LAST EFFORTS OF LOGISTICIANS. 183 



found the same mistake in the orthodox logis- 

 ticians. I have not seen it in the pages I have read, 

 but I do not know whether I should find it in the 

 three hundred pages they have written that I have no 

 wish to read. 



Only, they will have to commit the error as soon 

 as they attempt to make any sort of an application 

 of mathematical science. The eternal contemplation 

 of its own navel is not the sole object of this science. 

 It touches nature, and one day or other it will come 

 into contact with it. Then it will be necessary to 

 shake off purely verbal definitions and no longer to 

 content ourselves with words. 



Let us return to Mr. Hilbert's example. It is still 

 a question of reasoning by recurrence and of knowing 

 whether a system of postulates is not contradictory. 

 M. Couturat will no doubt tell me that in that case 

 it does not concern him, but it may perhaps interest 

 those who do not claim, as he does, the liberty of 

 contradiction. 



We wish to establish, as above, that we shall not 

 meet with contradiction after some particular number 

 of arguments, a number which may be as large as you 

 please, provided it is finite. For this purpose we 

 must apply the principle of induction. Are we to 

 understand here by finite number every number to 

 which the principle of induction applies? Evidently 

 not, for otherwise we should be involved in the most 

 awkward consequences. 



To have the right to lay down a system of postu- 

 lates, we must be assured that they are not contra- 

 dictory. This is a truth that is admitted by the 

 majority of scientists ; I should have said all before 



