1 88 SCIENTIFIC METHOD IN PHILOSOPHY 



at the beginning of his discussion of number, " rests 

 the possibility of spontaneously prolonging the series 

 of numbers ad infinitum." It is this view of number 

 as generated by counting which has been the chief 

 psychological obstacle to the understanding of infinite 

 numbers. Counting, because it is familiar, is erroneously 

 supposed to be simple, whereas it is in fact a highly 

 complex process, which has no meaning unless the 

 numbers reached in counting have some significance 

 independent of the process by which they are reached. 

 And infinite numbers cannot be reached at all in this 

 way. The mistake is of the same kind as if cows were 

 defined as what can be bought from a cattle-merchant. 

 To a person who knew several cattle-merchants, but had 

 never seen a cow, this might seem an admirable defini- 

 tion. But if in his travels he came across a herd of wild 

 cows, he would have to declare that they were not cows 

 at all, because no cattle-merchant could sell them. So 

 infinite numbers were declared not to be numbers at all, 

 because they could not be reached by counting. 



It will be worth while to consider for a moment what 

 counting actually is. We count a set of objects when 

 we let our attention pass from one to another, until we 

 have attended once to each, saying the names of the 

 numbers in order with each successive act of attention. 

 The last number named in this process is the number 

 of the objects, and therefore counting is a method of 

 finding out what the number of the objects is. But this 

 operation is really a very complicated one, and those 

 who imagine that it is the logical source of number show 

 themselves remarkably incapable of analysis. In the 

 first place, when we say " one, two, three . . . " as we 

 count, we cannot be said to be discovering the number 

 of the objects counted unless we attach some meaning 



