Are There Objective Grounds |
Karel Berka
Czechoslovak Academy of Sciences, Prague
TOWARDS A PHILOSOPHY OF MEASUREMENT
History of science testifies to the growth of measurement procedures in various branches of science and technology. This tendency was expressed by the well-known dictum of Galileo: “To measure what is measurable and to try to render measurable what is not so as yet.” It was further strengthened by the development of natural science, especially physics, and during the second half of this century extended to behavioral and social sciences as well.
The common practice of utilizing measurement procedures as widely as possible, even beyond their justifiable range, is closely connected with extensions of mathematical concepts and methods in social practice, and is one important aspect of contemporary mathematization. One can, therefore, assume that measurement theory and philosophy of measurement, which until recently were elaborated only very partially, will undergo a development similar to mathematical metatheory and philosophy of mathematics. What I have in mind is the transition from Hilbert’s program of formalization to Tarski’s emphasis on semantics, and from semantics to Quine’s ontological commitment implied by the adopted language of mathematics.
Naturally this is only a very broad analogy. What is relevant for present purposes is the overt tendency for certain philosophical or metaphysical assumptions to express themselves in measurement theory, although often in positivist terminology. This tendency can be roughly characterized by the change from the position advocated in 1966 by B. Ellis (1966 p. 3) who based his “consistent positivist account of the nature of measurement” on the thesis “that certain metaphysical presuppositions, made by positivists and nonpositivists alike, have played havoc with our understanding of many of the basic concepts of measurement, and concealed the existence of certain more or less arbitrary conventions”—to the modification of his view, as presented in this volume.
In pointing to the increased interest in philosophical analyses of measurement, I cannot neglect mentioning the previous, rather methodologically oriented approaches: homocentric operationalism as represented by H. Dingle (1950), methodological operationalism as advocated by P. W. Bridgman (1959), instrumentalism, conventionalism, and the formalism closely connected with a model platonism (Radnitzky, 1973, p. 40). The realistic conception of Byerly and Lazara (1973), which stresses the importance of philosophical foundations in measurement theory for the elucidation of its basic concepts and procedures, represents the starting point for a new systematic philosophy of measurement.
The variety of partial or more complex approaches to philosophical foundations of measurement reflect the perennial struggle between materialism and idealism in history of philosophical thought, and could easily induce scientists to adopt a neutral standpoint toward philosophical problems of measurement, of course without intending to follow the positivists in this regard. It is clear that, when deciding whether this or that philosophical position is correct and appropriate for solving basic problems of some scientific theory, one cannot proceed with the precision and strength available in logical or mathematical proofs. What is feasible, however, is to adopt the method of Aristotle, who, in his first philosophy, supported the fundamental principles of cognition and knowledge by pointing out the absurd consequences that would follow, if they were violated.
MATERIALISTIC VERSUS IDEALISTIC FOUNDATIONS OF MEASUREMENT THEORY
In the present chapter, I shall defend an ontologically committed conception of measurement, in agreement with the materialistic solution of the fundamental philosophical question of whether matter is prior to mind. The fruitfulness of this materialistic position will be demonstrated by an analysis of various controversial topics in measurement theory.
The advocated standpoint has the following basic components:1
1. Measurement is ontologically committed (i.e., rooted in and, hence, grounded by objective reality).
2. Magnitudes are historically and theoretically determined reflections of quantitative aspects of objectively existing entities and not merely the outcome of metricization or measuring procedures.
3. The object of measurement exists prior to metricization or measuring procedures.
4. In agreement with the historical determination of every phenomenon, a transfer of methods from one universe of discourse into another one is adequate only on the objective condition that certain structural similarities hold between the domains in question.
In its accentuation of the ontological grounding of measurement, the materialistic standpoint clearly opposes various shades of positivism, logical positivism, and postpositivism. Presupposing a dialectical unity of qualitative and quantitative aspects of measured entities, it aims toward a unification of the empirical operations used in measuring procedures and the conceptual operations utilized in metricization. Such an approach may overcome the one-sidedness of both operationalism and formalism, which in fact represent the Charybdis and Scylla of contemporary philosophy of measurement.
