The Nr.MBEH Concept. 33 



*' To count things means to consider them iis alike, to take them 

 tofjether, and to associate with them, singly, other things which are 

 also considered alike. Each of the things with which other things are 

 associated in counting is called a unit; each of the things which in 

 counting are associated with other things is called a one. The result of 

 counting is called imuiber. On account of the likeness (Gleichartigkeit) 

 of the iDiits among themselves and of the ones among themselves, the 

 number is independent of the order in which the ones are associated 

 with the jtnils. 



"If in a number we specify to what extent the units were consid- 

 ered as being alike, by assigning a collective name to the units, then we 

 have a concrete number. By completely ignoring the nature of the 

 things counted we pass from the notion of a concrete number to the 

 notion of an abstract number. 



"By the word number, unqualified, we shall always understand an 

 abstract number." — Burkhardt and Meyer's Encyklopadie der Mathe- 

 matischen Wissenschaften. Part I, Vol. I, 1898, pp. 1-3. 



"To count distinct things means to make of them an artificial indi- 

 vidual or group, and then to identify its elements with those of a familiar 

 group."— Prof. G. B. Halsted, in Science. N. S., Vol. VIII, 1898, 

 p. 909. 



" Counting presupposes the comparison of multitudes (Vielheiten). 

 By multitudes we mean a group of objects alike among themselves; 

 that is, we convey the notion of disjunctive things, the differences of 

 which remain unnoticed, and the mode of arrangement of which is 

 disregarded. Two multitudes are called equal when with each object 

 of the first there is associated, singly, one of the second and none remain 

 disassociated. That multitude is called greater than another which has 

 some objects left disassociated after every object of the second (smaller) 

 multitude is associated singly with some object of the first. The com- 

 mon mark (Merkmal) of all multitudes which are equal to a definite one 



is expressed by a numeral word The natural number is a 



multitude of units, that is of ones."— Orro Stolz, of the University of 

 Innsbruck, in Vorlesungen uber Allyemeine Arithuwlik. 188"), pp. 9. 10. 



