tively recent years that mathematics has made most of its gains towards being recognized as a fundamental science, and the study of advanced mathematics in our universities had a still later origin.
The rapid recent advances in various fields of mathematics have given rise to a very optimistic spirit as to the future. Although we still hold in high esteem the brilliant discoveries of the Greeks, we are inclined to give much more thought and attention to recent work, as may be seen from the references in the extensive German and French mathematical encyclopedias which are in the process of being published. The history of mathematics furnishes many instances of the vanishing of apparently insurmountable barriers. We need only recall the barrier created by the Greek custom of confining oneself to the rule and circle in the most acceptable geometric constructions, and the very formidable barrier furnished by the imaginary, and even by the negative and the irrational roots of a quadratic equation.
Those who fixed their attention upon these barriers in the past have naturally been led to think that the days of important advances in mathematics were about ended and that it only remained to fill in details. Such predictions had few supporters when new methods led over these barrier and turned them into steps to richer mathematical domains. As this process has been repeated so often it has gradually reduced the number of those to whom the future of mathematics looked dark. In fact, Poincaré, in his address[1] before the Fourth International Congress of Mathematicians, which was held at Rome, in April, 1908, said that all those who held these views are dead.
These facts seem to justify a very hopeful spirit as regards future progress, but it is necessary to examine them with great care in order to deduce from them any helpful suggestions as to the probable nature of this progress. Such prognostications clearly demand a mind that can deal with big problems as well as a thorough acquaintance with the past and the present developments in mathematics, to insure that the results obtained by a kind of extrapolation may be worthy of confidence. It is doubtful whether any living mathematician would be more generally regarded as qualified to make reliable predictions along this line than Poincaré, of Paris. The address to which we referred in the preceding paragraph was devoted to this subject and we proceed to give some of the main results.
The objects of mathematical thought are so numerous that we cannot expect to exhaust them. This appears the more evident since the mathematician creates new concepts from the elements which are presented to him by nature. Hence there must be a choice of subject matter, but who is to do the choosing? Some are inclined to think that the mathematician should confine himself to those problems which
- ↑ Bulletin des Sciences Mathématiques, Vol. 32 (1908), p. 168.
