viii.] PRINCIPLES OF NUMBER. 157 



the existence of difference where there is no difference. 1 

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

 over. But according to one of the conditions of logical 

 symbols, which I have called the Law of Unity (p. 72), 

 the same name repeated has no effect, and 



A -I- A = A. 



We must apply the Law of Unity, and must reduce all 

 identical alternatives before we can count with certainty 

 and use the processes of numerical calculation. Identical 

 alternatives are harmless in logic, but are wholly inad- 

 missible in number. Thus logical science ascertains the 

 nature of the mathematical unit, and the definition may 

 be given in these terms A unit is any object of thought 

 which can be discriminated from every other object treated as 

 a unit in the same problem. 



It has often been said that units are units in respect of 

 being perfectly similar to each other ; but though they 

 may be perfectly similar in some respects, they must be 

 different in at least one point, otherwise they would be 

 incapable of plurality. If three coins were so similar that 

 they occupied the same space at the same time, they 

 would not be three coins, but one coin. It is a property 

 of space that every point is discriminable from every other 

 point, and in time every moment is necessarily distinct 

 from any other moment before or after. Hence we 

 frequently count in space or time, and Locke, with some 

 other philosophers; has held that number arises from 

 repetition in time. Beats of a pendulum may be so 

 perfectly similar that we can discover no difference except 

 that one beat is before and another after. Time alone is 

 here the ground of difference and is a sufficient foundation 

 for the discrimination of plurality ; but it is by no means 

 the only foundation. Three coins are three coins, whether 

 we count them successively or regard them all simul- 

 taneously. In many cases neither time nor space is the 

 ground of difference, but pure quality alone enters. We 

 can discriminate the weight, inertia, and hardness of gold 

 as three qualities, though none of these is before nor after 

 the other, neither in space nor time. Every means of 

 discrimination may be a source of plurality. 



1 Pure Logic, Appendix, p. 82, 192. 



