viii.] PRINCIPLES OF NUMBER. 169 



crcte term in the plural, as men, without implying that 

 there are marks of difference. But when we use an 

 abstract term, we deal with unity. 



The origin of the great generality of number is now 

 apparent. Three sounds differ from three colours, or three 

 riders from three horses ; but they agree in respect of the 

 variety of marks by which they can be discriminated. The 

 symbols i+i+i are thus the empty marks asserting the 

 existence of discrimination. But in dropping out of sight 

 the character of the differences we give rise to new 

 agreements on which mathematical reasoning is founded. 

 Numerical abstraction is so far from being incompatible 

 with logical abstraction that it is the origin of our widest 

 acts of generalization. 



Concrete and Abstract Number. 



The common distinction between concrete and abstract 

 number can now be easily stated. In proportion as we 

 specify the logical characters of the things numbered, we 

 render them concrete. In the abstract number three 

 there is no statement of the points in which the three 

 objects agree ; but in three coins, three men, or three horses, 

 not only are the objects numbered but their nature is re- 

 stricted. Concrete number thus implies the same con- 

 sciousness of difference as abstract number, but it is 

 mingled with a groundwork of similarity expressed in the 

 logical terms. There is identity so far as logical terms 

 enter ; difference so far as the terms are merely numerical. 



The reason of the important Law of Homogeneity will 

 now be apparent. This law asserts that in every arith- 

 metical calculation the logical nature of the things num- 

 bered must remain unaltered. The specified logical 

 agreement of the things must not be affected by the un- 

 specified numerical differences. A calculation would be 

 palpably absurd which, after commencing with length, 

 gave a result in hours. It is equally absurd, in a purely 

 arithmetical point of view, to deduce areas from the 

 calculation of lengths, masses from the combination of 

 volume and density, or momenta from mass and velocity. 

 It must remain for subsequent consideration to decide in 

 what sense we may truly say that two linear feet multi- 



