History and Nature of Measurement Theory

properties of these procedures md observations and then to show mathematically that these axioms permit the construction of a numerical assignment in which familiar abstract relations and operations, such as “is greater than or equal to” (symbolized 2 ) and ““pus” (+), correspond structurally to the empirical (or concrete) relations and operations. The second fundamental problem is that of the uniqueness of the representation-i.e., how close it is to being the only possible representation of its type. The representation of mass, for example, is unique in every respect except the choice of unit; e.g., the representation is different for pounds than for grams or grains. Ordinary measxements of temperature, however, are unique in everything except the choice of both unit and origin-the Celsius and Fahrenheit scales differ not only in the size of unit but also in the zero point. A U.S. psychophysicist, S. Smith Stevens, was among the first to place great emphasis on the uniqueness of measurement in relation to the problem of units and the consequences that this uniqueness has for data handling. A subject closely related to the uniqueness of physical measurement is dimensional analysis, which, in a broad sense, attempts to explain why the various physical measures exhibit simple relations in the fundamental equations of classical physics. If length, time, and mass are taken to be the fundamental dimensions of mechanics, for example, then all other quantities, such as force or momentum, can be represented simply as products of powers of these dimensions-a fact that has strong implications for the forms of physical laws (see below Universal, system. and material constants). A third central problem for the theory of measurement is error. In spite of the early development of precise astronomical measurement, no systematic theory of observational error appears to have been developed until the 18th century, first in an article by Thomas Simpson, an English mathematician, in 1757, and then in the fundamental work of two French mathematicians, Joseph-Louis, comte de Lagrange, who worked on the theory of numbers and celestial mechanics, and Pierre-Simon, marquis de Laplace, also an astronomer, famous for his fundamental work, Traité de méchanique céleste ( 1798-1 827; Celestial Mechanics, 1966). Today, any really significant physical measurement is routinely reported together with some indication of the probable error, and theories are tested by confirming them within the errors of measurement. In the classical tests of Albert Einstein’s general theory of relativity, for example, the discrepancies between predictions and observations fell mainly within the estimated errors of measurement (for further discussion of errors of measurement, see below The problem of error). AXIOMATIC BASIS OF MEASUREMENT Axiom systems for measurement differ in the amount and type of structure involved. All include an ordering relation, but that alone is not enough to formulate many scientific laws because its numerical representation is not closely prescribed. Additional structure and an increased uniqueness of the representation arise either by having an empirical operation of addition (extensive measurement), or by having entities that have several independent components (difference and conjoint measurement), or by other primitives that lead to a geometric representation. A X ~ Q ~ S of order. Whenever a measurement is made, it is done in such a way that the order induced on the objects by the assigned measure is the same as that obtained by the basic empirical operation in question. Such measures are said to be order preserving. Of course, every numerical inequality involved in measurement is transitive (i.e., such that x 2 y and y 2. z implies that x 2. z) and connected ( i e . , either x 2. y or y 2 x ) . It is therefore necessary that the empincal orderingsymbolized 2 (with instead of tol suggest real entitles)-be transitive and connected in order for a representation of it to be possible by means of 2. Relations exhibltmg these two properties are called weak orders, in which “weak” simply means that indifference (symbolized -)-in which x-~y = x 2 y and y 2 x-is not necessarily equality (using = to mean “if and only if”). Not every weak order, however, has an orderpreserving numerical representation; the lexicographic Axioms for binary relations order in the plane (x,y) ,> (x’,y’) x > x’ or x = x’ and y > y’, is a counterexample. A second, independent property of numbers, which must in turn be reflected in the empirical ordering, is that rational numbers-i.e., numbers expressible as the quotient of two integers-are order-dense (i.e., they are such that between any two distinct real numbers lies a rational number) and countable (Le., they can be placed in one-to-one correspondence with the integers). In 1895 it was shown by Georg Cantor, a German