LECTURE VI 



THE PROBLEM OF INFINITY 

 CONSIDERED HISTORICALLY 



It will be remembered that, when we enumerated the 

 grounds upon which the reality of the sensible world 

 has been questioned, one of those mentioned was the 

 supposed impossibility of infinity and continuity. In 

 view of our earlier discussion of physics, it would seem 

 that no conclusive empirical evidence exists in favour of 

 infinity or continuity in objects of sense or in matter. 

 Nevertheless, the explanation which assumes infinity and 

 continuity remains incomparably easier and more natural, 

 from a scientific point of view, than any other, and since 

 Georg Cantor has shown that the supposed contradictions 

 are illusory, there is no longer any reason to struggle 

 after a finitist explanation of the world. 



The supposed difficulties of continuity all have their 

 source in the fact that a continuous series must have an 

 infinite number of terms, and are in fact difficulties con- 

 cerning infinity. Hence, in freeing the infinite from 

 contradiction, we are at the same time showing the logical 

 possibility of continuity as assumed in science. 



The kind of way in which infinity has been used to 

 discredit the world of sense may be illustrated by Kant's 

 first two antinomies. In the first, the thesis states : 

 " The world has a beginning in time, and as regards 

 space is enclosed within limits"; the antithesis states: 



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