ORDER, NUMBER, QUANTITY 267 



into the empirical concept of number through an operation 

 of ordering. In fact the negatives, fractions, and irrationals 

 may themselves be considered as serial extensions rather 

 than as the results of operational generalization. As was 

 pointed out, alternative routes of derivation are often 

 possible. But what is important is that some notion of order 

 is frequently considered essential to the definition of num- 

 ber, i.e., numbers are considered to be not merely a collection 

 of entities but an ordered collection. This gives rise to the 

 notion of ordinal numbers, and justifies the act of counting. 

 It gives rise also to elaborate techniques by which order 

 may be set up among fractions, for example, or among irra- 

 tionals. The general techniques for the establishment of 

 series have already been discussed ; the problem in each case 

 is the discovery of a relation which will satisfy the general 

 postulates defining order. For example, it is possible to 

 order the natural numbers on the basis of the relation "less 

 than"; all of the general postulates defining order would be 

 satisfied, and all of the special postulates defining the dis- 

 crete series; hence the result would be the ordinary number 

 system, 1,2,3,4,5, . . . , called earlier in the chapter the 

 progression. It is by means of this that counting is carried on. 



number: scientific content 



The generality involved in the scientific concept of number 

 suggests that it has become far removed from its empirical 

 foundation. With the introduction of 1, 0, the negatives, 

 the fractions, the irrationals, and the imaginaries, it seems 

 that mathematics is no longer concerned with events. "In 

 accepting these symbols as its numbers, arithmetic ceases 

 to be occupied exclusively or even principally with the 

 properties of numbers in the strict sense. It becomes an 

 algebra whose immediate concern is with certain opera- 

 tions . . . defined formally only, without reference to the 

 meaning of the symbols operated on." x Instead of defining 

 number as the property of groups, mathematics considers it 



1 H. B. Fine, The Number-System of Algebra (Boston: Heath, 1907), p. 26. 



