TEE FUTURE OF MATHEMATICS 117 



THE TUTUEE OF MATHEMATICS 



By Peofessoh G. A. MILLER 



UNIVERSITY OF ILLINOIS 



PROFESSOR A. VOSS, of the University of Munich, recently made 

 the following statement : " Our entire present civilization, as far 

 as it depends upon the intellectual penetration and utilization of nature, 

 has its real foundation in the mathematical sciences." 1 He adds that 

 this truth finds expression in the ever-increasing appreciation of the 

 educational value of mathematics, notwithstanding the fact that it is 

 the most unpopular of all the sciences. This unpopularity is natural 

 since "unpopularity is an essential feature of a real science/' because 

 such a science can be comprehended only through tireless and continued 

 efforts. 



An intelligent expression as regards the future of mathematics must 

 be based not only upon the past and present state of this science, but 

 also upon its real essence. One of those elements which mathematics 

 has in common with some of the other sciences, but which are more 

 prominent in mathematics than in any of the others, is the tendency 

 to use thought in the most economical manner. When one considers 

 the extent to which efforts to simplify methods, theorems and formulas 

 direct mathematical endeavor, one must admit that the statement 

 " Mathematics is the science of saving thought " expresses a great 

 truth, even if it is too sweeping to serve as a definition. 



That mathematics is the science which is preeminently devoted to 

 the discovery and mapping of routes along which thought may ascend 

 securely and with the greatest ease, is supported by the fact that it 

 has the oldest and the most extensive symbolical language. In the 

 introduction to his classic history of mathematics, Moritz Cantor asks, 

 " Why has mathematics, since the remotest times, found support, 

 simplification and advancement by means of word symbols, whether 

 these are number symbols or other mathematical symbols?" Although 

 the oldest of these word symbols are probably relics of a very ancient 

 picture language, yet it is of great interest that in mathematics the 

 picture language was retained and used side by side with an alphabetic 

 and syllabic language, while the latter displaced the former elsewhere. 

 Even those who have mastered only the elements of algebra and the 



1 Voss, " Ueber das Wesen der Mathematik." Rede gehalten om 11. Marz, 

 1908, in der oeffentlichen Sitzung der k. bayerischen Akademie der Wissen- 

 schaften; Teubner, 1908, p. 4. 



