1 86 SCIENTIFIC METHOD IN PHILOSOPHY 



Between philosophy and pure mathematics there is a 

 certain affinity, in the fact that both are general and a 

 priori. Neither of them asserts propositions which, like 

 those of history and geography, depend upon the actual 

 concrete facts being just what they are. We may illustrate 

 this characteristic by means of Leibniz's conception of 

 many possible worlds, of which one only is actual. In all 

 the many possible worlds, philosophy and mathematics 

 will be the same ; the differences will only be in respect of 

 those particular facts which are chronicled by the descrip- 

 tive sciences. Any quality, therefore, by which our 

 actual world is distinguished from other abstractly possible 

 worlds, must be ignored by mathematics and philosophy 

 alike. Mathematics and philosophy differ, however, in 

 their manner of treating the general properties in which 

 all possible worlds agree ; for while mathematics, starting 

 from comparatively simple propositions, seeks to build up 

 more and more complex results by deductive synthesis, 

 philosophy, starting from data which are common know- 

 ledge, seeks to purify and generalise them into the 

 simplest statements of abstract form that can be obtained 

 from them by logical analysis. 



The difference between philosophy and mathematics 

 may be illustrated by our present problem, namely, the 

 nature of number. Both start from certain facts about 

 numbers which are evident to inspection. But mathe- 

 matics uses these facts to deduce more and more com- 

 plicated theorems, while philosophy seeks, by analysis, to 

 go behind these facts to others, simpler, more fundamental, 

 and inherently more fitted to form the premisses of 

 the science of arithmetic. The question, " What is a 

 number?" is the pre-eminent philosophic question in 

 this subject, but it is one which the mathematician as such 

 need not ask, provided he knows enough of the properties 



