III. 



MATHEMATICS AND LOGIC. 



Introduction. 



Can mathematics be reduced to logic without having 

 to appeal to principles peculiar to itself? There is a 

 whole school full of ardour and faith who make it 

 their business to establish the possibilit}'. They have 

 their own special language, in which words are used 

 no longer, but only signs. This language can be 

 understood only by the kw initiated, so that the 

 vulvar are inclined to bow before the decisive affirma- 

 tions of the adepts. It will, perhaps, be useful to 

 examine these affirmations somewhat more closely, in 

 order to see whether they justify the peremptory tone 

 in which they are made. 



But in order that the nature of the question should 

 be properly understood, it is necessary to enter into 

 some historical details, and more particularly to review 

 the character of Cantor's work. 



The notion of infinity had long since been introduced 

 into mathematics, but this infinity was what philoso- 

 phers call a becomins;. Mathematical infinity was only 

 a quantity susceptible of growing beyond all limit ; it 

 was a variable quantity of which it could not be said 

 that it had passed^ but only that it would pass, all limits. 



