vn:.] PRINCIPLES OF NUMBER. 161 



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2x3 = 3x2, or X X y = y y. 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 -I- B = B -I- A, 



so in quantity 2 + 3 = 3 + 2, 



or generally x + y = y + x. 



The symbol -I- 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 



1/^ Px t// —— • fcC 



is true only in the two cases when a; = r, or a; = 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 tivo involves a different discrimination from the second 

 tioo. 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, 

 liglit heavy = o, 

 light light = light. 

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

 In sliort, tvnce two is two unless we take care that the 

 second two has a different meaning from the first. But 

 und(!r similar circumstances logical terms give the like 

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

 different in meaning from A'. 



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