PRINCIPLES OF N UMBER. 1 7 5 



mathematical science, and it is no wonder that we ever 

 meet with the traces of those laws from the domain of 

 which we can never emerge. 



The Nature of Number. 



Number is but another name for diversity. Exact 

 identity is unity, and with difference arises plurality. 

 An abstract notion, as was pointed out (p. 33), possesses 

 a certain oneness. The quality of justice, for instance, is 

 one and the same in whatever just acts it be manifested. 

 In justice itself there are no marks of difference by which 

 to discriminate justice from justice. But one just act can 

 be discriminated from another just act by many circum- 

 stances of time and place, and we can count and number 

 many acts each thus discriminated from every other. In 

 like manner pure gold is simply pure gold, and is so far 

 one and the same throughout. But besides its intrinsic 

 and invariable qualities, gold occupies space and must 

 have shape or size. Portions of gold are always mutually 

 exclusive and capable of discrimination, at least in respect 

 that they must be each without the other. Hence they 

 may be numbered. 



Plurality arises when and only when we detect differ- 

 ence. For instance, in counting a number of gold coins 

 I must count each coin once, and not more than once. Let 

 C denote a coin, and the mark above it the position in 

 the order of counting. Then I must count the coins 



If I were to make them as follows 



I should make the third coin into two, and should imply 

 the existence of difference where there is not difference 1 *. 

 C'" and C'" are but the names of one coin named twice 



b 'Pure Logic,' Appendix, p. 82, 192. 



