120 SCIENTIFIC METHOD IN PHILOSOPHY 



of the given event. It will be found that this class of events 

 is the first instant at which the given event exists, provided 

 every event wholly after some contemporary of the given 

 event is wholly after some initial contemporary of it. 



Finally, the series of instants will be compact if, given 

 any two events of which one wholly precedes the other, 

 there are events wholly after the one and simultaneous 

 with something wholly before the other. Whether this 

 is the case or not, is an empirical question ; but if it is not, 

 there is no reason to expect the time-series to be compact. 1 



Thus our definition of instants secures all that mathe- 

 matics requires, without having to assume the existence 

 of any disputable metaphysical entities. 



1 The assumptions made concerning time-relations in the above are as 

 follows : 



I. In order to secure that instants form a series, we assume : 



(a) No event wholly precedes itself. (An " event " is defined as 



whatever is simultaneous with something or other.) 



(b) If one event wholly precedes another, and the other wholly 



precedes a third, then the first wholly precedes the third. 



(c) If one event wholly precedes another, it is not simultaneous 



with it. 



(d) Of two events which are not simultaneous, one must wholly 



precede the other. 

 II. In order to secure that the initial contemporaries of a given event 



should form an instant, we assume : 

 (<?) An event wholly after some contemporary of a given event is 



wholly after some initial contemporary of the given event. 

 III. In order to secure that the series of instants shall be compact, 



we assume : 

 (/) If one event wholly precedes another, there is an event 



wholly after the one and simultaneous with something 



wholly before the other. 



This assumption entails the consequence that if one event covers the 

 whole of a stretch of time immediately preceding another event, then 

 it must have at least one instant in common with the other event ; i.e. it 

 is impossible for one event to cease just before another begins. I do 

 not know whether this should be regarded as inadmissible. For a 

 mathematico-logical treatment of the above topics, cf. N. Wilner, "A 

 Contribution to the Theory of Relative Position," Proc. Camb. Phil. Soc, 

 xvii. 5, pp. 441-449. 



