174 SOME PRor.LEMS OF IMINSICAL CONTINUITV. 



(time) and not in a finite (time), and we touch infinite things 

 with infinite things, not with finite things." Hence, if it could 

 really be proved that the continuum is infinite because it is 

 infinitely divisible, it would follow for the same reason that all 

 time is similarly infinite. The two difficulties would cancel one 

 another. For there is no intellectual difficulty in measuring 

 infinite distance by infinite time. 



Yet this only brings the difficulty back to us in a more 

 modern form. If we agree to think of continuous (juantity a:, 

 an infinite series of terms, we are faced with all the confusion 

 that arises when we try to imagine what an infinite number i> 

 like. It is no (|uestion of imagining a series which begins in 

 sight and travels beyond our ken. The infinite series which 

 constitute a definite space of some continuum are all before us 

 at the same time. 



A most ingenious attempt has recently been made by Mr. 

 Bertrand Russell* to give an ade([uate answer to these queries 

 by means of a new theory of infinite numbers. He finds the 

 root of the difficulty in the common notion that we must be 

 able to count a number. "If you set to work to count the 

 terms in an infinite collection, you will never have completed 

 your task" (p. t8i). But this possil)iHty of counting is not 

 essential to the reality of number. We know many finite 

 collections, such as " mankind," without being able to count the 

 whole collection one by one ; and so, too, infinite collections " may 

 be known by their characteristics, although their terms cannot 

 be enumerated." 



After this preliminary statement Mr. Russell sets out to^ 

 establish what he calls his positive theory on infinity. The 

 need of it arises, or appears to Mr. Russell to arise, from the 

 emergence of infinite numbers in the arithmetic of the continuous. 



The supposed difficulties of continuity all have their source in the fact 

 tliat a continuous scries must have an inlinite number of terms, and the\ 

 are in fact difficulties c( ncerning inlinit\ . Hence in freeing the infinite 

 from contradiction, we are at the same time showing the logical possibility 

 of continuity as assumed in science.! 



" What is a number?" Mr. Russell asks. If we count 

 out a certain number of objects, their number is commonly 

 thought to be that of the last object reached in consecutive 

 order. But, says Mr. Russell, that is only true for finite 

 numbers. Where infinite numbers are concerned, counting, 

 even if it were practically possible, would not be ii valid method 

 of discovering the number of terms in an infinite collection, 

 and would in fact give different results according to the manner 

 in which it is carried out. 



Hence he calls our attention to two differences, which he 

 discerns, between finite and infinite numbers. The latter have 

 a property of " reflexiveness " which the former have not, and" 



* ■' Our Knowledge of the External World." Lecture \'I1, pp. 18^-208 

 t"P. 15.=. 