The materialistic standpoint differs from the idealistic one in several respects. The most important respect, which will be unacceptable to many measurement theorists, is its ontological grounding. The view that measurement is ontologically grounded has direct consequences for the scope of measurement. Opposing thinkers will simply “reject the view that quantities have a kind of primary ontological status”, (Ellis, 1968, p. 38), because they are well aware that, by admitting such a view, the range of measurability cannot be conventionally extended. The ontological status of magnitudes determines objectively the possibility of measurement and at the same time narrows the concept of measurement into an objectively restricted frame.
This consequence of course vitiates the attempt of many proponents of extra-physical measurement to apply measurement in various fields of behavioral and social sciences, and even in humanities. Having in mind “the dignity implied by the term measurement” (Stevens, 1960, p. 45), they strive to achieve a very broad conception of measurement. This goal requires also a very vague definition of measurement, according to which measurement is defined as “the assignment of numerals according to rule,” or more explicitly as the “assignment of numerals to things so as to represent facts and conventions about them” (Stevens, 1960, pp. 145, 148).2
What follows from this viewpoint? We can, according to some rule, assign numerals to practically everything so as to create a nominal scale; thus, the range of measurement is not objectively limited. This boundless extension of measurement is supported by the fact that, for a nominal assignment of numerals, it is very easy to formulate some rule, for example, “Do not assign the same numeral to different classes or different numerals to the same class” (Stevens, 1960, p. 145). Such assignment fulfills the requirement of operationalism according to which measurement is concerned only with operations.
This conception together with its consequences contradicts the materialistic, foundational view of measurement. If we are to conclude that the object of measurement exists independently and prior to measuring procedures, we have to refute the view that measurement is just created by these procedures. The theoretical reproduction of quantitative aspects of objectively existing entities requires the assignment of numbers and not the assignment of numerals, which, in the criticized conception, are merely labels, names of numbers. The results so far attained by measurement procedures in social practice yield sufficient evidence that the broad conception is incorrect and misleading. The value of measurement consists in its ability to generate numerical data that help us to formulate numerical laws or to test empirical hypotheses.
It has to be further pointed out that operationalism ignores the fact that every empirical operation, if it has to convey theoretically relevant results, presupposes various hypotheses that depend on the objective nature of the measured entities. That a magnitude cannot be measured by a single operational procedure in the whole range of its occurrence is again objectively determined by the specificity of nature. This variability of measurement procedures is of course also dependent on the attained level of our theoretical knowledge. These circumstances, not the measuring operations or the nature of the measuring instruments, primarily determine the process of measurement. I need not mention that the construction of measuring instruments is based on numerical and empirical laws as well as on theoretical considerations, again a fact that operationalism does not take into account. Every change in the theoretical level modifies the operations used. The differentiation of operational procedures when measuring some magnitude under different conditions and in various ranges (e.g., when measuring length in very short and very far distances, on Earth or in the Universe) is objectively determined and has important methodological features that are closely connected not only with the theoretical framework of a given scientific branch but also with broader philosophical conceptions. This does not, however, imply fragmentation of magnitudes into various observationally reproducible instances. If it is possible, when measuring some magnitude, to obtain approximately identical numerical values by different operations, this outcome testifies to the fruitfulness of these operations and enables us to make the numerical results more precise but does not support the operationalist standpoint. The selection of operations is not primarily a matter of some pragmatic advantage but a result of theoretical insight and substantive adequacy rooted in the nature of the object of measurement.
A quite different standpoint, which, however, has very similar consequences, is defended by adherents of formalism. In their attempt to develop a theory of measurement, they concentrate their investigations on purely formal problems. On the formalist conception, the ultimate goal is the proof of two theorems, axiomatically deduced:3 a representation theorem and a uniqueness theorem. They assume that the conditions of metricization4—in the weaker case, only those of topologization—are sufficient to constitute special magnitudes of various kinds of nonextensive measurement.
