158 SCIENCE AND METHOD. 



ture to risk a criticism, but I am very much afraid 

 that this definition contains a petitio principii, seeing 

 that I notice the figure i in the first half and the 

 word One in the second. 



However that may be, Signor Burali-Forti starts 

 with this definition, and, after a short calculation, 

 arrives at the equation 



(27) I e No, 



which teaches us that One is a number. 



And since I am on the subject of these definitions 

 of the first numbers, I may mention that M. Couturat 

 has also defined both o and i. 



What is zero ? It is the number of elements in the 

 class nil. And what is the class nil ? It is the class 

 which contains none. 



To define zero as nil and nil as none is really an 

 abuse of the wealth of language, and so M. Couturat 

 has introduced an improvement into his definition by 

 writing 



o = ' A • ^^ = A- ^- A = (^c<^^), 



which means in English : zero is the number of the 

 objects that satisfy a condition that is never fulfilled. 

 But as never means in no case, I do not see that any 

 very great progress has been made. 



I hasten to add that the definition M. Couturat 

 gives of the number i is more satisfactory. 



One, he says in substance, is the number of the 

 elements of a class in which any two elements are 

 identical. 



It is more satisfactory, as I said, in this sense, 

 that in order to define I, he does not use the word 

 one ; on the other hand, he does use the word two. 



