to determine in advance, but, on the contrary, it asserts their existence and nourishes their unlimited development. These things the philosopher must learn along with his apprehension of modern science and all it, too, has to say about the world, knowledge, and truth. "The consideration of mathematics is at the base of knowledge of the mind as it is at the base of the natural sciences, and for the same reason: the free and fertile work of thought dates from that epoch when mathematics brought to man the true norm of truth.”2 1 Les étapes de la philosophie mathématique, pp. 567–568. Finally we find a domain for the validity of mathemathics in a region that might seem at first remote indeed. But nevertheless the truth in mathematics, a free creation of the imagination incarnated in forms of the reason, guarantees the truth of other free creations of the imagination when they are set forth in the realities of life. Poetry, music, painting, sculpture, architecture-may we call them the other fine arts ?-create the beautiful and give expression to the longings and hopes of man. But they have been told for centuries that these were but dreams, visions of that which did not exist, sad to say, fictions that one could but view for awhile, then, with a sigh, return to cold reality. Mathematics vindicates the right of all these to stand in the front rank of the pioneers that search the real truth and find it crystallized forever in brilliant gems. The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth. The scientist studies nature for the same reason, and in its harmonies finds also everlasting truth. But the nature he studies is the creature of his own construction. His conceptions and theories and scientific systems he really builds himself. So, too, the. artist sees beauty and constructs imperishable forms which also have everlasting truth. Many mathematicians have borne witness to the element of beauty in mathematics: Poincaré, high priest of beauty in mathematics and science, Sylvester, who wrote rhapsodies in the midst of his mathematical memoirs, Pringsheim, Kummer, Kronecker, Helmholtz, Bôcher, B. Peirce, Russell, Hobson, Picard, Hadamard-why prolong the list? And because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination. Because it contains freedom, it guarantees freedom to the whole realm of art. Because it is not primarily utilitarian, it validates the joy of imagination for the pure pleasure of imagination. "Not in the ground of need, not in bent and painful toil, but in the deep-centred play-instinct of the world, in the joyous mood of the eternal Being, which is always young, science has her origin and root; and her spirit, which is the spirit of genius in moments of elevation, is but a sublimated form of play, the austere and lofty analogue of the kitten playing with the entangled skein or of the eaglet sporting with the mountain winds.” REFERENCES Keyser, The Human Worth of Rigorous Thinking, 1916. I Keyser, Mathematics, Columbia University Lectures, 1907. |