176 THE PRINCIPLES OF SCIENCE. 



over. But according to one of the conditions of logical 

 symbols, which I have called the Law of Unity (p. 86), 

 the same name repeated has no effect, and 



Al 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 produce deadly 

 error 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. 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 fre 

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

 other philosophers, has even held that number arises from 

 repetition in time. Beats of a pendulum might be so 

 perfectly similar that we could 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 simultaneously. In many cases neither time nor space 

 is the ground of difference, but pure quality alone enters. 

 AVe can discriminate for instance the weight, inertia, and 



