i82 SCIENCE AND METHOD. 



just as a prisoner is presumed to be innocent, until the 

 contrary is proved. 



It is unnecessary to add that I do not acquiesce 

 f in this claim. But, you say, the demonstration you 

 demand of us is impossible, and you cannot require 

 us to " aim at the moon." Excuse me; it is impossible 

 for you, but not for us who admit the principle of 

 induction as an a priori synthetic judgment. This 

 would be necessary for you as it is for us. 



In order to demonstrate that a system of postulates 

 does not involve contradiction, it is necessary to apply 

 the principle of complete induction. Not only is there 

 nothing " extraordinary " in this method of reasoning, 

 but it is the only correct one. It is not " incon- 

 ceivable " that any one should ever have used it, and 

 it is not difficult to find "examples and precedents." 

 In my article I have quoted two, and they were 

 borrowed from Hilbert's pamphlet. He is not alone 

 in having made use of it, and those who have 

 not done so have been wrong. What I reproach 

 Hilbert with, is not that he has had recourse to it 

 (a born mathematician such as he could not but see 

 that a demonstration is required, and that this is the 

 only possible one), but that he has had recourse to it 

 without recognizing the reasoning by recurrence. 



IV. 



The Second Objection. 



I had noted a second error of the logisticians in 

 Hilbert's article. To-day Hilbert is excommuni- 

 cated, and M. Couturat no longer considers him as 

 a logistician. He will therefore, ask me if I have 



