Measurement Theory Research Paper

1. Measurement Theory and Problems Internal to Quantitative Science

1.1 Measurement of Extensive Magnitudes

The first treatise touching on the theory of measurement was Book V of Euclid’s Elements (Heath 1908), written about 300 BC and attributed to Eudoxus. It defined the central concepts of magnitude and ratio and explained the place of numbers in measurement. This explanation was accepted as standard for more than 2,000 years. The early Greeks divided quantity into multitude (or discrete quantity, e.g., the size of a football crowd) and magnitude (or continuous quantity, e.g., the length of a stadium). The measure of a quantity was given relative to numbers of appropriately defined units. Number, then, was what today is called ‘whole number.’ This posed a problem because it was known that magnitudes could be mutually incommensurable (two magnitudes are mutually incommensurable if and only if the ratio of one to the other cannot be expressed as a ratio of whole numbers, e.g., as is the case with the lengths of the side and diagonal of a square). The problem for Euclid was to explain how his discrete concept of measure applied to continuous magnitudes.

It was solved by liberalizing the concept of measure to that of ratio and by requiring that for each specific magnitude (say, each specific length) there is the series of multiples of that magnitude (e.g., for any specific length, L, there exists mL, for each whole number, m). Considering just a single pair of lengths, say L and L , and letting n and m stand for any two whole numbers, the ratio between L and L is given by the following three classes of numerical ratios:

(a) The class of all n/m such that mL₁ ˂ nL₂;

(b) The class of all n/m such that mL₁ = nL₂; and

(c) The class of all n/m such that mL₁ ˃ nL₂.

Defined this way, a ratio exists between each magnitude and any arbitrary unit, whether incommensurable or not, and, in principle, this ratio may be estimated. This is the classical theory of measurement. Of course, unless there is some physical operation for obtaining multiples of magnitudes it is not possible to estimate these ratios. Hence, this elegant and powerful solution was not taken to apply beyond the range of extensive magnitudes (i.e., those for which some operation of addition can be defined). At the time this included only the geometric magnitudes, weight and time.

1.2 Measurement of Intensive Magnitudes in Physics

The early Greeks recognized that other attributes, for example, temperature, admitted of degrees and Aristotle even discussed these hypothetically in terms of ratios (On Generation and Corruption 334 b, 5–20). This raised the possibility of nonextensive magnitudes (i.e., magnitudes for which an operation of addition was not known), a possibility explored during the Middle Ages via the concept of intensive magnitude. For example, Oresme thought intensive magnitudes similar enough to line lengths to assert that ‘whatever ratio is found to exist between intensity and intensity, … a similar ratio is found to exist between line and line and vice versa’ (Clagget 1968, p. 167), thereby attributing to them an internal structure sufficient to sustain ratios.

The medieval philosophers, however, did not attempt to show that their intensive magnitudes really were, like length, additive in structure. While they contemplated intensive magnitudes as theoretical possibilities, they were not interested in actually measuring them. Attempts to expand the range of measurable attributes were more successful from the Scientific Revolution of the seventeenth century onwards, when science became more closely identified with quantitative methods. This success raised another problem: it was one thing to distinguish intensive from extensive magnitudes in theory; it was quite another to show that a particular procedure actually measured any of the former. Gradually, the issue dawned of what was to count as evidence that hitherto unmeasured attributes are measurable.

Most of the physical magnitudes found measurable after the Scientific Revolution were not extensive and were measured indirectly. For example, velocity was measured via distance and time, density, via mass and volume. In each case, the magnitude measured in this derived way was thought to be some quantitative attribute of the object or system involved, but one for which a physical operation directly reflecting the supposed underlying additive structure of the attribute could not be (or, possibly, had not yet been) identified. Because quantitative physics, as a body of science, was held in high regard, other disciplines aped its quantitative character. For example, Hutcheson (1725) proposed a quantitative moral psychology. He was criticized by Reid (1748) for ‘applying measures to things that properly have not quantity’ (p. 717), a charge that was to become increasingly familiar as further attempts were made to integrate quantitative thinking into psychology. The problem is how to tell quantitative from nonquantitative attributes. Solving it requires specifying quantitative structure.

