132 SCIENTIFIC METHOD IN PHILOSOPHY 



same number of husbands as of wives, without having to 

 arrange them in a series. But continuity, which we are 

 now to consider, is essentially a property of an order : it 

 does not belong to a set of terms in themselves, but only 

 to a set in a certain order. A set of terms which can be 

 arranged in one order can always also be arranged in 

 other orders, and a set of terms which can be arranged 

 in a continuous order can always also be arranged in 

 orders which are not continuous. Thus the essence 

 of continuity must not be sought in the nature of the 

 set of terms, but in the nature of their arrangement 

 in a series. 



Mathematicians have distinguished different degrees 

 of continuity, and have confined the word " continuous," 

 for technical purposes, to series having a certain high 

 degree of continuity. But for philosophical purposes, all 

 that is important in continuity is introduced by the lowest 

 degree of continuity, which is called " compactness." A 

 series is called " compact ' when no two terms are 

 consecutive, but between any two there are others. One 

 of the simplest examples of a compact series is the series 

 of fractions in order of magnitude. Given any two frac- 

 tions, however near together, there are other fractions 

 greater than the one and smaller than the other, and 

 therefore no two fractions are consecutive. There is no 

 fraction, for example, which is next after ^ : if we choose 

 some fraction which is very little greater than J, say j^, 

 we can find others, such as -J-^, which are nearer to |-. 

 Thus between any two fractions, however little they 

 differ, there are an infinite number of other fractions. 

 Mathematical space and time also have this property of 

 compactness, though whether actual space and time have 

 it is a further question, dependent upon empirical evidence, 

 and probably incapable of being answered with certainty. 



