178 THE PRINCIPLES OF SCIENCE. 



number three asserts the existence of marks without 

 specifying their kind. 



Numerical abstraction is then a totally different process 

 from logical abstraction (see p. 33), for in the latter 

 process we drop out of notice the very existence of 

 difference and plurality. In forming the abstract notion 

 hardness, for instance, I drop out of notice altogether 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 concrete term in the 

 plural, as men, without implying that there are marks of 

 difference. Only when we use a term in the singular 

 and abstract sense man, do we deal with unity, unbroken 

 by difference. 



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 fact of discrimination which may apply to objects 

 wholly independently of their peculiar nature. 



Concrete and Abstract Numbers. 



The common distinction between concrete and ab- 

 stract numbers can now be easily stated. In proportion 

 as we specify the logical character of the things num- 

 bered, we render them coilcrete. In the abstract num- 

 ber 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 variety of objects defined, 



