148 SCIENCE AND METHOD. 



to a minimum the number of the fundamental axioms 

 of geometry, and to make a complete enumeration of 

 them. Now, in the arguments in which our mind 

 remains active, in those in which intuition still plays 

 a part, in the living arguments, so to speak, it is 

 difficult not to introduce an axiom or a postulate that 

 passes unnoticed. Accordingly, it was not till he had 

 reduced all geometrical arguments to a purely me- 

 chanical form that he could be certain of having 

 succeeded in his design and accomplished his work. 



What Hilbert had done for geometry, others have 

 tried to do for arithmetic and analysis. Even if they 

 had been entirely successful, would the Kantians be 

 finally condemned to silence? Perhaps not, for it is 

 certain that we cannot reduce mathematical thought 

 to an empty form without mutilating it. Even admit- 

 ting that it has been established that all theorems can 

 be deduced by purely analytical processes, by simple 

 logical combinations of a finite number of axioms, and 

 that these axioms are nothing but conventions, the 

 philosopher would still retain the right to seek the 

 origin of these conventions, and to ask why they were 

 iudged preferable to the contrary conventions. 



And, further, the logical correctness of the argu- 

 ments that lead from axioms to theorems is not the 

 only thing we have to attend to. Do the rules of 

 perfect logic constitute the whole of mathematics? 

 As well say that the art of the chess-player reduces 

 itself to the rules for the movement of the pieces. 

 A selection must be made out of all the construc- 

 tions that can be combined with the materials 

 furnished by logic. The true geometrician makes 

 this selection judiciously, because he is guided by 



