172 SCIENCE AND METHOD. 



is only acceptable if it is established that it does not 

 involve contradiction. We have also shown that, in 

 the case of the first definition, this demonstration is 

 impossible ; while in the case of the second, on the 

 contrary, we have just recalled the fact that Hilbert 

 has given a complete demonstration. 



So far as the third is concerned, it is clear that it 

 does not involve contradiction. But does this mean 

 that this definition guarantees, as it should, the 

 existence of the object defined ? We are here no 

 longer concerned with the mathematical sciences, but 

 with the physical sciences, and the word existence has 

 no longer the same meaning ; it no longer signifies 

 absence of contradiction, but objective existence. 



This is one reason already for the distinction I make 

 between the three cases, but there is a second. In 

 the applications we have to make of these three 

 notions, do they present themselves as defined by 

 these three postulates ? 



The possible applications of the principle of induc- 

 tion are innumerable. Take, for instance, one of those 

 we have expounded above, in which it is sought to 

 establish that a collection of axioms cannot lead to 

 a contradiction. For this purpose we consider one of 

 the series of syllogisms that can be followed out, start- 

 ing with these axioms as premises. 



When we have completed the n^^ syllogism, we see 

 that we can form still another, which will be the 

 («-t-i)''^: thus the number n serves for counting a 

 series of successive operations ; it is a number that 

 can be obtained by successive additions. Accordingly, 

 it is a number from which we can return to unity by 

 successive subtractiotis. It is evident that we could 