Before analysing this formal standpoint, let me note that axiomatization in logic or mathematics serves the goal of deductive systematization of some theory previously studied only intuitively, together with the clarification of the deductive and definitional structure of the theory in question. An axiomatic system in this case contains, besides axioms, primitive terms, inference rules and definitions, and a very large subset of theorems.
Such axiomatic systems can be found in works dealing with the systematization of extensive magnitudes, where various axiomatizations are isomorphic with axiomatic systems elaborated in arithmetics. As an example, consider the axiomatic systems of O. Hölder (1901) or E. V. Huntington (1902), which are explicitly related to mathematics.
Axiomatizations in the field of nonextensive magnitudes or measurements in behavioral and social sciences as developed by adherents of the formal approach are different. Axiomatic systems of intensive magnitudes (e.g., expected utility) or of various kinds of measurement (e.g., difference measurement, bisection measurement, or conjoint measurement) are not intended to reveal the structural features of some theory. They have been explicitly elaborated for one reason: to deduce the representation theorem and the uniqueness theorem.
Both theorems describe the metricization conditions for some nonextensive magnitude and a corresponding measurement scale whose numerical values are expected to be obtained. The real reason why axiomatization is so widely utilized in measurement theory of intensive magnitudes flows from a desire to justify the concept of measurement by formal means in a very broad sense and to overcome the well-known difficulties of behavioral and social measurement.
Against this specific, axiomatic approach, several objections can be raised. Many of these have been made by M. Allais (1953) in connection with axiomatized expected utility theory.
It is not difficult to show that the axioms are provided neither with a sufficient syntactical and semantical justification nor with a convincing empirical interpretation that would serve as a basis for actual measurements. One should recall the discussions concerning the relationship of structural and behavioral axioms, and the attempts to prove the empirical nature of the strong independence axiom of P. A. Samuelson (1952, p. 672) and the sure-thing principle of L. J. Savage (1954, pp. 21ff).
Another objection is based on the argument of conceptual economy. It seems rather “expensive” to use a very complicated, formal procedure with the humble result of deducing only two theorems. However, what is more, the whole, deductive chain leading to them is burdened with serious doubts whether the deduction is correct from the logical or methodological point of view. The empirical relational systems are assumed to be logically prior to the numerical ones; therefore, the axioms cannot refer to numerical expressions. Nonetheless the theorems deduced from these nonnumerical axioms deal with numerical expressions. Both horns of the dilemma resulting from this situation are apparent: Either the representation theorem is not solely deduced from the axioms, or the axioms contain, at least implicitly, numerical connotations. If the derivation of the basic theorems is not exclusively determined by the axioms, then it is incorrect from the logical point of view. If some numerical assumptions are implicitly contained in the axioms, then the deduction is invalid for methodological reasons.
It is further questionable whether the deduction of both theorems, if one concedes their validity from the logical and methodological points of view, can be considered as a sufficient grounding of the measurability of some nonextensive magnitude. The theoretical conditions under which a magnitude can be measured do not yet guarantee its measurability by empirical procedures. The construction of a mathematical model must be tested in practice. Such a test requires a significant empirical interpretation of its concepts and a feasible operationalization of the metricization conditions.
Another critical remark amounts to the following argument. In order to check the theoretical appropriateness and the practical relevance of both basic theorems, one does not need to construct an axiomatic system at all. Instead of this approach (which is difficult, due to lack of justification for the choice of the primitive notions and axioms), it suffices to substitute the axiomatic justification of the representation theorem by an immediate formulation of correspondence rules holding between the empirical and numerical characteristics of both relational systems, whereas the uniqueness theorem can be supplanted by the property of invariance of a scale with regard to its admissible transformations. This direct procedure is very simple and plausible. An empirical interpretation and operational verification of the conditions required to obtain numerical results, by measuring some intensive magnitude, will immediately show whether or in what sense it is correct to assume that these conditions obtain.
