248 MAN AND THE WORLD. 



two circles intersect at four points, etc. And, of 

 course, so long as these symbols are recognized as 

 fictions convenient, and even necessary, for the 

 technical purposes of mathematicians, nobody need 

 complain (cp. ch. vi. § 3), but unfortunately mathe- 

 maticians, like other mortals, are apt to forget this, 

 and frequently require a gentle reminder of their 

 logical absurdity. When, e.g., they say that two 

 parallel straight lines meet at infinity, they really 

 mean that they do not meet at all, or that we can 

 continue to conceive ourselves as prolonging them, 

 without their approaching. Or, again, the doctrine 

 that one infinity can be greater than another, is, to 

 say the least, inaccurate. For if infinity be taken 

 positively, it must mean something out of relation to 

 quantity, and different in kind, to which, therefore, 

 phrases like ''greater and less than" are totally 

 inapplicable. If, e.g., one of two straight lines may 

 be produced indefinitely in one direction and the 

 other in both, the mathematical doctrine is that the 

 second infinity is greater than the first. But the 

 question whether one will at any time be greater or 

 less than the other will depend on the rate at which 

 they are produced and the size of the "successive 

 syntheses," and not on their being infinite in one or 

 two directions. But in order to measure them at 

 all, and so to be able to speak of greater or less 

 with respect to them, they must both be limited 

 first, which is ex hypothesi impossible. Hence the 

 category of quantity is inapplicable to the case, and 

 the positive conception of infinity is absurd, an 

 infinite quantity being a contradiction in terms. For 

 being infinite, no measure can exhaust it, while a 

 quantity is that which is composed of units of 

 measurement. 



