260 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



should always be at least one element and therefore an 

 infinity of elements between any two. Empirical events 

 could not constitute such a series, for at the limits of the 

 perceptually discernible adjacency would be an observed 

 fact. It is easy, however, at least with the techniques of 

 mathematics, to construct series satisfying these postulates. 

 One of the most useful is that of the class of proper fractions 

 arranged in the usual order. 1 



An even more compact type of series is that called the 

 continuum, which has important applications in the analysis 

 of time and space. Its postulates are as follows: 



Postulate CI. (Dedekind's postulate.) 



Postulate C2. (Postulate of density.) 



Postulate C3. (Postulate of linearity.) The class K contains a 

 denumerable subclass R in such a way that between any two elements 

 of the given class K there is an element of R. 2 



Some of the commonest examples of the continuum are the 

 points on a line, the instants in a duration, the real numbers, 

 and the non-terminating decimal fractions. It is clear that 

 these are not empirical orders in the strict sense of the word ; 

 they are definable by means of operations having empirical 

 application within a given range, but extended beyond that 

 range through a principle justifying indefinite repetition. 



number: empirical foundation 



That there are numbers, or at least numbered groups, 

 seems obvious; yet the problem of the precise empirical 

 foundation of the concept of number is one involving some 

 difficulty. This is due at least partly to the fact that the 

 concept of number in ordinary speech is ambiguous. It may 

 refer, on the one hand, to a certain property of groups, or 

 pluralities; in this sense it applies only to collections of 

 events, and not to a single event. But it may refer, on the 

 other hand, to the property of individuality, distinguishabil- 

 ity, or discreteness, which is possessed by an event; in this 



1 Ibid., p. 14. *Ibid„ p. 44. 



