THE POSITIVE THEORY OF INFINITY 189 



to the words one, two, three, ... A child may learn 

 to know these words in order, and to repeat them 

 correctly like the letters of the alphabet, without attach- 

 ing any meaning to them. Such a child may count 

 correctly from the point of view of a grown-up listener, 

 without having any idea of numbers at all. The 

 operation of counting, in fact, can only be intelligently 

 performed by a person who already has some idea what 

 the numbers are ; and from this it follows that counting 

 does not give the logical basis of number. 



Again, how do we know that the last number reached 

 in the process of counting is the number of the objects 

 counted ? This is just one of those facts that are too 

 familiar for their significance to be realised ; but those 

 who wish to be logicians must acquire the habit of 

 dwelling upon such facts. There are two propositions 

 involved in this fact : first, that the number of numbers 

 from 1 up to any given number is that given number 

 for instance, the number of numbers from 1 to 100 is 

 a hundred ; secondly, that if a set of numbers can be 

 used as names of a set of objects, each number occurring 

 only once, then the number of numbers used as names 

 is the same as the number of objects. The first of 

 these propositions is capable of an easy arithmetical 

 proof so long as finite numbers are concerned ; but with 

 infinite numbers, after the first, it ceases to be true. The 

 second proposition remains true, and is in fact, as we 

 shall see, an immediate consequence of the definition 

 of number. But owing to the falsehood of the first 

 proposition where infinite numbers are concerned, count- 

 ing, even if it were practically possible, would not be a 

 valid method of discovering the number of terms in an 

 infinite collection, and would in fact give different results 

 according to the manner in which it was carried out. 



