182 THE PRINCIPLES OF SCIENCE. 



under similar circumstances logical terms would give 

 exactly the like result, and it is not true that A' A" = A', 

 identically where A" is different in meaning from A'. 

 In an exactly similar manner it may be shown that 



the Law of Unity 



A I A = A 



holds true alike of logical and mathematical terms. It is 

 absurd indeed to say that 



x + x = x 



except in the one case when x = absolute zero. But this 

 contradiction x -f x = x arises from the fact that we have 

 already defined the unit in one x as differing from those in 

 the other. Under such circumstances the Law of Unity 

 does not apply. For if in 



A'tA" = A' 



we mean that A" is in any way different from A' the 

 assertion of identity is evidently false. 



The contrast then which seems to exist between logical 

 and mathematical symbols is only apparent. It is because 

 the Law of Simplicity and Unity must always be ob- 

 served in the operation of counting that those laws can 

 no longer be operative. This is the understood condition 

 under which we use all numerical symbols. Whenever 

 I use the symbol 5 I really mean 



i + i + i -f- i + i, 



and it is perfectly understood that each of these units is 

 distinct from each other. If requisite I might mark them 

 thus 



l' + l"+i'"+l^+i*'" 



Were this not the case and were the units really 



i'+i" + i" + i"' + i"", 

 the Law of Unity would, as before remarked, apply, and 



l"+l"=l". 



Mathematical symbols then obey all the laws of logical 



