TEE FUTURE OF MATHEMATICS 119 



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, Poincare, in his address 2 before the Fourth Inter- 

 national 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 gen- 

 erally regarded as qualified to make reliable predictions along this line 

 than Poincare, 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 can- 

 not 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 



2 Bulletin des Sciences MathSmatiques, Vol. 32 (1908), p. 168. 



