MATHEMATICAL INSTRUCTION 161 



SOME RECENT TENDENCIES IN MATHEMATICAL 



INSTRUCTION 



By Professor G. A. MILLER 



LELAND STANFORD JUNIOR UNIVERSITY 



SEVEEAL prominent writers have suggested that ' pure mathe- 

 matics ' should be called ' free mathematics ' in view of the great 

 latitude of freedom both in subject matter and in methods of work. 1 

 This view is diametrically opposed to the one commonly held. The 

 student of elementary text-books on mathematics can not fail to be 

 impressed by the close similarity in subject matter and in methods. 

 In our arithmetics and algebras we find problems which are quite similar 

 to those in the work written by Ahmes, an Egyptian priest living 

 seventeen hundred years before Christ. Our geometries bear such a 

 close resemblance to Euclid's, written three hundred years before 

 Christ, that the terms Euclid and elementary geometry are still prac- 

 tically synonymous in the minds of many teachers. 



Moreover, there are numerous a priori reasons to regard mathe- 

 matics as a slave rather than as a free, living, pulsating being, ex- 

 hibiting the many exhilarating changes characteristic of youthful 

 development. The reasonableness of the main postulates of mathe- 

 matics has never been questioned. In most cases contrary hypotheses 

 would appear perfectly absurd to those who have not been trained to 

 mistrust their intuitions. The main function of mathematics has been, 

 and probably always will be, to draw necessary conclusions from such 

 postulates. Whether the postulates have been explicitly stated or not 

 is a secondary matter. Hence mathematics has become a vast struc- 

 ture which is perfectly invulnerable except possibly at its foundation, 

 and here the attack seems merely a matter of words. The most that 

 has been done in an effectual manner is to erect structures on other 

 sets of postulates. It should be emphasized that the main duty of 

 the mathematician is to build upon given postulates. In these opera- 

 tions he will always be free from attack, since he can not arrive at any 

 conclusion without proving that it is the only possible one. 2 



1 Cf. Liebmann, Jahresbericht der Deutschen Mathematiker-Vereinigung, 

 Vol. 14 (1905), p. 231. 



2 No attempt is here made to define the term mathematics. The views ex- 

 pressed are equally true whether mathematics is defined as a method, or 

 whether the objects which it considers should enter into the definition. From 



VOL. LXVITI.— 11. 



