162 THE PRINCIPLES OF SCIENCE. [CHAP. 



In a similar manner it may be shown that the Law of 

 Unity A -|- 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 + x = x arises from the fact that we have 

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

 the other. Under such circumstances the Law of Unity 

 does not apply. For if in 



A' -|. A" = 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 Laws of Simplicity and Unity must always be observed 

 in the operation of counting that those laws seem no further 

 to apply. This is the understood condition under which 

 we use all numerical symbols. Whenever I write the 

 symbol 5 I really mean 



I + I 4 I 4- I + I, 



and it is perfectly understood that each of these units is 

 distinct from each other. If requisite I might mark them 

 thus 



i' + i" + i"' -f i"" + 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 

 i" + i" = i". 



Mathematical symbols then obey all the laws of logical 

 symbols, but two of these laws seem to be inapplicable 

 simply because they are presupposed in the definition of 

 the mathematical unit. Logic thus lays down the con- 

 ditions of number, and the science of arithmetic developed 

 as "it is into all the wondrous branches of mathematical 

 calculus is but an outgrowth of logical discrimination. 



Principle, of Mathematical Inference. 



The universal principle of all reasoning, as I have 

 asserted, is that which allows us to substitute like for like. 

 I have now to point out how in the mathematical science? 



