124 SCIENCE AND METHOD. 



cannot be established in the arguments unless it is 

 first introduced into the definitions. 



For a long time the objects that occupied the atten- 

 tion of mathematicians were badly defined. They 

 thought they knew them because they represented 

 them by their senses or their imagination, but they 

 had only a rough image, and not a precise idea such 

 as reasoning can take hold of. 



It is to this that the logicians have had to apply their 

 efforts, and similarly for incommensurable numbers. 



The vague idea of continuity which we owe to 

 intuition has resolved itself into a complicated system 

 of inequalities bearing on whole numbers. Thus it 

 is that all those difficulties which terrified our ances- 

 tors when they reflected upon the foundations of the 

 infinitesimal calculus have finally vanished. 



In analysis to-day there is no longer anything but 

 whole numbers, or finite or infinite systems of whole 

 numbers, bound together by a network of equalities 

 and inequalities. Mathematics, as it has been said, 

 has been arithmetized. 



5. But we must not imagine that the science of 

 mathematics has attained to absolute exactness with- 

 out making any sacrifice. What it has gained in 

 exactness it has lost in objectivity. It is by with- 

 drawing from reality that it has acquired this perfect 

 purity. We can now move freely over its whole 

 domain, which formerly bristled with obstacles. But 

 these obstacles have not disappeared ; they have only 

 been removed to the frontier, and will have to be 

 conquered again if we wish to cross the frontier and 

 penetrate into the realms of practice. 



