2 4 o DEVELOPMENT AND PURPOSE CHAP. 



calculus, is that a summation of quantitative changes pro- 

 longed to infinity amounts to a qualitative change. This 

 result may be resolved into three propositions, (i) No 

 quantitative extension of the series yields the change of 

 quality. (2) Every such extension makes the summed-up 

 series approximate more closely to the different quality, and 

 there is no barrier to the approximation short of the 

 limiting quality itself. (3) If such a series represents 

 successive points in a physical continuum, that continuum 

 may extend up to and beyond the limit without any breach 

 in it. 



We have seen above how this conception is applied to 

 the division of the continuous. The point is a part of 

 space which dwindles as division continues. At the limit 

 in which the number of points is infinite its dimensions 

 are also zero. That is, the conception has undergone a 

 qualitative change whereby, instead of conceiving the 

 space as an aggregate of points, we conceive it as a con- 

 tinuum. As we touch the limit we reach a new concep- 

 tion. Now whether the result so exemplified in the case 

 of the infinitely little would have similar application to the 

 infinitely great is a further question. But at least, in 

 expecting that we should find infinite space something 

 qualitatively different from finite space, and eternity some- 

 thing qualitatively different from time, we should be 

 moving in accordance with philosophical tradition. 



Before considering this possibility further, let us note 

 the bearing of the discussion on the question of the validity 

 of thought and its relation to reality. Whether we accept 

 the mathematics of the transfinite as philosophy, or merely 

 recall what has been said of the development of the theory 

 of the Calculus, we have equally to recognise the transfor- 

 mation of conceptions by contact with the infinite. From 

 this transformation we learn, first, that the discrete treat- 

 ment of space, time and quantity is inadequate. It does 

 not represent and cannot adequately express continuity of 

 process, of motion, of transition, for when we represent 

 space, time, motion or anything physically continuous by 

 a number, we take it at a certain point, not as in process 

 through that point. But, secondly, a method thus faulty 