Neither operationalism nor the formal approach recognize that measurement is ontologically committed and that this objective determination constitutes the foundation of its conceptual and theoretical framework. Seen from this point of view, it is obvious that the assignment of numbers to empirical entities—in agreement with the conditions of metricization—is appropriate, only if (and because) these entities have some quantitative aspects.
The thesis that ontological assumptions are necessary preconditions for measurability neither implies nor requires an unmediated, direct relationship. We can significantly assign numbers to empirical entities also in a mediated way (as, e.g., in the case of associative measurement). However, a purely qualitative aspect of empirical entities that cannot be related somehow to a quantitative aspect cannot be measured in the strict sense of the term. Without such a relation, the assignment of numbers would not be adequate; it would neither reflect the degree nor the size of the quantitative aspects in question.
The core of the materialistic position is that objective reasons make it impossible to change the ontological status of a property by metricization or measuring operations. The assignment of numbers cannot be understood in the sense of reductionism, for one cannot reduce qualities to quantities by measurement procedures. Neither conventions nor postulates can, thus, replace the constraint of what is objectively measurable and what is not.
The thesis of the ontological dependency of measurement procedures of course cannot be absolutized, because measurement is also historically determined by the evolution of social practice, by the growth of knowledge. The dialectics of the process of cognition include the active role of man, the socially conditioned reflection of his activities, in the given case those of various measurement procedures and measurement conceptions. It would be a mistake, or rather a concession to mechanistic or vulgar materialism, to deny the influence of man’s activities when constructing and using diverse measurement procedure or the relative impact of measuring instruments and techniques on measuring operations and their results. From the standpoint of dialectical and historical materialism, the thesis of the priority of the ontological status of magnitudes yields a theoretically adequate and methodologically fruitful conception of measurement only when it is connected with the thesis of the historically determined nature of this method.
SOME SPECIAL PROBLEMS OF MEASUREMENT
I will now consider the implications of the materialistic conception of measurement for various miscellaneous topics in measurement theory.
The distinction between fundamental and derived measurement, and between fundamental and derived magnitudes, is acknowledged by all measurement theorists. However, its elucidation is—in regard to the ontological and antiontological standpoints in the philosophical foundations of measurement—a matter of dispute. The dispute arises from the view of operationalism that magnitudes depend on measuring operations described in a purely phenomenological manner. According to operationalism, if one succeeds in constructing some measuring instrument by means of which a derived magnitude can be directly measured, then the derived magnitudes changes into a fundamental one. (E.g., the density of a liquid is considered as a derived magnitude, if its numerical values are obtained by calculation from the numerical values of volume and mass, but, if measured directly by some instrument—for example, a hydrometer—it will be considered a fundamental magnitude.) Thus, every change in instrumental techniques modifies the division between basic and nonbasic magnitudes. However, these modifications are determined only operationally, not objectively.
A similar problem arises with regard to temperature measurement on Stevens’ theory of scales. If we measure temperature by using a scale with an arbitrary zero, as is the case of Reaumur or Celsius scales, it will be considered, by antiontological measurement theorists, as a magnitude measurable by means of an interval scale. However, when utilizing the Kelvin scale, which employs an absolute zero, temperature has to be classified as a magnitude measurable by means of a ratio scale. The difference between these types of scales is often related to the difference between nonextensive and extensive magnitudes, or between nonadditive and additive ones. In consequence, Stevens’ view implies that a magnitude can be both extensive and nonextensive.
