MONISM IN ARITHMETIC. C, 



the boughs and branches of which may still increase in size or even 

 give forth fresh sprouts, but whose main trunk has attained its full- 

 est development. 



Since arithmetic has arrived at its maturity, the more profound 

 investigation of the nature of numbers and their combinations shows 

 that a unitary conception of arithmetic is not only possible but also 

 necessary. If we logically abide by this unitary conception, we ar- 

 rive, starting from the notion of counting and the allied notion of 

 addition, at all conceivable operations and at all possible extensions of 

 the notion of number. Although previously expressed by Grassmann, 

 Hankel, E. Schroder, and Kronecker, the author of the present ar- 

 ticle, in his "System of Arithmetic," Potsdam, 1885, was the first 

 to work out the idea referred to, fully and logically and in a form 

 comprehensible for beginners. This book, which Kronecker in his 

 "Notion of Number," an essay published in Zeller's jubilee work, 

 makes special mention of, is intended for persons proposing to learn 

 arithmetic. As that cannot be the object of the readers of these es- 

 says, whose purpose will rather be the study of the logical construc- 

 tion of the -science from some single fundamental principle, the fol- 

 lowing pages will simply give of the notions and laws of arithmetic 

 what is absolutely necessary for an understanding of its develop- 

 ment. 



The starting-point of arithmetic is the idea of counting and of 

 number as the result of counting. On this subject, the reader is re- 

 quested to read the first essay of this collection. It is there shown 

 that the idea of addition springs immediately from the idea of count- 

 ing. As in counting it is indifferent in what order we count, so in 

 addition it is indifferent, for the sum, or the result of the addition, 

 whether we add the first number to the second or the second to the 

 first. This law, which in the symbolic language of arithmetic, is 



expressed by the formula 



a -}- b = b -f- a, 



is called the commutative law of addition. Notwithstanding this law, 

 however, it is evidently desirable to distinguish the two quantities 

 which are to be summed, and ou-t of which the sum is produced, by 

 special names. As a fact, the two summands usually are distin 



