Theories of Meaningfulness

Description

Written by one of the masters of the foundation of measurement, Louis Narens' new book thoroughly examines the basis for the measurement-theoretic concept of meaningfulness and presents a new theory about the role of numbers and invariance in science. The book associates with each portion of mathematical science a subject matter that the portion of science is intended to investigate or describe. It considers those quantitative or empirical assertions and relationships that belong to the subject matter to be meaningful (for that portion of science) and those that do not belong to be meaningless. The first two chapters of the Theories of Meaningfulness introduce meaningfulness concepts, their place in the history of science, and some of their traditional applications. The idea that meaningfulness will have different, but interrelated uses is then introduced. To provide formal descriptions of these, the author employs a powerful framework that incorporates pure mathematics, provides for qualitative objects and relations, and addresses the relationships between qualitative objects and pure mathematics. The framework is then applied to produce axiomatic theories of meaningfulness, including generalizations and a new foundation for the famous Erlanger Program of mathematics. The meaningfulness concept is further specialized with the introduction of intrinsicness, which deals with meaningful concepts and relations that are lawful and qualitativeness, which is concerned with qualitative concepts. The concept of empiricalness is then introduced to distinguish it from meaningfulness and qualitativeness. The failure to distinguish empiricalness from meaningfulness and qualitativeness has produced much confusion in the foundations of science literature and has generated many pseudo-controversies. This book suggests that many of these disappear when empiricalness is intersected with the other concepts to produce "meaningful and empirical relations," "empirical laws," and "qualitative and empirical concepts." A primary goal of this book is to show that the new theories of meaningfulness and intrinsicness developed in this book are not only descriptive but are also potent. Asserting that they do more than codify already existing concepts the book: *works out logical relationships between meaningfulness concepts that were previously unrecognized; *clarifies certain well-known and important debates by providing rich languages with new concepts and technical results (theorems) that yield insights into the debated issues and positions taken on them; and *provides new techniques and results in substantive scientific areas of inquiry. This book is about the role of mathematics in science. It will be useful to those concerned with the foundations of science in their respective fields. Various substantive examples from the behavioral sciences are presented. Foreword by R. Duncan Luce/xiii Acknowledgments/xvii 1 Introduction and Historical Background/1 1.1 Introduction/1 1.1.1 Plan and Objectives of This Book/3 1.2 Philosophies of Mathematics/5 1.3 Pythagorism/9 1.4 Invariance in Geometry/22 1.5 Dimensional Analysis/30 1.6 Eddington's “Method of Pure Numbers"/35 1.7 Ideal Numbers/38 1.8 Actual Infinity and the Axiom of Choice/41 2 Intuitive Theories of Meaningfulness/45 2.1 Overview/45 2.2 Stevens' Theory of Scales