738 SCIENTIFIC THOUGHT. 



side with philosophy in regard to our notions of space 

 and time, and in the questions which have arisen as 

 to the universality and necessity of the truths of 

 mathematics and the foundation of our knowledge of 

 them " ; and he subsequently refers specially to the 

 " notion which is really the fundamental one under- 

 lying and pervading the whole of modern analysis and 

 geometry," meaning the complex magnitude, as deserv- 

 ing to be specially discussed by philosophers. Be- 

 ginnings of the philosophical treatment of this and 

 other questions indeed exist. The questions are still 

 sub judice, and the historian can merely refer to their 

 existence and importance. 



There is, however, one controversy which has arisen 

 out of these and similar speculations, and out of the 

 desire to bring unity and consistency into the funda- 

 mental notions of elementary as well as higher mathe- 

 matics, which deserves to be specially mentioned, because 

 it occupies a prominent place in foreign literature, hav- 

 ing given rise to a special term, and thus commanding 

 more general attention. Prof. Klein of Gottingen, under 

 whose master-hand many abstract and obscure subjects 

 have become plain and transparent, has prominently 

 brought the subject before the scientific public in a 

 68. recent address. 1 I refer to the tendency represented 



Arithmetis- 



mg tendency m its extreme form by the late rror. Kronecker 01 



in mathe- 

 matics. Berlin, to reduce all mathematical conceptions to the 



fundamental arithmetical operations with integral num- 

 bers, banishing not only all geometrical and dynamical 

 conceptions, such as those of continuity and flow, but 



1 ' Ueber Arithmetisirung der Mathematik ' (Gottingen, 1895). 



