Invariance and Structural Dependence

This is a revised version of a doctoral thesis, submitted in mimeographed fonn to the Faculty of Arts, Uppsala University, 1988. It deals with the notions of struc­ tural dependence and independence, which are used in many applications of mathe­ matics to science. For instance, a physical law states that one physical aspect is structurally dependent on one or more other aspects. Structural dependence is closely related to the mathematical idea of functional dependence. However, struc­ tural dependence is primarily thought of as a relation holding between aspects rather than between their measures. In this book, the traditional way of treating aspects within measurement theory is modified. An aspect is not viewed as a set-theoretical structure but as a function which has sets as arguments and set-theoretical structures as values. This way of regarding aspects is illustrated with an application to social choice and group deci­ sion theory. Structural dependence is connected with the idea of concomitant variations and the mathematical notion of invariance. This implies that the study of this notion has roots going back to Mill's inductive logic, to Klein's Erlangen Program for geome­ try and to Padoa's method for proving the independence of symbols in formal logic.

Contenu

1. Problem Area and Basic Formal Apparatus.- 1. The Concept of Dependence in Applied Mathematics; a First Account.- 1.0 Introduction.- 1.1 Determination and relevance.- 1.2 Partial determination.- 1.3 Structural dependence.- 1.4 Dependence and concomitant variations.- 1.5 Supervenience and dependence.- 1.6 Invariance and dependence.- 1.7 Independence of primitive symbols.- 1.8 Relations as functions.- 1.9 Notions of independence in modern measurement and decision theory.- 1.10 Applications of structural dependence.- 1.11 Summing up.- 2. Basic Formal Concepts and Terminology.- 2.0 Introduction.- 2.1 Relations and functions.- 2.2 Properties of binary relations.- 2.3 Order relations.- 2.4 Two lemmas on weak orders.- 2.5 Semiorders.- 2.6 Correspondences.- 2.7 Invariance.- 2.8 Relational structures.- 2.9 Isomorphisms and homomorphisms.- 2.10 Congruence relations.- 2.11 Lattices.- 2. An Informal Presentation of the Main Themes.- 3. Relationals.- 3.0 Introduction.- 3.1 The fundamentals of relational.- 3.2 Formal properties of relationals.- 3.3 Some examples.- 3.4 Finitary systems of relationals.- 3.5 Historical and bibliographical remarks.- 4. Subordination, Uncorrelation and Derivation.- 4.0 Introduction.- 4.1 Isomorphism preservation and transitions.- 4.2 Subordination and definability.- 4.3 Uncorrelation.- 4.4 The dependence between R and its regionalization R*.- 4.5 Equality and decision methods for relationals.- 4.6 Derived and derivable relationals.- 4.7 Stability of transitions.- 4.8 The structural character of transitions and subordination.- 4.9 Significance.- 5. An Example: Social Choice.- 5.0 Introduction.- 5.1 The notion of dependence in social choice theory.- 5.2 Preference relationals and collective choice rules.- 5.3 Isomorphism preservation, subordination and social choice.- 5.4 Relative effectiveness, derivability and social choice.- 5.5 Stability and background for collective choice rules.- 5.6 Structural dependence and aggregation; a preliminary remark.- 6. Conformity and Measures.- 6.0 Introduction.- 6.1 Equality preservation and independent realizability.- 6.2 Congruence relational, conformity and import.- 6.3 Homomorphic representations.- 6.4 Measures.- 6.5 Numerical measures and representations.- 6.6 Connections between relational defined by measures.- 3. Formal Treatment of Basic Topics.- 7. Transitions Between Systems of Relationals.- 7.0 Introduction.- 7.1 Relational systems.- 7.2 Transitions and subordination.- 7.3 Transitions and uncorrelation.- 7.4 Concatenation and transition.- 7.5 Significance.- 7.6 Stability and monotonicity of s-functions.- 8. The Structure of Subordination.- 8.0 Introduction.- 8.1 Subalternation and rank.- 8.2 The lattice of subalternation.- 8.3 Correlation and collaterally.- 8.4 Semiranks.- 8.5 On equality preservation and independent realizability of structures.- 9. Isomorphic Mappings and Invariance.- 9.0 Introduction.- 9.1 Mappings.- 9.2 Isomorphic mappings.- 9.3 Automorphic mapping invariance.- 9.4 Global isomorphic mappings and global subordination.- Final remarks.- References.