MATHEMATICS AND LOGIC. 145 



psychology. It is certainly not in this manner that 

 the human mind proceeded to construct mathematics, 

 and I imagine, too, its authors do not dream of intro- 

 ducing it into secondary education. But is it at least 

 logical, or, more properly speaking, is it accurate? 

 We may well doubt it. 



Nevertheless, the geometricians who have employed 

 it are very numerous. They have accumulated formulas 

 and imagined that they rid themselves of all that is not 

 pure logic by writing treatises in which the formulas 

 are no longer interspersed with explanatory text, as in 

 the ordinary works on mathematics, but in which the 

 text has disappeared entirely. 



Unfortunately, they have arrived at contradictory 

 results, at what are called the Cantorian antinomies^ 

 to which we shall have occasion to return. These 

 contradictions have not discouraged them, and they 

 have attempted to modify their rules, in order to 

 dispose of those that had already appeared, but with- 

 out gaining any assurance by so doing that no new 

 ones would appear. 



It is time that these exaggerations were treated as 

 they deserve. I have no hope of convincing these 

 logicians, for they have lived too long in this atmo- 

 sphere. Besides, when we have refuted one of their 

 demonstrations, we are quite sure to find it cropping 

 up again with insignificant changes, and some of them 

 have already risen several times from their ashes. 

 Such in old times was the Lerna;an hydra, with its 

 famous heads that always grew again. Hercules was 

 successful because his hydra had only nine heads 

 (unless, indeed, it was eleven), but in this case there are 



too many, they are in England, in Germany, in Italy, 

 (1,777) 10 