and Meaningful Statistics/46 2.2.1 Stevens' Theory/46 2.2.2 Comments/50 2.3 A Formal Theory of Scales/50 2.4 Intuitive Applications of Stevens' Meaningfulness Con/55 2.4.1 An Elementary Application/55 2.4.2 An Example Concerning Perceived Risk/56 2.5 Luce's Possible Psyciiophysical Laws/60 2.5.1 Luec's 1959 Paper/60 2.5.2 Rozeboom's Criticism/62 2.5.3 Luce's Reply/63 2.6.4 Some Observations/63 2.5.5 Generalizations to Several Independent Variables/66 2.6 Fulmagne's and Narens' Meaningful Quantitative Laws/68 2.6.1 Comments/81 2.7 Roberts' and Rosenbaum's Possible Psychophysical Laws/81 2.8 Applications of the Lawfulness/Meaningfulness Concept/86 2.8.1 Magnitude Estimation/86 2.8.2 Meaningful Averaging of Rating Data/89 2.8.3 A Psychophysical Application/93 2.9 Conclusions/97 3 Axiomatic Set Theory/99 3.1 Introduction/99 3.2 A Language for the Theory of Sets/99 3.3 The Axiom System ZFA/102 3.4 Consequences of the Axiom System ZFA/107 3.4.1 Elementary Algebra of Sets/107 3.4.2 Relations, Functions, and Cartesian Products/107 3.4.3 Ordering Relations/108 3.4.4 Cardinal Numbers/109 3.4.5 Ordinal Numbers/110 3.5 The Rank Function/112 3.6 Permutations of Atoms/116 3.7 Pure Set Theory and Axiom System ZF/118 4 Axiomatic Generalizations of the Erlanger Program/123 4.1 A Formal Approach to Meaningfulness/123 4.1.1 Introduction/123 4.1.2 The Language L(∈, A, ∅, M)/123 4.1.3 The Axiom System ZFA/125 4.1.4 Pure versus Applied Mathematics/127 4.1.5 The Intended Use and Scope of the Meaningfulness Concept/127 4.2 The Erlanger Program: Transformational Meaningfulness/130 4.3 Definitional Generalizations of the Erlanger Program/142 4.3.1 Axiom System D'/142 4.3.2 Axiom System D/144 4.3.3 Axiom System D*/144 4.3.4 Axiom System \bar{D}/145 4.3.5 Axiom System D"(a)/146 4.3.6 Interrelationships Among the Definitional Meaningfulness Concepts/152 4.4 Transformational Generalizations of the Erlanger Program/154 4.5 Consequences of D', D*, and TM/157 4.5.1 Some Basic Concepts/157 4.5.2 Homogeneity/158 4.5.3 Meaningful Cardinals/159 4.5.4 Meaningful "Set Theory"/161 4.6 Formulations in Second-Order Languages/166 4.6.1 Introduction/166 4.6.2 The Language L²/167 4.6.3 Axiom System D²/168 4.6.4 What Has Been Accomplished/170 4.7 Additional Approaches to Invariance and Definability/170 4.7.1 Weakening Axiom System ZFA/170 4.7.2 Changing Axiom System ZFA/171 4.7.3 Weakening Axioms MC' and MC/171 4.7.4 Weakening Axiom MP/171 4.7.5 Weakening Axiom TM/172 4.7.6 Using Infinitary Languages and Logics/172 4.8 Conclusions/173 4.9 Summary of Axioms and Axiom Systems/175 4.9.1 Axioms/175 4.9.2 Axiom Systems/176 4.9.3 Theorems Interrelating the Axiom Systems/176 4.10 Additional Proofs and Results/177 4.10.1 Introduction/177 4.10.2 Basic Concepts I/177 4.10.3 Basic Concepts II/178 4.10.4 Sufficient Conditions for TM/181 4.10.5 Preliminary Lemmas Rom Set Theory/181 4.10.6 The logical equivalence of M and TM/184 4.10.7 For Externally Finite a, D"(a) Implies TM/187 4.10.8 Homogeneity/188 4.10.9 Meaningful Cardinals/189 4.10.10 Generalized Transformational Meaningfulness/193 4.10.11 Independence Results/199 5 Representational Theory of Measurement/205 5.1 Introduction/205 5.1.1 Overview/205 5.1.2 Preliminaries/206 5.2 Representational Theories of Measurement/211 5.3 Criticisms of the Representational Approach/213 5.3.1 Adams' Criticisms/213 5.3.2 Niedcrec's Criticisms/215 