Holder (1901) achieved this, specifying conditions defining the concept of an unbounded, continuous quantitative attribute. Call the attribute Q and let its different levels (the specific magnitudes of Q) be designated by a, b, c, …. For any three levels, a, b, and c, of Q, let a + b = c if and only if c is entirely composed of discrete parts a and b. Then according to Holder, Q being quantitative means that the following seven conditions hold.

(a) Given any two magnitudes, a and b, of Q, one and only one of the following is true:

(i) a is identical to b (i.e., a = b and b = a);

(ii) a is greater than b and b is less than a (i.e., a ˃ b and b ˂ a); or

(iii) b is greater than a and a is less than b (i.e., b ˃ a and a ˂ b).

(b) For every magnitude, a, of Q, there exists a b in Q such that b ˂ a.

(c) For every pair of magnitudes, a and b, in Q, there exists a magnitude, c, in Q such that a + b = c.

(d) For every pair of magnitudes, a and b, in Q, a + b ˃ a and a + b ˃ b.

(e) For every pair of magnitudes, a and b, in Q, if a ˂ b, then there exists magnitudes, c and d, in Q such that a + c = b and d + a = b.

(f) For every triple of magnitudes, a, b, and c, in Q, (a + b) + c = a + (b + c).

(g) For every pair of classes, φ and ψ, of magnitudes of Q, such that

(i) each magnitude of Q belongs to one and only one of φ and ψ;

(ii) neither φ nor ψ is empty; and

(iii) every magnitude in φ is less than each magnitude in ψ, there exists a magnitude x in Q such that for every other magnitude, x′, in Q, if x′ ˂ x, then x ɛ φ and if x′ ˃ x, then x′ ɛ ψ (depending on the particular case, x may belong to either class).

Holder had one eye on Euclid’s concept of ratio and, so, what he was able to prove from these conditions was that if an attribute has this sort of structure, then the system of ratios of its magnitudes is isomorphic to the system of positive real numbers. Among other things, this entails that each level of the attribute may be measured by any other level taken as the unit.

Using Holder’s conditions, the issue of the sort of evidence needed to confirm the hypothesis that some attribute is quantitative may be considered. If, relative to the attribute in question, a concatenation operation upon objects can be identified which directly reflects quantitative additivity, then one has evidence that the attribute is quantitative. For example, rigid straight rods of various lengths may be joined lengthwise so that the resulting length equals the sum of the lengths of the rods concatenated. This issue was considered in a systematic way by Helmholtz (1887) and Campbell (1920) and Campbell’s term, ‘fundamental measurement,’ has become the standard one to describe this way of establishing measurement.

Campbell (1920) also identified the category he called ‘derived measurement.’ He held that ‘the constant in a numerical law is always the measure of a magnitude’ (p. 346). For example, the ratio of mass to volume is a constant for each different kind of substance, say, one constant for gold, another for silver, and so on. That such system-dependent constants identify magnitudes of quantitative attributes is perfectly sound scientific thinking. In the case of density, the fact that this constant is always observed must have a cause within the structure of the system or object involved. Since the effect is quantitative, likewise must be the cause. Hence, the magnitude identified by the constant, or as we call it, the density of the substance, is quantitative and measured by this constant.

