measurement

The fundamental concepts of the theory of measurement are that of a quantity being measured, an empirical determination for a lesser, equal, or greater amount of the quantity, and then a rule assigning numerical values to the quantities empirically determined. Different quantities are therefore represented by different numbers. The rule must require procedures for assigning the same numerals to the same things under the same conditions, and it must be non-degenerate, in the sense that the rule allows for the possibility of assigning different numerals to different things, or the same thing under different conditions. The rule then defines a scale from the least value of the quantity to the greatest. Moh's hardness scale is an ordering of minerals from the softest (talc, 1) to the hardest (diamond, 10). Such a scale gives no sense to the idea of one point on the scale (say, orthoclase, 6) being twice as hard as another (calcite, 3), nor to the question of whether the difference between one pair of intervals on the scale is the same as that of another. In the terminology of Brian Ellis, Basic Concepts of Measurement (1966), Moh's scale is simply an ordering, or an ordinal scale. Features that can be ordered but no more, are sometimes called qualitative, or non-metric. If, in addition, formulae providing for the absolute sameness or difference of intervals on the scale can be interpreted, we have an interval scale; if such intervals can be compared we have an ordinal-interval scale; and if a = nb can be interpreted where (a,b) are numbers on the scale and n is any positive integer, we have a ratio scale. The date scales of the calendar are ordinal-interval scales, whereas ordinary scales of mass, length, and time are ratio scales. Questions for the philosophy of science include the nature and objectivity of measurement, the question of whether the same quantity can be measured by scales which are not simple transformations of one another (as the Celsius and Fahrenheit scales of temperature are), and the nature of the considerations, such as mathematical simplicity, that guide the choice of fundamental scales for measuring physical quantities. In particular disciplines, for instance economics, the question of whether a quantity such as utility or welfare is purely qualitative, or is susceptible of more structured measurement, may assume great importance. Similarly philosophical, scientific, and pragmatic considerations affect finding the correct quantities to measure in order to understand multi-dimensional complexes such as an economy or a society.