THE MEANINGS OF INFINITY. 



247 



numerable," etc., all of which originally implied 

 infinity. From this point of view infinity is the 

 jast straggler of a whole host of words, which under 

 Ithe persuasive influence of popular usage have long 

 come to mean nothing more than great magnitude, 

 ;and is distinguished from them merely by the 

 precarious allegiance it still owns to the technical 

 terminology of the learned. 



§ 3. From this wholly improper and positive use 

 of infinity we may pass to one wholly proper, when 

 used in its strictness, but negative. This is the 

 mathematical use, which asserts that there can be no 

 end to the successive synthesis of unity in measuring 

 a quantity. We can never in our thought arrive at 

 a point when the addition of unity to a quantity, 

 however large, is impossible. 



Now as to this, it is noticeable (i) that the 

 definition is purely negative, and makes the con- 

 ception of infinity the conception of a limit, and (2) 

 that it is purely subjective. The definition makes 

 no reference to reality, but merely asserts that ''we 

 cannot help thinking. ..." 



We seem thus to receive a hint that the idea of 

 infinity indicates a defect, imperfection or limitation 

 of our thought, to which reality is only subjected 

 in so far as we must interpret it by our thought. 



§ 4. From this, the true conception of infinity, is 

 derived the mathematical doctrine of infinity, that 

 since infinity contains a number of given units greater 

 than all number, all finite quantities maybe neglected 

 in comparison with it. This reasoning involves 

 a subtle transition from the negative to a positive 

 conception, which finally results in infinity becoming 

 a kind of mathematical topsyturvydom, where two 

 parallel straight lines meet and enclose spaces, and 



