170 SCIENCE AND METHOD. 



IX. 



The end of Mr. Hilbert's treatise is altogether 

 enigmatical, and I will not dwell upon it. It is full 

 of contradictions, and one feels that the author is 

 vaguely conscious of the petitio principii he has been 

 guilty of, and that he is vainly trying to plaster up 

 the cracks in his reasoning. 



What does this mean ? It means that when he 

 comes to demo7tstrate that the definition of the whole 

 number by the axiom of complete induction does not 

 involve contradiction, Mr. Hilbert breaks down, just as 

 Mr. Russell and M. Couturat broke down, becatise the 



difficulty is too great. 



X. 



Geometry. 



Geometry, M. Couturat says, is a vast body of 

 doctrine upon which complete induction does not 

 intrude. This is true to a certain extent : we cannot 

 say that it does not intrude at all, but that it intrudes 

 very little. If we refer to Mr. Halsted's " Rational 

 Geometry " (New York : John Wiley and Sons, 

 1904), founded on Hilbert's principles, we find the 

 principle of induction intruding for the first time 

 at page 1 14 (unless, indeed, I have not searched care- 

 fully enough, which is quite possible). 



Thus geometry, which seemed, only a few years 

 ago, the domain in which intuition held undisputed 

 sway, is to-day the field in which the logisticians 

 appear to triumph. Nothing could give a better 

 measure of the importance of Hilbert's geometrical 

 works, and of the profound impression they have left 

 upon our conceptions. 



