that creation of this critical character is not so fundamental as that of a synthetic character, but it is in the end necessary and extremely useful. Other cases might be cited, such as the curves that fill up an area, the Jordan curves, the monogenic non-analytic functions of Borel, etc. These are no more artificial than were in their day the negative and the imaginary. Indeed, the time may come when the demands of physics may make it necessary to consider the path of an electron to be a continuous non-differentiable curve, and the Borel function may become a necessity to explain the fine-grained character of matter. "To the evolution of Physics should correspond an evolution in Mathematics, which, of course, without abandoning the classic and well-tried theories, should develop, however, with the results of experiment in view."

The origin of these creations is a most interesting question for the psychologist and is buried in the mysterious depths of the mind. An interesting account of it is given by Poincaré in a description of some of his own creations, to be found in his book, Science et méthode. His conclusion may be stated briefly thus: the mind is in a state of evolution of new ideas and new mental forms, somewhat continuously. Of those that come to the front some will have a certain relation of harmony and fitness for the problem at hand, which secures for them keen attention. They may turn out to be just what is wanted, sometimes they may turn out to be unfit or even contradictory. There seems little to add to this statement, for it pretty accurately describes what every reflective mathematician has observed in his own mental activity. A little emphasis may be laid, however, on the

1 Borel, Lecture at Rice Institute, 1912; Introduction géométrique à quelques théories physiques, pp. 126–137.

significance of the fact that sometimes the newborn notions are contradictory to the known theorems, because this fact shows conclusively that the mind is not impelled to its acts by a blind causality. In that case the new forms would have to be always consistent. This faculty is analogous to that possessed by the artist. Indeed, many have noted the numerous relations of mathematics to the arts that create the beautiful. Sylvester1 said: "It seems to me that the whole of aesthetic (so far as at present revealed) may be regarded as a scheme having four centers, which may be treated as the four apices of a tetrahedron, namely, Epic, Music, Plastic, Mathematic.” Poincaré was specially the advocate of the aesthetic character of mathematics, and reference may be made to his many essays. Many others have mentioned the fact in their addresses. The cultivation of the aesthetic sensitiveness ought, therefore, to assist the creative ability of the mind.

Poincaré points out that these flashes of inspiration usually follow long and intense attention to a problem. That is, one must endeavor to generalize, to turn the searchlight of intuition on the problem, to deduce from every phase of it all the consequences that follow, and then he must trust to the spontaneity of the mind some day to furnish the newborn creature that is engendered by these processes. The process of maturing the conception may even take years. This fourth method is the culmination, the crown, of the others and of the acquisitions of the mathematical student. He must read widely, scrutinize intently, reflect profoundly, and watch for the advent of the new creatures resulting. If he is of a philosophic

1 Collected Papers, 3, p. 123.

turn, he will have the satisfaction of knowing that he is able to see knowledge in the process of creation, and that of all reality he has the most secure. He will know that the flowers of thought whose growth and bloom he superintends are immortelles and the infinite seasons of the ages will see them in everlasting fragrance and beauty.

REFERENCES

Milhaud, Études sur la pensée scientifique, 1906.

Nouvelles études sur l'histoire de la pensée scientifique,

1911.

CHAPTER XVI

VALIDITY OF MATHEMATICS

We have surveyed the whole of mathematics, finding it to be a constantly growing creation of the intellect, constructed primarily for its own sake. The mathematician builds because he enjoys the building, and the fascination of his creation is the impetus that keeps him creating. It is not the usefulness of what he creates, but the innate beauty of it that he is forever thirsting for. Poincaré,1 the subtlest of the mathematical philosophers, said:

The scientist does not study nature because it is useful; he studies it because it pleases him, and it pleases him because it is beautiful. Were nature not beautiful, it would not be worth knowing, life would not be worth living. I do not mean here, of course, that beauty which impresses the senses, the beauty of qualities and appearances; not that I despise it— far from it; but that has nought to do with science; I mean that subtler beauty of the harmonious order of the parts which pure intellect perceives. This it is which gives a body, a skeleton as it were, to the fleeting appearances that charm the senses, and without this support the beauty of these fugitive dreams would be but imperfect, because it would be unstable and evanescent. On the contrary, intellectual beauty is selfsufficient, and for its sake, rather than for the good of humanity, does the scientist condemn himself to long and tedious labors.

The Greeks studied conic sections two thousand years before they were of use to anyone at all, and the imaginary and complex functions were developed long before they were of use to the wireless telegrapher. Nevertheless, • Science et méthode, p. 15.

the tree Yggdrasil has its roots in the earth and thence draws sustenance for its growth. As Poincaré and Borel point out, many of the notions of mathematics had their origin in the demands of physics for a scheme by which it could think the material world—such notions as continuity, derivative, integral, differential equation, vector calculus, and the integral and integro-differential equation. Equally the learning of mankind in its efforts to understand its own whence, why, and whither has furnished sustenance for another root of the trunk. Pythagoras, Plato, Leibniz, Kant, Poincaré merely to mention these names brings to mind the debt of mathematics to the philosophic thought of the centuries. The search for that in life which was definitive, for freedom of the intellect, for the unity and harmony of the spirit as well as of nature-all these have contributed to the sustenance of the trunk, even if they could not be part of the tree. And the other root was the accumulated learning of the past. We are today heirs of the whole past in mathematics. Nothing is wasted, nothing is dissipated, but the wealth, the flashing gems of learning, which are the reward of painful toil of men long since dead, are ours today, a capital which enables us to advance the faster and to increase the riches all the more. With such a source of power we must then inquire where the mathematician will find a valid domain in which to justify to the rest of the world his right to exist. What is its fruitage either in bloom or in mature fruit for the sustenance of the nations? What storms will its foliage protect from, and what distant peaks with their glistening slopes are visible from its lofty summit? Since it rears its head so proudly into the rarefied upper atmosphere where only the privileged few can ever go, what can it bring down for the inspiration