6 . NOTION AND DEFINITION OF NUMBER. 



As we see, the languages of humanity now no longer possess 

 natural number-signs and number-words, but employ names and 

 systems of notation adopted subsequently to this first stage. Ac- 

 cordingly, we must add to the definition of counting above given a 

 third factor or element which, .though not absolutely necessary, is 

 yet important, namely, that we must be able to express the results 

 of the above-defined associating of certain other things with the 

 things to be counted, by some conventional sign or numeral word. 



Having thus established what counting or numbering means, 

 we are in a position to define also the notion of number, which we 

 do by simply saying that by number we understand the results of 

 counting or numeration. These are naturally composed of two ele- 

 ments. First, of the ordinary number-word or number-sign ; and 

 secondly, of the word standing for the specific things counted. For 

 example, eight men, seven trees, five cities. When, now, we have 

 counted one group of things, and subsequently also counted another 

 group of things of the same kind, and thereupon we conceive the 

 two groups of things combined into a single group, we can save 

 ourselves the labor of counting the things a third time by blending 

 the number-pictures belonging to the two groups into a single num- 

 ber-picture belonging to the whole. In this way we arrive on the 

 one hand at the idea of addition, and on t-he other, at the notion of 

 "unnamed" number. Since we have no means of telling from the 

 two original number-pictures and the third one which is produced 

 from these, the kind or character of the things counted, we are ulti- 

 mately led in our conception of number to abstract wholly from the 

 nature of the things counted, and to form the definition of unnamed 

 number. 



We thus see that to ascend from the notion of named number 

 to the notion of unnamed number, the notion of addition, joined to 

 a high power of abstraction, is necessary. Here again our theory 

 is best verified by observations of children learning to count and 

 add. A child, in beginning arithmetic, can well understand what 

 five pens or five chairs are, but he cannot be made to understand 

 from this alone what five abstractly is. But if we put beside the 

 first five pens three other pens, or beside the five chairs three other 





