THE PHILOSOPHICAL BASIS 31 



with the touch of genius which often accompanies 

 sober scientific insight and imagination — some one 

 who is able to brush aside for a time the non- 

 essential, and to rise above the confusion of 

 detail — is inspired with a conception of order in 

 the multiplicity of the phenomena : order to be 

 seen when some simple principle is borne in mind, 

 and is expressed in a formula, which, in terms 

 of our conceptual shorthand, enables us to re- 

 member and to predict the sequence of phenomena. 

 If the formula is expressed in terms of simple con- 

 ceptions, already known and often used in other 

 branches of knowledge, the mind at once looks 

 on it as an *' explanation " of the phenomena, 

 though it is evident on further thought that the 

 phenomena are no more fully understood than 

 are the fundamental conceptions — mass, space, 

 time, whatever they be — in which the "explana- 

 tion " is expressed. 



The next step consists in deducing new conse- 

 quences of the hypothesis ; and here the methods 

 of mathematical analysis are usefullyapplied. The 

 science of mathematics as such has nothing to do 

 with natural phenomena. Like physical science it 

 is concerned with ideal conceptions ; but neither 

 does it seek to gain those conceptions from 

 an examination of Nature, nor to check their 

 correspondence by the methods of experiment. 

 Mathematics may borrow subject-matter from 

 observational science, or may acquire by pure 

 mental processes subject-matter, such as the 

 geometry of four dimensional space, which may 

 or may not have a counterpart in Nature. In 

 either case, mathematics deals with the concep- 

 tions as such, and traces their results and the 



