THE NEW LOGICS. 171 



But we must not deceive ourselves. What is, in 

 fact, the fundamental theorem of geometry ? It is that 

 the axioms of geometry do not involve contradiction, and 

 this cannot be demonstrated without the principle of 

 induction. 



How does Hilbert demonstrate this essential point ? 

 He does it by relying upon analysis, and, through it, 

 upon arithmetic, and, through it, upon the principle 

 of induction. 



If another demonstration is ever discovered, it will 

 still be necessary to rely on this principle, since the 

 number of the possible consequences of the axioms 

 which we have to show are not contradictory is 

 infinite. 



XL 



Conclusion. 



Our conclusion is, first of all, that the principle of 

 induction cannot be regarded as the disguised definition 

 of the whole number. 



Here are three truths : — 



The principle of complete induction ; 

 Euclid's postulate ; 



The physical law by which phosphorus melts 

 at 44° centigrade (quoted by M. Le Roy). 



We say : these are three disguised definitions — the 

 first that of the whole number, the second that of the 

 straight line, and the third that of phosphorus. 



I admit it for the second, but I do not admit it 

 for the two others, and I must explain the reason of 

 this apparent inconsistency. 



In the first place, we have seen that a definition 



