Page:Popular Science Monthly Volume 82.djvu/222

218 difference of opinion which there is among mathematicians regarding the existence of a definable infinity as due to the difference in the psychology of the two classes. One, the idealistic, feeling that everything we define is due to the mind, and finite; the other, the realistic, feeling that there is an external world which may well contain an infinity. The idealistic class, to which Poincaré belonged, would consider that these extensions to which we referred are in a sense creations.

It is scarcely necessary to enumerate the creations of Poincaré. They are many, for he was gifted with extraordinary originality. The account given above of the creation of the fuchsian functions is an example of one of his most important. It opened an immense field of investigation. He created a type of arithmetic invariants expressible as series or definite integrals, which opened a new field in theory of numbers. His investigations of ordinary differential equations which are not linear, such as those in dynamics and the problem of N bodies, created an extensive class of new functions which (I believe) are yet without special names, as well as suggesting the existence of classes of functions for which we have, as yet, even no means of expression. The investigations of asymptotic expansions opened paths to dizzy heights. Fundamental functions in partial differential equations also open a region now under development. We may say that the most marvelous of his creations rise from the general field of differential equations. We might cite further his researches in analysis situs, the realm of the invariants of a battered continuity. His double residues and studies in functions of many real variables are creations from which will spring a noble progeny. Even the lectures in which he presented the results of others scintillate with original thoughts.

To generalize in mathematics and science it is not enough simply to get together facts or ideas and to put them into new combinations. Most of these combinations would be useless. The real investigator does not form the useless combinations at all, but unconsciously rejects the unprofitable combinations. It is as if he were an examiner for a higher degree; only the candidates who have passed the lower degrees ever appear before him at all. Often domains far distant furnish the useful combinations, as in the account given of the genesis of the fuchsian functions, the theory of arithmetic forms through the roundabout route of non-euclidean geometry furnished the generalization of the first fuchsian functions to the complete class. This was of the first type. But how are those of the second type born?

We come thus to the heart of the matter. Merely to say that we discover laws is not sufficient. How do we discover extensions? How devise new formulas? Make new constructions? The answer to this question is, for Poincaré, found in psychology. It is necessary to get together many facts, but this does not insure that we shall build with