THE NUMBER CONCEPT. 



I5Y FLORIAN CAJOUI. 



For the benetit of teachers who do not have access to a 

 modern mathematical library it is proposed in this compila- 

 tion to present the views of mathematicians of the present 

 time relating to number, its origin and nature. We shall 

 begin by giving what is considered to be the primary num- 

 ber concept; then we shall briefly indicate how the original 

 and primitive idea of number is extended so as to apply to 



measurement. 



Primary Number. 



" Separateness or distinctness is a primary cognition, being neces- 

 sary even to the cognition of things as individuals, as distinct from 

 other things. The notion of number is based immediately on this 

 primary cognition. 



'• Number is that property of a group of distinct things which remains 

 unchanged during any change to which the group may be subjected 

 which does not destroy the distinctness of the individual things. Such 

 changes are changes of the characteristics of the individual things or 

 of their arrangement; for these do not cause one thing to split up into 



more than one, nor more than one to merge in one The 



number of things in any two groups of distinct things is the same, when 

 for each thing in the first group there is one in the second, and recipro- 

 cally, for each thing in the second group, one in the first. 



"Thu.?, the number of letters in the two groups, A, B, C: D, E, F, 

 is the same. In the second group there is a letter which may be assigned 

 to each of the letters in the first: as D to A, E to B, F to C: and recip- 

 rocally, a letter in the first which may be assigned to each in the second: 

 as Ato D, B to E, C to F. 



" Two groups thus related are said to be in one-to-one (1 — 1) corre- 

 spondence The fundamental operation of arithmetic is 



counting. To count a group is to set up a one-to-one correspondence 

 between the individuals of this group and the individuals of some rep- 

 resentative group. Counting leads to an expression for the number of 

 things in any group in terms of the representative group: if the repre- 

 sentative group be the fingers, to a group of fingers; if marks, to a group 

 of marks: if the numeral words, or symbols in common use, to one of 

 these words or symbols." — Prof. H. B. Fine, of Princeton, in The Num- 

 ber-System of Algebra. 1891, pp. 3, 4, 5. 



