LECTURE V 



THE THEORY OF CONTINUITY 



The theory of continuity, with which we shall be occupied 

 in the present lecture, is, in most of its refinements 

 and developments, a purely mathematical subject very 

 beautiful, very important, and very delightful, but not, 

 strictly speaking, a part of philosophy. The logical basis 

 of the theory alone belongs to philosophy, and alone 

 will occupy us to-night. The way the problem of con- 

 tinuity enters into philosophy is, broadly speaking, the 

 following : Space and time are treated by mathematicians 

 as consisting of points and instants, but they also have a 

 property, easier to feel than to define, which is called 

 continuity, and is thought by many philosophers to be 

 destroyed when they are resolved into points and instants. 

 Zeno, as we shall see, proved that analysis into points and 

 instants was impossible if we adhered to the view that the 

 number of points or instants in a finite space or time must 

 be finite. Later philosophers, believing infinite number 

 to be self-contradictory, have found here an antinomy : 

 Spaces and times could not consist of a finite number of 

 points and instants, for such reasons as Zeno's ; they 

 could not consist of an infinite number of points and 

 instants, because infinite numbers were supposed to be 

 self-contradictory. Therefore spaces and times, if real at 

 all, must not be regarded as composed of points and 



instants. 



129 9 