A similar line of thought suffices for all cases of indirect quantification in physics, and because Campbell (1920) thought that physics was ‘the science of measurement’ (p. 267), he concluded that the issue of the sort of evidence needed to show that an attribute is quantitative is exhausted by the categories of fundamental and derived measurement. Campbell did not know that Holder (1901) had already refuted this conclusion using a simple model. If all that could be known of any two points on a line is their order and of any two intervals between such points, whether they are equal, then Holder showed this knowledge sufficient to prove distance quantitative. His 10 axioms applying just to order between points and equality between intervals exploit the fact that these nonadditive relations may indirectly capture the underlying additivity between distances (viz., if A B C (where A, B, and C are points), then the distance from A to C equals that from A to B plus that from B to C). Applied to other attributes, this result entails that there are ways other than those standard in physics for showing attributes as quantitative. Hence, the possibility of measurement in disciplines lacking fundamental measurement cannot be ruled out.

1.3 Measurement of Psychological Magnitudes

Attempts at measurement in psychology presented entirely new challenges for measurement theory. These attempts are hampered by the fact that there are no fundamentally measured, extensive psychological magnitudes upon which to base the measurement of theoretical, psychological attributes. Hence, in attempting to gain evidence that psychological attributes of interest are quantitative, Campbell’s categories of fundamental and derived measurement cannot be utilized. To cope with this, a new approach to measurement had to be devised.

This challenge was met by the theory of conjoint measurement. Unlike fundamental and derived measurement, conjoint measurement does not depend upon finding extensive magnitudes. This theory (Krantz et al. 1971) applies in circumstances where a dependent attribute is a noninteractive function of two independent attributes and where distinct levels of the independent attributes can at least be classified (as same or different) and the dependent attribute weakly ordered. Since, in psychology, such classifications and orderings are sometimes possible, this theory is applicable. If the attributes involved are quantitative, then this is detectable via trade-offs between the contributions of the conjoining independent attributes to the order upon levels of the dependent attribute. The theory has been applied to the measurement of psychological attributes and has been extended to situations involving more than two independent attributes.

2. Measurement Theory and Problems External to Quantitative Science

2.1 The Philosophy of Mathematics

The most potent external factor affecting measurement theory has been the philosophy of mathematics, especially as it relates to the concept of number. Holder’s (1901) proof of an isomorphism between ratios of magnitudes of a continuous quantitative attribute and the positive real numbers appeared to secure the view, popular at least since the Scientific Revolution (and, arguably, implicit in Euclid) that numbers are such ratios (see, e.g., Frege 1903). However, at least since Russell (1897), some philosophers sought to divorce the concept of number from that of quantity. The view that came to dominate twentieth century philosophy was the thesis that numbers are ‘abstract entities’ and able to be conceptualized as elements of formal systems. Accordingly, they were thought not to be empirically located, as are quantities. As a result, measurement ceased to be defined in the classical manner, i.e., as the numerical estimation of ratios of magnitudes. Russell (1903) proposed that measurement be defined instead as the representation of magnitudes by numbers. This representational approach has dominated thinking about measurement this century (e.g., Campbell 1920, Nagel 1931, Ellis 1966), being developed most comprehensively by Suppes, Luce, Krantz, and Tversky (e.g., Krantz et al. 1971, Suppes et al. 1989, Luce et al. 1990) and their associates.

The application of this approach to the measurement of any attribute reduces to four steps. First, an empirical system is specified as an empirically identifiable set of some kind (e.g., of objects or attributes) together with a finite number of empirical relations between its elements. Second, a set of axioms is stated for this empirical system. As far as possible these should be empirically testable. Third, a numerical structure is identified such that a set of many-to-one mappings (homomorphisms) between the empirical system and this numerical structure can be proved from the axioms (the ‘representation theorem’). Fourth, inter-relations between the elements of this set of homomorphisms are specified, generally by identifying the class of mathematical functions which includes all transformations of any one element of this set into the other elements (the ‘uniqueness theorem’). This allows the distinctions between types of scales, popularized by Stevens (1951), to be precisely defined. This approach to measurement theory has produced, among other important developments, the theory of conjoint measurement.

