THE THEORY OF CONTINUITY 131 



outline what the mathematical theory of continuity is in 

 its philosophically important essentials. The application 

 to actual space and time will not be in question to begin 

 with. I do not see any reason to suppose that the points 

 and instants which mathematicians introduce in dealing 

 with space and time are actual physically existing entities, 

 but 1 do see reason to suppose that the continuity of 

 actual space and time may be more or less analogous to 

 mathematical continuity. The theory of mathematical 

 continuity is an abstract logical theory, not dependent for 

 its validity upon any properties of actual space and time. 

 What is claimed for it is that, when it is understood, 

 certain characteristics of space and time, previously very 

 hard to analyse, are found not to present any logical 

 difficulty. What we know empirically about space and 

 time is insufficient to enable us to decide between various 

 mathematically possible alternatives, but these alternatives 

 are all fully intelligible and fully adequate to the observed 

 facts. For the present, however, it will be well to forget 

 space and time and the continuity of sensible change, in 

 order to return to these topics equipped with the weapons 

 provided by the abstract theory of continuity. 



Continuity, in mathematics, is a property only possible 

 to a series of terms, i.e. to terms arranged in an order, so 

 that we can say of any two that one comes before the 

 other. Numbers in order of magnitude, the points on a 

 line from left to right, the moments of time from earlier 

 to later, are instances of series. The notion of order, 

 which is here introduced, is one which is not required in 

 the theory of cardinal number. It is possible to know 

 that two classes have the same number of terms without 

 knowing any order in which they are to be taken. We 

 have an instance of this in such a case as English husbands 

 and English wives : we can see that there must be the 



