THE PROBLEM OF INFINITY 159 



assumes absolute space ; but it is spatial relations that are 

 alone important, and they cannot be reduced to points. 

 This ground for his view depends, therefore, upon his 

 ignorance of the logical theory of order and his oscilla- 

 tions between absolute and relative space. But there is 

 also another ground for his opinion, which is more 

 relevant to our present topic. This is the ground 

 derived from infinite divisibility. A space may be halved, 

 and then halved again, and so on ad infinitum^ and at 

 every stage of the process the parts are still spaces, not 

 points. In order to reach points by such a method, it 

 would be necessary to come to the end of an unending 

 process, which is impossible. But just as an infinite class 

 can be given all at once by its defining concept, though 

 it cannot be reached by successive enumeration, so an 

 infinite set of points can be given all at once as making 

 up a line or area or volume, though they can never be 

 reached by the process of successive division. Thus the 

 infinite divisibility of space gives no ground for denying 

 that space is composed of points. Kant does not give his 

 grounds for this denial, and we can therefore only con- 

 jecture what they were. But the above two grounds, 

 which we have seen to be fallacious, seem sufficient to 

 account for his opinion, and we may therefore conclude 

 that the antithesis of the second antinomy is unproved. 



The above illustration of Kant's antinomies has only been 

 introduced in order to show the relevance of the problem 

 of infinity to the problem of the reality of objects of sense. 

 In the remainder of the present lecture, I wish to state and 

 explain the problem of infinity, to show how it arose, and 

 to show the irrelevance of all the solutions proposed by 

 philosophers. In the following lecture, I shah 1 try to 

 explain the true solution, which has been discovered by 

 the mathematicians, but nevertheless belongs essentially 