5.3.3 hlichell's Criticisms/216 5.3.4 Criticisms of the Representational Meaningfulness Concept/221 5.4 Continuous Tleasurement Structures/222 5.4.1 Introduction/222 5.4.2 Continua/222 5.4.3 Continuous Threshold Structures/223 5.4.4 Continuous Extensive Structures/225 5.4.5 Continuous Scalar Structures/229 5.4.6 Continuous Additive Conjoint Structures/233 5.4.7 Continuous Linear Structures/236 5.4.8 Homogeneous Measurement Structures/237 5.5 Representational Concepts of Meaningfulness/240 5.6 Meaningful Scales/251 5.7 Possible Psychophysical Laws Revisited/254 5.8 Magnitude Estimation Revisited/257 5.8.1 Introduction/257 5.8.2 Ratio Magnitude Estimation/259 5.8.3 Behavioral Axiomatization/259 5.8.4 Cognitive Axiomatization/261 5.8.5 Additive Scales/266 5.8.6 Numeral Multiplicative Scales/268 5.8.7 Magnitude Estimation with Generalized Numerals/270 5.9 Weber's Law/271 5.9.1 Weber Representations/271 5.9.2 Weber's Law and Meaningfulness/273 5.10 Dimensional Analysis/274 5.10.1 Overview/274 5.10.2 Dimensional Analysis: Quantitative Theory/276 5.10.3 Distributive Triples with Associative Operations/278 5.10.4 Generalized Distributive Triples/282 5.10.5 Qualitative Dimensional Structures/286 5.10.6 Alternative Physical Measurements/291 5.10.7 Scale Types of Derived Physical Qualities/296 5.10.8 Dimensionless Quantities/298 5.10.9 Summary for Dimensional Analysis/300 5.11 Discussion and Conclusions/301 5.12 Additional Proofs and Theorems/305 6 Intrinsicness/313 6.1 Overview/313 6.2 ℰ-Intrinsicness/315 6.3 Intrinsicness Relative to M, {S_j)j∈J/319 6.3.1 Definition of Intrinsicness Relative to M, {Sj}j∈J/319 6.3.2 Enervation of Inferential Techniques Due to the Overspecification of Primitives/321 6.3.3 The Relativity of Meaningfulness and Intrinsicness/323 6.4 Lawfulness/323 6.4.1 Introduction/323 6.4.2 Possible Psychophysical Laws/323 6.4.3 Weber's Law/329 6.4.4 Stevens' Psychophysical Power Law/330 6.4.5 Luce's Possible Psychophysical Laws, 1990/332 6.4.6 Meaningfulness Versus Intrinsicness/335 6.5 A Theory of the Psychological-Physical Relationship/335 6.5.1 Separable Psychophysical Situations/336 6.5.2 The Equivalence Principle/338 6.5.3 Applications of the Equivalence Principle/339 6.6 Structural Archimedeanness/344 6.7 Dichotomous Data Analysis/356 6.7.1 Boolean Equivalent Sets of Properties/356 6.7.2 An Illustrative Example/357 6.7.3 Conclusions/358 6.8 Conclusions/360 6.9 Additional Proofs and Theorems/361 7 Qualitativeness/371 7.1 Introduction/371 7.2 Axiom System Qw(a)/372 7.3 Integral Domains and Fields/377 7.4 Qualitative Systems of Magnitude Numbers/381 7.5 Qualitative Homogeneity/383 7.6 Qualitative Canonical Measurement/387 7.7 Qualitative Magnitude Estimation/392 7.8 Method of Standard Sequences/399 7.9 Qualitative Homogeneity: Other Cases/403 7.10 Qualitative Numbers and Metaphysical Reduction/404 7.11 Meaningfulness Versus Qualitativeness Versus Empiricalness/407 7.11.1 Meaningfulness Versus Qualitativeness/407 7.11.2 Empiricalness/408 7.11.3 Conclusions/410 7.12 Summary of Main Points/411 7.13 Additional Proofs and Theorems/413 8 Meaningfulness and the Axiom of Choice/427 8.1 Introduction/427 8.2 The Axiom of Choice/428 8.3 Lebesgue's Measure Problem/434 8.4 Hausdorff's Measure Problem/436 8.5 Results by Banach and Ularn/437 8.6 Discussion/438 8.7 Lebesguc Measurability and Meaningfulness/441 References/445 Index/457