PRINCIPLES OF NUMBER. 181 



Similarly, just as in logic 



triangle or square = square or triangle, 

 or generally A I B = B I A, 



so in quantity 2+3 = 3 + 2, 



or generally x + i y = y + x. 



The symbol I is not identical with + , but it is so far 

 analogous. 



How far, now, is it true that mathematical symbols 



v 



obey the law of simplicity expressed in the form 



AA = A, 

 or the example 



Round round = round ? 



Apparently there are but two numbers which obey this 

 law ; for it is certain that 



x x x x 

 is true only in the two cases when x = i or o. 



In reality all numbers obey the law, for 2 x 2 = 2 is not 

 really analogous to AA = A. According to the definition 

 of a unit already given, each unit is discriminated from 

 each other in the same problem, so that in 2 x 2 , the 

 first two involves a different discrimination from the 

 second ttvo. I get four kinds of things, for instance, if I 

 first discriminate heavy and light and then cubical and 

 spherical/ for we now have the following classes- 

 heavy, cubical. light, cubical, 

 heavy, spherical. light, spherical. 

 But suppose that my two classes are in both cases 

 discriminated by the same difference of light and heavy, 

 then we have 



hearvy heavy = heavy, 

 heavy light - o, 

 light heavy = o, 

 light light = light. 



In short, twice two is two unless we take care that the 

 second two has a different meaning from the first. But 



