ORDER, NUMBER, QUANTITY 269 



might leave the operations themselves in an undefined form, 

 and thus define a highly abstract group. Or one might inter- 

 pret them in a manner which suggests certain empirical 

 operations performable on certain empirical objects, and 

 thus define a less abstract group. The postulate system which 

 is here offered as a definition of the field of numbers employs 

 the notions of addition and multiplication as the basic 

 operations. Then the number system may be defined by the 

 following postulates : 



1. If a and b are numbers, then a X b and a + b are numbers. 



2. (Associative law.) a X (b X c) = (a X b) X c, and a -j- 

 (6 + C ) = (a + b) + c. 



3. (Commutative law.) a X b = b X a, and a + b = b + a. 



4. (Definition of identity element.) There exists a number 1 such 

 that aXl = lXa = a, and a number such that a + = + 

 a = a, for every number a of the class of numbers. 



5. (Definition of inverse element.) There exists, corresponding to 

 any number a, another — a such that a + — a = 0, and another 

 1 la such that a X 1/a = 1, except that no inverse of is required. 



6. (Distributive law.) If a, b, c are numbers, a X (b + c) = 

 a X b + a X c, and (b + c) Xa = bXa-\-cXa. 1 



Nothing is said in this postulate scheme concerning the 

 scope of the class of numbers; the only two elements whose 

 existence is assured are 1 and 0. But if one supposes, say, 

 that successive additions of 1 determine distinct elements, 

 the class becomes infinite. The operations of subtraction and 

 division may be defined, respectively, in terms of addition 

 and multiplication. Through ^these operations the class 

 becomes indefinitely extended to include the negatives, the 

 fractions, the irrationals, and the imaginaries. The prin- 

 ciples by which the class of numbers may be ordered are not 

 given in this set, though they may be added as further 

 postulates if they are required. 



The important feature to be noted in connection with 

 this abstract definition of number is the discrepancy be- 



1 This set is a modification of that given by J. W. Young, Fundamental Concepts of 

 Algebra and Geometry, p. 94. See also R. B. Lindsay and H. Margenau, Founda- 

 tions of Physics (New York: Wiley, 1926), pp. 9-10. 



