1 82 SCIENTIFIC METHOD IN PHILOSOPHY 



known to us. We can do this because we know of 

 various characteristics which every individual has if he 

 belongs to the collection, and not if he does not. And 

 exactly the same happens in the case of infinite collections : 

 they may be known by their characteristics although their 

 terms cannot be enumerated. In this sense, an unending 

 series may nevertheless form a whole, and there may be 

 new terms beyond the whole of it. 



Some purely arithmetical peculiarities of infinite 

 numbers have also caused perplexity. For instance, an 

 infinite number is not increased by adding one to it, or 

 by doubling it. Such peculiarities have seemed to many 

 to contradict logic, but in fact they only contradict con- 

 firmed mental habits. The whole difficulty of the subject 

 lies in the necessity of thinking in an unfamiliar way, 

 and in realising that many properties which we have 

 thought inherent in number are in fact peculiar to 

 finite numbers. If this is remembered, the positive 

 theory of infinity, which will occupy the next lecture, 

 will not be found so difficult as it is to those who cling 

 obstinately to the prejudices instilled by the arithmetic 

 which is learnt in childhood. 