mathematician, that, for any empirical ordering, the existence of a numerical representation that preserves its order is equivalent to the ordering’s being a weak one that includes in its domain a countable, order-dense subset. Any two such representations are so related that one can be mapped upon the other in a strictly increasing fashion. Though various ordinal categorizations are widely used (e.g., brightness of stars, magnitude of earthquakes, hardness of minerals), the fact that equivalent ordinal representations are seldom in simple proportion to one another makes them unsatisfactory in the statement of many scientific laws. To increase the uniqueness of the representation, some structure in addition to order must be preserved by the representation. Axioms of extension. For many physical attributesincluding mass, length, time duration, and probabilityOrder the objects or events exhibiting the attribute may be comand bined, or concatenated, to form new objects or events concatethat also exhibit the attribute. Both alone and combined, nation for example, objects have mass. Denoting the concatenation of a and b by a o b, the assignment (designated G) of a given set of numbers to the objects is called an extensive representation of the empirical ordering 2 and the concatenation o, provided that it is not only order preserving but also additive in the sense that +(a o b ) = Theories that are intended to account for the existence of such numerical representations must state empirical laws about ordering and concatenation separately, as well as how they interrelate. The key property of an empirical ordering is that it is a weak order, and that of a concatenation is that it is insensitive to the order of combination-concatenation is weakly commutative (i.e., its order can be reversed), a o b b o a, and weakly associative (i.e., its terms can be regrouped), ( a o b ) o c a o ( b o c)-in which indicates an equivalent formulation. The key property relating them, weak monotonicity, is that the empirical ordering a ,> b holds if and only if the ordermg of its concatenation with any object c, a o c > b o c, also holds. As in ordinal measurement, a furh e r and more subtle property, called Archimedean, is needed; this asserts that the elements within the structure are commensurable with one another. This property is formally analogous to the numerical Archimedean property that if x > y > O, then for some integer n, ny > x (meaning y is not “infinitesimally” small relative to x). With these stipulations, the numerical representation can be constructed. The basic idea of the construction is simple and, in somewhat modified form, is widely used to carry out fundamental measurement. The measurer first chooses some object u, such as a foot rule, to be the unit of measurement. For any object a (say a house stud), he then finds other objects (studs) that are equivalent to a in the attribute in question (length). He concatenates these by laying them end-to-end in a straight line. Denoting by na the concatenation of any n of them, he now finds how many copies of u, say m ( n ) , are needed to approximate na; i.e., [m ( n ) +l] u 2 na 2 m ( n ) a. The limit toward which m ( n ) / n tends as n approaches infinity exists and is defined to be # ( a ) , which is the measurement sought; it can be shown to be order preserving and additive. In contrast to ordinal measurement, such an extensive representation is almost unique: it is determined except for a positive multiplicative constant or coefficient setting the scale; or, what is the same thing, only the choice of the unit is arbitrary. As a result, such a representation is called a ratio scale. +(a) + + ( b ) Measurement, Theory of 741.11 Sequences such as na, n = 1,2, . . ., being ubiquitous in measurement practice, are called standard sequences. Practical examples are standard sets of weights in multiples of a gram and the metre rule, subdivided into millimetres. For some purposes, including the development of other kinds of measurement-e.g., difference and probability measurement--it is necessary tQ generalize the theory to cover concatenation operations (o) that are defined only for some pairs of objects and not for others. Probability, for example, is not additive over all pairs of events, but only over those that are disjoint. Such theories, developed during the 1950s and 1960s, all assume that certain empirical inequalities have solutions. One form of such solvability is: if a > b, then there exists a c such that c and b can be concatenated and a ,> c o b. Axioms d difAFerence. Length, but not mass, may be Axioms treated also in terms of intervals (a, b ) on a line. Here the for empirical ordering ,> is a quaternary rather than a binaquaterry relation, since it involves the comparison of two nary pairs, and so four points, such as the end points of two relations intervals; and concatenation is defined only for adjacent intervals; i.e.,