MEASUREMENT
BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE EDITED BY ROBERT S. COHEN AND MARX W. WARTOFSKY VOLUME 72
KAREL BERKA Institute for Philosophy and Sociology, Czechoslovak Academy of Sciences MEASUREMENT Its Concepts, Theories and Problems Translated from the Czech by Augustin Riska D. REIDEL PUBLISHING COMPANY DORDRECHT: HOLLAND / BOSTON: U.S.A. LONDON: ENGLAND
Library of Congress Cataloging in Publication Data Berka, Karel. Measurement: its concepts, theories, and problems. (Boston studies in the philosophy of science; v. 72) Translation of: Mereni. Bibliography: p. Includes index. I. Physical measurements. I. Title. II. Series. Q174.B67 vol. 72 [QC39] SOls [530.1'6] 82-20428 ISBN-13: 978-94-009-7830-0 e-isbn-13: 978-94-009-7828-7 DOl: 10.1007/978-94-009-7828-7 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland Sold and distributed in the U.S.A. and Canada by Kluwer Boston Inc., 190 Old Derby Street, Hingham, MA 02043, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group P.O. Box 322, 3300 AH Dordrecht, Holland D. Reidel Publishing Company is a member of the Kluwer Group Original title Mereni: pojmy/teorie/probll?my All Rights Reserved Copyright 1983 by D. Reidel Publishing Company, Dordrecht, Holland Sotkover reprint of the hardcover I 5t edition 1983 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
EDITORIAL PREFACE For many years, Karel Berka has worked at some of the central problems of the theory of the sciences. At once a logician, a mathematician, a careful student of the physical sciences and the social sciences, and a sharp but sympathetic critic of the major philosophies of science in this century, Berka brings to this treatise on measurement both his technical mastery and his historical sensitivity. We appreciate his careful analysis of his predecessors, notably Helmholtz, Campbell, Holder, Bridgman, Camap, Hempel, and Stevens, and of his contemporaries such as Brian Ellis and also Patrick Suppes and J. L. Zinnes. The issues to be clarified are familiar but still troubling: how to justify the conceptual transition from classification to a metric; how to explore ways to provide a quantitative understanding of a qualitative concept; indeed how to understand, and thereby control, the Galilean enthusiasm "to measure what is measurable and to try to render measurable what is not so as yet". Berka explores the vexed puzzle that arises when we explicate the so-called mapping of empirical relational systems onto numerical relational systems, empirical structures onto numerical structures: to what extent may we say that operations among the numbers tell us something about empirical operations, or other empirical relations? In fact, we see an epistemological and a historical discussion opening before us, as Berka proceeds to set forth the 'conceptual-mathematical' project of metrization in its several contexts of the empirical, operational, and theoretical practices of the methods and procedures of measurement in psychology, in physics and in all the intermediate scientific domains. "What then is the object of measurement?" With this question, Berka initiates his searching examination of quantities, magnitudes, and numbers, and illuminates the conditions for extending about physical magnitudes to a generalized theory of magnitude, which in turn may.allow us to formulate a clear and reliable theory of extra-physical magnitudes (and, in the process, to distinguish numbers from magnitudes, and magnitudes from quantities, and to sort out the epistemic nuances of scales, so easily construed as results of measurement but also as tools for measuring). Berka's critique of the additivity requirement, and of the proposal for an empirical operation of v
vi EDITORIAL PREF ACE 'concatenation' (whether in physics or economics), seems to us the philosophical nucleus of his materialist theory of measurement, linked as closely as it is with the debate over extensive and intensive magnitudes in the measuring processes. But the reader will find discussions of philosophical importance throughout Berka's book, as in the deceptively clear account of imprecision in measurement, the extended methodological and epistemological analysis of operationism, the interesting, almost playful examination of the grave problem of meaningfulness and reliability in the use of measurements, the sharp judgment of any 'overestimation' of a conventionalist interpretation of 'measurements units'; and indeed in Berka's argued dismissal of any 'real' ontological basis for measurement which relies on a merely contemplative (which is to say, passive) materialist epistemology. We may say that Berka brings us to the point of wishing another book, a fully developed