ORDER, NUMBER, QUANTITY 277 



selves. In the second place, as was pointed out a moment ago, 

 measured values are always numbers of something. Hence 

 what is properly attached to events in measurement is not a 

 number but a ratio, i.e., a fraction whose denominator is 

 the standard used and whose numerator is the number of 

 times that standard is contained in the quantity measured. 

 A length is not 5 but 5 inches, a weight is not 10 but 

 10 pounds, and an angle is not 30 but 30 degrees. 



With these reservations it becomes possible to say that 

 the field of scientific quantity is the field of numbers. As a 

 consequence, no specific set of postulates is required to 

 define quantities. Quantities are simply numbers with 

 which some qualitative notion is associated. It follows that 

 the features of the abstract postulate set defining numbers 

 have their correlative interpretations when applied to quan- 

 tities. Operations upon quantities are subject to specific 

 interpretations for different quantities. For example, the 

 product of two lengths is an area, the quotient of two lengths 

 is an angle, the quotient of space over time is a velocity, 

 and the product of mass and acceleration is a force. In 

 this way new quantities may be introduced by definition 

 in terms of elemental quantities. The identity elements, 

 1 and 0, are given correlative interpretations, the former 

 signifying usually the unit value of the quantity, and the 

 latter indicating the complete absence of intensive or ex- 

 tensive manifestation. The negatives, fractions, and irra- 

 tionals are sometimes subject to quantitative interpretation 

 and sometimes not. As quantities, however, they are subject 

 to more extensive empirical reference than as mere numbers. 

 It is, in fact, their applications as devices of measurement 

 which account for their importance in the natural sciences. 

 Some difficulty is found in trying to grasp the empirical 

 reference of fractions, negatives, irrationals, and even 1 and 

 0, so long as these are considered as numbers. But as quan- 

 tities, fractions have an interpretation which is almost uni- 

 versal; negatives have certain noteworthy interpretations, 

 as in the case of negative velocities, accelerations, spaces 



