PRINCIPLES OF NUMBER. 177 



hardness of gold as three qualities, though none of these 

 is before or after the other, either in space or time. 

 Every means of discrimination may be a source of 

 plurality. 



Our logical notation may be used to express the rise of 

 number. The symbol A stands for one thing or one class, 

 and in itself must be regarded as a unit, because no differ- 

 ence is specified. But the combinations AB and Kb are 

 necessarily two, because they cannot logically coalesce, and 

 there is a mark B which distinguishes one from the other. 

 A logical definition of the number four is given in the 

 combinations ABC, ABc, A&C, Abe, where there is a double 

 difference, and as Puck says 



' Yet but three ? Come one more ; 

 Two of both kinds makes up four.' 



I conceive that all numbers might be represented as 

 arising out of the combinations of the Abecedarium, more 

 or less of each series being struck out by various logical 

 conditions. The number three, for instance, arises from 

 the condition that A must be either B or C, so that the 

 combinations are ABC, ABc, A6C. 



Of Numerical Abstraction. 



There will now be little difficulty in forming a clear 

 notion of the nature of numerical abstraction. It consists 

 in abstracting the character of the difference from which 

 plurality arises, retaining merely the fact. When I speak 

 of three men I need not at once specify the marks by 

 which each may be known from each. Those marks must 

 exist if they are really three men and not one and the 

 same, and in speaking of them as many I imply the 

 existence of the requisite differences. Abstract number, 

 then, is the empty form of difference; the abstract 



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