NOTION AND DEFINITION OF NUMBER. 7 



chairs, we can usually bring the child to see that five things plus 

 three things are always eight things, -no matter of what nature the 

 things are, and that accordingly we need not always specify in 

 counting what kind of things we mean. At first we always make 

 the answer to our question of what five plus three is, easy for the 

 child, by relieving him of the process of abstraction, which is neces- 

 sary to ascend from the named to the unnamed number, an end 

 vnich we accomplish by not asking first what five plus three is, but 

 sjy associating with the numbers words designating things within 

 the sphere of the child's experience, for example, by asking how 

 many five pens plus three pens are. 



The preceding reflexions have led us to the notion of unnamed 

 or abstract numbers. The arithmetician calls these numbers posi- 

 tive whole numbers, or positive integers, as he knows of other kinds 

 of numbers, for example, negative numbers, irrational numbers, etc. 

 Still, observation of the world of actual facts, as revealed to us by our 

 senses, can naturally lead us only to positive whole numbers, such 

 only, and no others, being results of actual counting. All other kinds 

 of numbers are nothing but artificial inventions of mathematicians 

 created for the purpose of giving to the chief tool of the mathema- 

 tician, namely, arithmetical notation, a more convenient and more 

 practical form, so that the solution of the problems which arise in 

 mathematics may be simplified. All numbers, excepting the results 

 of counting above defined, are and remain mere symbols, which, 

 although they are of incalculable value in mathematics, and, there- 

 fore, can scarcely be dispensed with, yet could, if it were a ques- 

 tion of principle, be avoided. Kronecker has shown that any prob- 

 lem in which positive whole numbers are given, and only such are 

 sought, always admits of solution without the help of other kinds of 

 numbers, although the employment of the latter wonderfully sim- 

 plifies the solution. 



How these derived species of numbers, by the logical applica- 

 tion of a single principle, flow naturally from the notion of number 

 and of addition above deduced, I shall show in the next article en- 

 titled "Monism in Arithmetic. 5 ' 