The nonontological conceptions have unpleasant consequences, and they lead to a serious discrepancy between the theoretical framework of physics and dimensional analysis. By adopting the materialistic standpoint, these difficulties can easily be avoided. The decision, whether a magnitude belongs in the class of fundamental or derived magnitudes, has its objective grounds. The class of magnitudes that are fundamentally measured in the known system of physics and, hence, conceived as fundamental is determined by the structure and evolution of objective reality and its theoretical reflection by man. The distinction between fundamental and derived magnitudes, although it can per analogiam be explicated by the function of primitive and defined concepts of some axiomatic system, is neither conventional nor arbitrary. From the materialistic point of view, length is a fundamental magnitude for reasons stemming from the nature of objective reality, whereas density will remain a derived magnitude, even if it can be measured fundamentally.
Operational criteria violate fundamental measurement. Every magnitude (e.g., length), which, for ontological reasons, is the most basic fundamental magnitude, presupposes the possibility of assigning numerical values from the domain of real numbers. However, only rational numbers are attainable by actual measurement procedures. The set of empirically measured values of a magnitude is practically finite but theoretically infinite. Numerical laws of physics require values from the set of real numbers, which is infinite; therefore, these laws cannot rest on operationalist methods. A very simple example supports this criticism: From the operationalistic point of view, the length of the hypotenuse of a square whose side is equal to 1 meter cannot be determined. The use of operational procedures will have fruitful results only if there are objective conditions that are connected on one side with the nature of the measured entities independently (whether we have or have not constructed for them some measuring operation) and, on the other side, with calculations based on theoretical considerations. Without an objectively determined and theory-laden interpretation of the results obtained by measurement procedures, no scale value is actually significant.
Another problem of extraphysical measurement, whether based on Stevens’ theory of scales, Hempel’s conception of metricization, or the axiomatic deduction of representation and the uniqueness theorems, arises in connection with the ontological dependency or independency of concepts used when describing the nature of the empirical relational system. It is assumed that this empirical system is the starting point of the construction of scales of measurement, metricization or axiomatic deductions. In fact, however, the empirical nature of its basic concepts and operations is only pretended. This becomes obvious when we take into account the concept of equality as used in Stevens’ conception. One cannot maintain that “nominal scales … are merely scales for the measurement of identity and difference” (Ellis, 1968, p. 42), because identity is without any doubt just a purely logical concept. Similarly such operations as “determination of equality of ratios” (Stevens, 1960, p. 143); “forming the center of gravity” (Neumann & Morgenstern, 1953, p. 21) (or, to use verbal equivalents, “combination of alternatives,” “combined options,” “mixed prospects”), which have to be performed on elements of the empirical relational system, cannot be understood in pleno sensu as empirical operations. These seemingly empirical concepts and operations are in reality quasiempirical counterparts of numerical concepts and operations.
Another characteristic example of this problem is the search for some adequate empirical counterpart of the numerical operation of addition implied by the requirements of extensive measurement. There are great obstacles to finding some appropriate equivalent in the outer world to the concatenation operation, even in the measurement of length; thus, various suggestions have been made to replace or narrow these requirements. In J. Pfanzagl (1959, p. 284), additivity is replaced by a metric connection concerning the distance holding between scale numbers, which makes a regular division of scale intervals possible. However, this conception lacks an objectively justified background in those cases where there does not exist a really empirical equivalent to the numerical addition. Even if one acknowledges that the metric connection is theoretically and methodologically acceptable, it is weaker than the usual additivity condition of extensive measurement. Even the additivity condition is dropped in the case of non-extensive measurement, and empirical interpretation is required only for fundamentally measured magnitudes. Other magnitudes that are measured by derived or associative measurement procedures can be measured in the strict sense of this term (i.e., as instances of extensive measurement), if they are connected by theoretical, numerical, or empirical laws with fundamentally measured magnitudes. (This conception, which is illustrated by the measurement of temperature, has to be accepted.)