The numerical representation of empirical systems depends on identifying structures within those systems which possess numerical correlates. Insofar as mathematics may be thought of as the study of the general forms of structures (empirical or otherwise), proofs of representation theorems, therefore, rely upon identifying mathematical structure within empirical systems. Insofar as quantitative theories in science require something like Holder’s concept of continuous quantity as the empirical system to be numerically represented, it follows that the mathematical structure of the positive real numbers might be thought of as already present (as ratios of magnitudes) in that empirical system. It is along these lines that a reconciliation between classical and representational theory may be possible.

2.2 The Philosophy of Science

Developments in the philosophy of science have also affected measurement theory. For psychology, the most significant factor was operationism (Bridgman 1927). Operationism was developed into a theory of measurement by Stevens (1951). If, as Bridgman held, the meaning of a concept is the set of operations used to specify it, then measurement is the ‘assignment of numerals to objects or events according to rules’ (Stevens 1951, p. 1) and the attribute measured via any such assignment is defined by the rules used. This approach proved useful in the social and behavioral sciences before it was known how to test the hypothesis that such attributes are quantitative.

Michell (1999) contains further material on the history and philosophy of measurement in psychology.

Bibliography:

1. Aristotle 1941 On generation and corruption. In: McKeon R (ed.) The Basic Works of Aristotle. Random House, New York
2. Bridgman P W 1927 The Logic of Modern Physics. Macmillan, New York
3. Campbell N R 1920 Physics, the Elements. Cambridge University Press, Cambridge, UK
4. Clagget M (ed.) 1968 Nicole Oresme and the Medie al Geometry of Qualities and Motions. University of Wisconsin Press, Madison, WI
5. Ellis B B 1966 Basic Concepts of Measurement. Cambridge University Press, Cambridge, UK
6. Frege G 1903 Grundgesetze der Arithmetik. Hildesheim, Germany, Georg Olms
7. Heath T L 1908 The Thirteen Books of Euclid’s Elements. Cambridge University Press, Cambridge, UK, Vol. 2
8. Helmholtz H von 1887 Zahlen and Messen erkenntnistheortisch betrachtet. Philosophische Aufsatze Eduard Zeller zu seinum funfzigjahrigen Doktorjubilaum gewindmet. Fues’ Verlag, Leipzig [1971 An Epistemological Analysis of Counting and Measurement. In: Kahl R (ed.) Selected Writings of Hermannvon Helmholtz, 1st edn. Wesleyan University Press, Middletown, CT]
9. Holder O 1901 Die Axiome der Quantitat und die Lehre vom Mass. Berichte uber die Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 53: 1–46 [1996 The axioms of quantity and the theory of measurement, Part I. Journal of Mathematical Psychology 40: 235–52; 1997, Part II. Journal of Mathematical Psychology 41: 345–56]
10. Hutcheson F 1725 An Inquiry into the Original of our Ideas of Beauty and Virtue. Darby, London
11. Krantz D H, Luce R D, Suppes P, Tversky A 1971 Foundations of Measurement. Academic Press, New York, Vol. 1
12. Luce R D, Krantz D H, Suppes P, Tversky A 1990 Foundations of Measurement. Academic Press, San Diego, CA, Vol. 3
13. Michell J 1999 Measurement in Psychology: A Critical History of a Methodological Concept. Cambridge University Press, New York
14. Nagel E 1931 Measurement. Erkenntnis 2: 313–33
15. Reid T 1748 An essay on quantity. [1849 In: Hamilton W (ed.) The Works of Thomas Reid. Maclachlan, Stewart & Co, Edinburgh, UK]
16. Russell B 1897 On the relations of number and quantity. Mind 6: 326–41
17. Russell B 1903 Principles of Mathematics. Cambridge University Press, New York
18. Stevens S S 1951 Mathematics, measurement and psychophysics. In: Stevens S S (ed.) Handbook of Experimental Psychology. Wiley, New York
19. Suppes P, Krantz D H, Luce R D, Tversky A 1989 Foundations of Measurement. Academic Press, New York, Vol. 2