24 THE AIM AND ACHIEVEMENTS OF V SCIENTIFIC METHOD. 



addition of a new one. This process cannot lead to the 

 introduction of an infinite number of terms in the accurately ^ 

 defined sense in which philosophical mathematicians. use the 

 word.* As a matter of fact such a series is called by 

 Mr. Eussell compact] and must be distinguished from the 

 continuous series which implies an infinite number of terms 

 between any two. 



9. 



The most important single series for our purposes is the 

 series of numbers. Among all the classes of things in th-e 

 Objective we can detect classes of similar classes that is 

 classes between whose terms a " one-one correlation " can be 

 established. This is true, for example, of the days of the week, 

 the Hills of Rome, the Champions of Christendom, the Wise 

 Men of Greece and the Wonders of the World. These and 

 the other classes similar to them owe their similarity to the 

 possession of a common property or, as Mr. Eussell puts it, 

 to their relation to a common term their number. This is 

 the logical definition of the cardinal numbers and probably 

 indicates also the psychological process by which they we're 

 discovered. It is obvious that they form a discrete, series 

 infinite in number, and generated by the constantly repeated 

 addition of one to the first of them ; while no place in this 

 series is occupied by more than one terra. .It is clear that 

 such- a series may be correlated with any other discrete series. 

 This correlation the system of counting or marking with 

 consecutive numbers must at a very early stage of civilisation 

 have replaced simpler methods of correlation used for the same 

 purpose. A time came when the series of integers proved to 

 be insufficient for the practical purposes to which numbers 

 were applied, and had to be extended by the addition of the 

 terms with which we are familiar as " fractions." There can 



* Kussell, op. cit., pp. 356-7. 

 t By Cantor " iiberall dicht." 



