158 THE PRINCIPLES OF SCIENCE. [CHAP. 



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 

 difference is specified. But the combinations AB and A& 

 are necesssarily 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 ABO, ABc, AbC, Abe, where there is a 

 double difference. 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 Logical Alphabet, 

 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 0, so that 

 the combinations are ABC, ABc, AbG. 



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 number three asserts the ex- 

 istence of marks without specifying their kind. 



Numerical abstraction is thus seen to be a dif- 

 ferent process from logical abstraction (p. 27), for in the 

 latter process we drop out of notice the very existence of 

 difference and plurality. In forming the abstract notion 

 hardness, we ignore entirely the diverse circumstances in 

 which the quality may appear. It is the concrete notion 

 three hard objects, which asserts the existence of hardness 

 along with sufficient other undefined qualities, to mark out 

 three such objects. Numerical thought is indeed closely 

 interwoven with logical thought. We cannot use a con- 



