THE PROBLEM OF INFINITY 179 



Between any two of them, there are others, for example, 

 the arithmetical mean of the two. Thus no two fractions 

 are consecutive, and the total number of them is infinite. 

 It will be found that much of what Zeno says as regards 

 the series of points on a line can be equally well applied 

 to the series of fractions. And we cannot deny that there 

 are fractions, so that two of the above ways of escape 

 are closed to us. It follows that, if we are to solve 

 the whole class of difficulties derivable from Zeno's by 

 analogy, we must discover some tenable theory of infinite 

 numbers. What, then, are the difficulties which, until 

 the last thirty years, led philosophers to the belief that 

 infinite numbers are impossible ? 



The difficulties of infinity are of two kinds, of which 

 the first may be called sham, while the others involve, for 

 their solution, a certain amount of new and not altogether 

 easy thinking. The sham difficulties are those suggested 

 by the etymology, and those suggested by confusion of 

 the mathematical infinite with what philosophers im- 

 pertinently call the " true ' infinite. Etymologically, 

 " infinite ' should mean " having no end." But in fact 

 some infinite series have ends, some have not ; while 

 some collections are infinite without being serial, and can 

 therefore not properly be regarded as either endless or 

 having ends. The series of instants from any earlier one 

 to any later one (both included) is infinite, but has two 

 ends ; the series of instants from the beginning of time 

 to the present moment has one end, but is infinite. 

 Kant, in his first antinomy, seems to hold that it is harder 

 for the past to be infinite than for the future to be so, 

 on the ground that the past is now completed, and that 

 nothing infinite can be completed. It is very difficult to 

 see how he can have imagined that there was any sense in 

 this remark ; but it seems most probable that he was 