Setting aside the just-mentioned difficulties of extraphysical measurement, the advantage of the reversed order between the empirical and numerical relational systems is quite clear. It is well known under what conditions the assignment of numbers can be considered with full right as an instance of measurement; therefore, the proponents of the very broad measurement conception are striving to find for the relevant numerical concepts and operations some counterparts with a more or less plausible empirical interpretation. An overt admission of this intention in the case of utility, the measurement of which has become the standard model of extraphysical measurement, appears in the initial specification of expected utility theory by J. Neumann and O. Morgenstern (1953) in the following words: “We have practically defined numerical utility as being the thing for which the calculus of mathematical expectation is legitimate” (p. 28).
This approach is in principle admissible, if the following conditions are fulfilled. First, when introducing an empirical interpretation for numerical concepts and operations, we have to be well aware of the difference between the idealized mathematical concept and its fuzzy counterparts in reality, and draw from this fact the necessary conclusions (e.g., in respect to the empirical unrealizability of the transitivity property). Second, if measurement has to fulfill its practical goals, the empirical interpretation has to be further complemented by operational realizations. Finally, if the interpretation is to be a truly empirical one, we have to check its ontological footing. This last step is often avoided for various reasons, chiefly because it is intuitively known but not taken into account that, in the given instance, no feasible empirical equivalent with an operational realization can be correlated with the relevant numerical concept or operation. For example, there is usually no feasible empirical operation of addition in such cases.
An “experimentum crucis” to defend my standpoint against the proponents of extraphysical measurement can be devised for the zero point (scale zero) or the origin of a scale, and the existence of the measurement unit.
When dealing with different kinds of temperature scales, an arbitrary or conventional scale zero of the Celsius, Reaumur, or Fahrenheit scales is confronted with the natural or absolute origin of the Kelvin scale of temperature. Without going into details,5 I will only point out that the existence of an arbitrary origin of a scale cannot be considered as a confirmation of conventionalism in measurement theory. I do not deny that the scale zeros of temperature scales that admit “positive” and “negative” scales values are more conventional than the absolute zero of the Kelvin scale. However, in spite of this fact, and the term “arbitrary,” these scale zeros are not arbitrarily stipulated but ontologically determined by dependency on objective laws of nature and are simultaneously influenced by social practice as well. The boiling and freezing points of water under constant conditions in respect to other factors (as, e.g., pressure of air) that determine the calibration of thermometers in general use are due to the construction of such instruments, especially with regard to the thermometric substance represented by different scale numbers. This “convention” is evidently less arbitrary than it seems at first glance.
When analysing the question of measurement units, the conventions applied in selecting measurement units are also arbitrary only in a very limited sense. Their choice is historically determined and for this reason shows a considerable variability, but not arbitrariness, even when we take into account the variety of units that man has used in different epochs and countries for the measurement of length, mass, or volume. The choice of various units for the measurement of one and the same magnitude is never made without sufficient reason, at least from the standpoint of its originator. If the choice of units were really conventional, there would be no difficulty in introducing units for magnitudes assumed to be measurable in behavioral or social science. The lack of appropriate measurement units in this domain testifies convincingly that measurement units cannot be introduced arbitrarily. The existence of measurement units in physical measurement and the absence of them in most instances of extraphysical measurement must be considered the decisive argument against view that measurement units are conventional. This view has been adopted by adherents of the antiontological standpoint as rationalization for an unjustified extension of measurability. These facts concerning units are a clear confirmation of the objective nature of measurement procedures and of their ontological determination.
A CASE STUDY: EXPECTED UTILITY
The measurability of expected utility is commonly regarded as a model case of extraphysical measurement of intensive magnitudes. This justification is based, so to speak, on a relative proof, that is, on an analogy with temperature measurement.
Assuming the measurability of temperature by means of an interval scale, one attempts to show that the measurement of expected utility has the characteristic features of interval scales; namely (a) a constant unit of measurement, (b) an arbitrary scale zero, and (c) invariance of scale form under any linear transformation.
