mi.] PRINCIPLES OF NUMBER. 161 



2x3 = 3x2, or x x y = y x x. 



The properties of space are as indifferent in multiplication 

 as we found them in pure logical thought. 

 Similarly, as in logic 



triangle or square = square or triangle, 

 or generally A -|- B = B -|- A, 



so in quantity 2 + 3 = 3 + 2, 



or generally x + y = y + x. 



The symbol -|- is not identical with +, but it is thus far 

 analogous. 



How far, now, is it true that mathematical symbols obey 

 the Law of Simplicity expressed in the form 



AA = A, 

 or the example 



Eound 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 x = 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 

 two. I get four kinds of things, for instance, if 1 first dis- 

 criminate "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 dis- 

 criminated by the same difference of light and heavy, then 

 we have 



heavy heavy = heavy, 

 heavy light = o, 

 light heavy = o, 

 light light = light. 



Thus, (heavy or light) x (heavy or light) = (heavy or light). 

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

 second two has a different meaning from the first. But 

 under similar circumstances logical terms give the like 

 result, and it is not true that A' A" = A', when A" is 

 different in meaning from A'. 



M 



