i84 SCIENCE AND METHOD. 



reading M. Couturat's last article. But what does it 

 signify ? Does it mean that we must be sure of not 

 meeting with contradiction after a finite number of 

 propositions, the finite number being, by definition, 

 that which possesses all the properties of a recurrent 

 nature in such a way that if one of these properties 

 were found wanting — if, for instance, we came upon a 

 contradiction — we should agree to say that the number 

 in question was not finite ? 



In other words, do we mean that we must be sure 

 of not meeting a contradiction, with this condition, 

 that we agree to stop just at the moment when we are 

 on the point of meeting one ? The mere statement 

 of such a proposition is its sufficient condemnation. 



Thus not only does Mr. Hilbert's reasoning assume 

 the principle of induction, but he assumes that this 

 principle is given us, not as a simple definition, but 

 as an a priori synthetic judgment. 



I would sum up as follows : — 



A demonstration is necessary. 



The only possible demonstration is the demonstra- 

 tion by recurrence. 



This demonstration is legitimate only if the prin- 

 ciple of induction is admitted, and if it is regarded 

 not as a definition but as a synthetic judgment. 



V. 



The Cantorian Antinomies. 



I will now take up the examination of Mr. Russell's 

 new treatise. This treatise was written with the object 

 of overcoming the difficulties raised by those Cantorian 