This analogy, as I shall now briefly indicate, does not serve its purpose, because neither of the three conditions is satisfied. The proponents of the view that utility is measurable at least on an interval scale have not succeeded in producing a unit of measurement nor even an acceptable pseudounit. Some authors have introduced, as a unit, the utile, however, without sufficient reasons for this choice and without an explication or empirical reproducibilty. The proposal of S. S. Stevens (1959, p. 55) to define one utile as the utility of one U.S. dollar is absurd. To refute this view, it suffices to point out that this proposal contradicts Bernoulli’s conception of the “moral” value of money as opposed to its “numerical” value. Other authors6 try to find a way out by introducing arbitrary numbers without labeling them. If one employs this approach, one cannot avoid considering one of these arbitrarily chosen numbers as the unit of measurement and some other one as the origin of the scale. If there are no grounded reasons for these choices, why not take as a unit the number 3 or 121? The absurdity of this possible consequence is obvious. Neither will it help to follow the suggestion of E. W. Adams (1960, p. 183) to take, as a unit, the difference between 0 assigned to the least desirable alternative and 1 assigned to the most desirable one.
The arbitrariness of the zero of the utility scale cannot be understood in the same sense as the zero of temperature scales. For the origin of the utility scale— by contrast with that of the temperature scales—can be practically chosen without restriction, that of satisfying the ordinal property of expected utility, namely u(x) < u(y). This restriction in fact allows an unlimited choice. It is curious that the proponents of the analogy of temperature and utility scales do not see that it already fails, simply because the scale zero of temperature measured on an interval scale does not represent the smallest value. By contrast, the zero of the utility scale is usually selected in a manner analogous to that for the zero of the Kelvin scale. Can we, therefore, conclude that utility is a magnitude measurable on a ratio scale? The standard numerical representation of the utility scale in its canonical form—for example, limiting the utility interval by the numerical interval [0,1], where the number 0 functions as an absolute zero—is in fact determined by the requirement of isomorphism between the quasinumerical utility interval and the numerical probability interval, or rather by their illegitimate identification. Both intervals differ of course in essential ways: the elements of the utility interval are formed by desirabilities of alternative events (prospects, options, etc.), whereas those of the probability interval are determined by probabilities of the occurrence of the alternative events in question.
The first two conditions are not satisfied; thus, eo ipso neither is the third one fulfilled. The linearity of utility and the linear transformation of interval scales differ. For utility scales, there is no significant interpretation of the transformation formula.
y = ax + b a > 0
The transformation of two interval scales (e.g., represented by the Celsius and Fahrenheit scales) requires that the constants a and b have a semantically justified and operationally realizable sense: a indirectly designates the unit of measurement and b the scale zero. However, this requirement does not hold for utility scales.
The thesis that measurement has its objective grounds that cannot be neglected or arbitrarily surpassed supports a restricted concept of measurement. It will, therefore, disappoint all those who desire to extend measurement procedures to various branches of behavioral and social sciences in an extreme manner. This desire cannot be satisfied at any cost. There are objective limits for measurement procedures, as for other forms of mathematization. The utilization of measurement in physics cannot be taken as the paradigm for all sciences. It should not lead us to an uncritical standpoint that would ignore the ontological limitations and historical determination of measurement. If, in behavioral and social sciences, qualitative procedures predominate over quantitative ones, no derogatory evaluation of these disciplines is thereby implied. People are not rigid bodies. Hence, the analysis of their behavior cannot be made by procedures utilized in mechanics. Mathematical methods, including measurement, outside natural sciences will lead to fruitful outcomes, only if they are applied in accord with the regularities and evolution of objective reality and by means that are theoretically justified, empirically feasible, and developed, if not on materialistic foundations, at least on a realistic basis.
REFERENCES
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1 See (Berka, 1983, p. 206).
2These definitions have their origin in Campbell’s (1957, p. 267) following definition: “Measurement is the assignment of numerals to represent properties;” but Campbell used the term “numeral” in a different sense.
3See (Krantz, Luce, Suppes, & Tversky, 1971).
4See (Hempel, 1952).
5But see (Berka, 1983, pp. 87ff).
6E.g., (Luce & Raiffa, 1957, p. 33)