theory of knowledge. For he bluntly concludes that measurement will be comprehended only within an analysis of its limitations as well as of its conditions and possibilities, and for him this must go beyond the phenomenalism, as he sees it, of Bridgman and even of the Carnap who is cited by Berka: ".. it is we who assign numbers to nature" (p. 215). Indeed, Berka concludes: "some properties of real objects are not measurable in principle". And then, with that in mind, how will the social sciences fare with respect to measurements and scales? We look forward to further reflections from Karel Berka's skilled and profound philosophical labors. * * * We are most grateful to Professor Augustin Riska for his careful and graceful translation, and also for the very thorough index of subjects. Our thanks too go to Susan Riska for the index of names, and to Carolyn Fawcett for her skillful editorial assistance throughout. Center for Philosophy & History of Science Boston University October 1982 ROBERT S. COHEN MARX W. WARTOFSKY
T ABLE OF CONTENTS EDITORIAL PREF ACE TRANSLATOR'S REMARKS PREFACE v ix xi 1. INTRODUCTION 2. MEASUREMENT 14 2.1. The Explication of the Concept of Measurement 14 2.2. The Definition of the Concept of Measurement 20 2.3. The Subject Matter, Function and Scope of Measurement 28 3. MAGNITUDES 35 3.1. Quantities, Magnitudes, Numbers: A Historical Excursion 37 3.2. Quantities and Magnitudes 42 3.3. The Object of Measurement 46 3.4. Measurement Units, Naming and Dimension 55 3.5. The Classification of Magnitudes 73 4. SCALES 83 4.1. The Concept of a Scale 83 4.2. The Origin of a Scale 87 4.3. Distance 91 5. QUANTIFICATION 101 5.1. Scaling 101 5.2. Counting 105 6. THEORY OF MEASUREMENT 112 6.1. Representation Theories of Measurement 113 6.2. Kinds of Measurement 115 6.3. Metrization 133 6.4. The Representation Theorem 150 vii
viii TABLE OF CONTENTS 7. THEORY OF SCALES 7.1. The Classification of Scale Types 7.2. Scale Transformations and the Uniqueness Theorem 8. METHODOLOGICAL PROBLEMS OF MEASUREMENT 8.1. Axiomatization of the Systems of Measurement 8.2. Empirical Relations and Operations 8.3. The Precision of Measurement 8.4. Meaningfulness, Validity and Reliability 9. PHILOSOPHICAL PROBLEMS OF MEASUREMENT 9.1. Materialist Foundations of Measurement 9.2. The Possibilities and Limits of Measurement NOTES BIBLIOG R APHY INDEX OF PERSONAL NAMES INDEX OF SUBJECTS 158 158 168 178 178 184 192 199 205 205 214 218 226 234 237
TRANSLATOR'S REMARKS (1) The Czech term velicina, occurring so frequently in the present work, has been consistently translated as magnitude, although in the contemporary English literature on measurement (for example, in Krantz et al. (1971) or Ellis (1966» the term quantity is preferred. Yet the author needs the Czech term kvantita (translated as quantity) to be reserved for special philosophical purposes, as he explains in Section 3.1. Hence the use of the terms quantity and quantities instead of magnitude would conflict with the conceptual distinctions drawn by the author. At any rate, our use of the word magnitude is in accordance with the older usage, for instance with the terminology of Campbell (1920/1957), and also with the language adopted by Carnap (1966). (2) Since velicina is being translated as magnitude, the Czech term velikost is translated as size (the size of a magnitude). This somewhat cumbersome term must be introduced in order to avoid formulations such as 'the magnitude of a magnitude'. Of course, if the term quantity could replace our term magnitude, one would get a more satisfactory rendering of the above phrase, namely 'the magnitude of a quantity'. -(3) The author distinguishes the terms skala and stupnice (see Section 4.1); as he expresses it, the term skala signifies conceptual scales, while the term stupnice signifies material scales. Hence, whereas the term skala is translated simply as scale, the other term, stupnice, is translated consistently as gauge. (4) One has to be reminded of the important distinction between numeral and numerical; the adjective numeral pertains to the symbolic representation of numbers, whereas the adjective numerical refers to numbers themselves and their properties (the Czech equivalents are: cislice, cislicovy, for numeral; and cislo, ciselny, numericky, for number and numerical). The terms numerousness and numerosity (following Stevens) are used in an alternative way, rendering the meaning of the Czech term pocetnost (in the sense of a number of..., an amount). Of course, cardinality can be used in this sense as well. (5) The term gnosiological (occurring in the phrase 'ontologico-gnosiological') is not very common in the English literature, yet it is not replaced by the standard term epistemological, for it would then lose its special philosophical flavor. One may also take note of the frequently used phrase 'things, phenomena and processes of the objective reality'. ix AUGUSTIN RISKA
