156 THE PRINCIPLES OF SCIENCE. [CHAP. 



basis both of logic and of quantitative mathematics. But 

 I do not care to pursue the subject because I think that 

 in either case Boole was wrong. In my opinion logic is 

 the superior science, the general basis of mathematics as 

 well as of all other sciences. Number is but logical dis- 

 crimination, and algebra a highly developed logic. Thus 

 it is easy to understand the deep analogy which Boole 

 pointed out between the forms of algebraic and logical 

 deduction. Logic resembles algebra as the mould 

 resembles that which is cast in it. Boole mistook the 

 cast for the mould. Considering that logic imposes its 

 own laws upon every branch of mathematical science, it 

 is no wonder that we constantly meet with the traces of 

 logical laws in mathematical processes. 



The Nature of Number. 



Number is but another name for diversity. Exact iden- 

 tity is unity, and with difference arises plurality. An 

 abstract notion, as was pointed out (p. 28), possesses a 

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

 and the same in whatever just acts it is 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 circumstances of 

 time and place, and we can count many acts thus discri- 

 minated each from each. In like manner pure gold is 

 simply pure gold, and is so far one and the same through- 

 out. But besides its intrinsic qualities, gold occupies 

 space and must have shape and size. Portions of gold 

 are always mutually exclusive and capable of discrimina- 

 tion, in respect that they mnst 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 order of 

 counting. Then I must count the coins 



C' + C" + C'" + C"" + 



If I were to count them as follows 



C' + C" + C"' + C'" + C"" + . . ., 

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



