THE THEORY OF CONTINUITY 133 



In the case of abstract objects such as fractions, it is 

 perhaps not very difficult to realise the logical possibility 

 of their forming a compact series. The difficulties that 

 might be felt are those of infinity, for in a compact series 

 the number of terms between any two given terms must 

 be infinite. But when these difficulties have been solved, 

 the mere compactness in itself offers no great obstacle to 

 the imagination. In more concrete cases, however, such 

 as motion, compactness becomes much more repugnant 

 to our habits of thought. It will therefore be desirable 

 to consider explicitly the mathematical account of motion, 

 with a view to making its logical possibility felt. The 

 mathematical account of motion is perhaps artificially 

 simplified when regarded as describing what actually 

 occurs in the physical world ; but what actually occurs 

 must be capable, by a certain amount of logical manipula- 

 tion, of being brought within the scope of the mathematical 

 account, and must, in its analysis, raise just such problems 

 as are raised in their simplest form by this account. 

 Neglecting, therefore, for the present, the question of 

 its physical adequacy, let us devote ourselves merely to 

 considering its possibility as a formal statement of the 

 nature of motion. 



In order to simplify our problem as much as possible, 

 let us imagine a tiny speck of light moving along a scale. 

 What do we mean by saying that the motion is continuous ? 

 It is not necessary for our purposes to consider the whole 

 of what the mathematician means by this statement : only 

 part of what he means is philosophically important. One 

 part of what he means is that, if we consider any two 

 positions of the speck occupied at any two instants, there 

 will be other intermediate positions occupied at inter- 

 mediate instants. However near together we take the 

 two positions, the speck will not jump suddenly from 