PREFACE The historical process of the development and utilization of quantitative methods in science, particularly in connection with the extension of measuring procedures in various domains of the social sciences, has conditioned a great upsurge of methodological investigations dealirig with behavioral and social measurement. Starting with the contributions of S. S. Stevens toward the theory of measurement scales, problems of measurement have become the central theme of the methodology of empirical sciences, not only for the scientistic philosophy of science, but also for many psychologists, sociologists and economists. However, in a very extensive literature, which is focused on the practice and theory of measurement and scaling, two one-sided methodological positions are strongly exhibited: on the one hand, the position influenced by instrumentalism, operationalism and neopositivism; on the other hand, the viewpoint based on the fonnalistic philosophy of mathematics. From a purely empirical point of view, measurement is reduced only to the use of different scaling and measuring techniques. A purely mathematical doctrine of measurement, which prevails today, is reduced to the construction of various scales of measurement, defmed merely by purely formally invariant transfonnations under which their fonn is unchanged, or, alternatively, to the derivation of the representation and uniqueness theorems from axiomatically defined relational structures. A common denominator of both these antagonistic doctrines is a very broad explication of the concept of measurement, encompassing a mere numbering, as well as an uncritical application of measuring procedures to the widest possible extent. Such explication disregards the general, objectively existing conditions of measurability, as well as the specific nature of the social sciences with respect to the natural sciences, especially physics. In the present publication, which is the result of extensive work connected with the theory of measurement, we strive to analyze the problems of measurement scales on the basis of the methodological principles of dialectical and historical materialism. In this we start from the evident fact that measuring procedures have developed in the process of a practical appropriation of the world by man. From its very origin, measurement and its quantitative results have constituted a complex of mutually conditioned empirical and xi
xii PREFACE mathematical aspects, reflecting the dialectical unity of qualitative and quantitative characteristics of objectively existing objects, phenomena, and processes. The conceptual reflection and theoretical foundation of these mutual connections must fully respect not only the ontologico-gnosiological ground of measurement, but also the concrete conditions under which this empirico-mathematical method may be applied in various disciplines. The boundaries between what can and cannot be measured, are therefore, in our view, marked out contingently upon the historical level of our knowledge of the laws of the objective reality. Hence, these boundaries cannot be changed in an arbitrary manner. Certainly, we encounter many problems in various doctrines of behavioral and social measurement as well as in the theories of scales which are tied up with these doctrines. Nevertheless, these problems definitely cannot be short-circuited by a conventional choice of scaling techniques that are not theoretically justified, or by a construction of scales which cannot be empirically realized in a satisfactory way, or, fmally, by formal means, however refmed, with the help of which one cannot in fact solve any problem of content in a meritorious manner. An unfounded utilization of mathematical methods has a harmful rather than a beneficial effect; in other words, such utilization diverts us from the elucidation of problems which deserve priority from the theoretical point of view, and puts an additional burden on the solution of questions which are a necessary precondition of appropriate quantification. It is exactly this standpoint from which one is to understand our discussion of the concepts of the origin of a scale, distance, scale types and transformation criteria. Yet, should the reader of our exposition get the impression that by our critical objections agamst the 'wider' doctrine of measurement we wished in principle to cast doubt on the possibility of social measurement, such impression would amount to sheer misunderstanding. KAREL BERKA