THE FUTURE OF MATHEMATICS 123 



The future of mathematics appears bright, both for the investigator 

 and for the teacher. When a country Avhich has such an enlight- 

 ened educational system as France increased the amount of time de- 

 voted to secondary mathematics so recently as 1902 and again in 1906, 

 it furnishes one of the strongest possible encouragements to the teacher 

 who may have been troubled by the thought that the educational value 

 of mathematics was not being as fully appreciated as in earlier years. 

 Naturally we may expect that there will be local changes of view as 

 regards the value of mathematics as an educational subject, and these 

 changes will not always be for the better, but the civilized world, as a 

 whole, is learning to appreciate more and more the fundamental im- 

 portance of early mathematical training, so that we should not be too 

 much perturbed by local steps backwards, but we should move ahead 

 with the assurance that we are engaged in a work of the highest peda- 

 gogical importance. 



The boundless confidence in the importance of early and extensive 

 mathematical training should, however, not blind us to the need of 

 changes and new adaptations. As an important function of mathe- 

 matical training is the furnishing of the most useful and the most 

 powerful tools of thought, it is evident that the choice of these tools 

 will vary with the advancement of general knowledge. All admit that 

 the concept of a derivative is one of the most useful elementary tools 

 of thought, and in a number of countries this concept has been intro- 

 duced into secondary mathematics and used with success. At the last 

 International Mathematical Congress, held at Borne, M. Borel, of 

 Paris, reported that the notion of derivative had been introduced into 

 French secondary education in 1902 and that it had led to satisfactory 

 results. At the same meeting M. Beke, of Budapest, stated that this 

 notion, together with the notion of function and graph, had been intro- 

 duced into the courses of secondary education in Hungary. 



At the recent joint conference of the Mathematical Association and 

 the Federated Association of London Non-Primary Teachers, the chair- 

 man remarked : " I have always thought that a mathematician was a 

 man who when he wants to find anything out, uses his brains for that 

 purpose, whereas a physicist, when he wants to find out anything, re- 

 sorts to experiment." Although this statement is not to be construed 

 literally, yet it does involve a great partial truth and it calls attention 

 to elements which insure mathematical appreciation as long as there 

 is scientific thought. " It is the mind that sees as well as the eye," 

 and the mind sees some of the greatest truths most clearly by means of 

 mathematical symbolism. In fact, mathematical symbols serve both 

 as a telescope and also as a microscope for mental vision, and as long 

 as such vision is demanded the teacher of mathematics will be ap- 

 preciated. 



