Ebook: 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.

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This book deals with the notices of structural dependence and independence, which are used in many applications of mathematics 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, functional dependence expresses a kind of complete dependence while structural dependence includes partial dependence. Further, structural 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 regardingaspects is illustrated with an application to social choice and group decision theory. Structural dependence is connected with the idea of concomitant variations and the mathematical notion of invariance. A distinction between dependence in the sense of determination and in the sense of relevance is drawn. Structural dependence both in the sense of determination and of relevance can be found in degrees from complete determination and relevance to complete undetermination and irrelevance. The possibility of different "scales" of determination and relevance is considered. One chapter of the book is devoted to a study of the dependence relation between the social ordering and the individual orderings in social choice and group decision theory.

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This book deals with the notices of structural dependence and independence, which are used in many applications of mathematics 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, functional dependence expresses a kind of complete dependence while structural dependence includes partial dependence. Further, structural 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 regardingaspects is illustrated with an application to social choice and group decision theory. Structural dependence is connected with the idea of concomitant variations and the mathematical notion of invariance. A distinction between dependence in the sense of determination and in the sense of relevance is drawn. Structural dependence both in the sense of determination and of relevance can be found in degrees from complete determination and relevance to complete undetermination and irrelevance. The possibility of different "scales" of determination and relevance is considered. One chapter of the book is devoted to a study of the dependence relation between the social ordering and the individual orderings in social choice and group decision theory.
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
The Concept of Dependence in Applied Mathematics; a First Account....Pages 3-39
Basic Formal Concepts and Terminology....Pages 40-70
Front Matter....Pages 71-71
Relationals....Pages 73-95
Subordination, Uncorrelation and Derivation....Pages 96-134
An Example: Social Choice....Pages 135-151
Conformity and Measures....Pages 152-172
Front Matter....Pages 173-173
Transitions Between Systems of Relationals....Pages 175-200
The Structure of Subordination....Pages 201-218
Isomorphic Mappings and Invariance....Pages 219-234
Back Matter....Pages 235-248

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This book deals with the notices of structural dependence and independence, which are used in many applications of mathematics 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, functional dependence expresses a kind of complete dependence while structural dependence includes partial dependence. Further, structural 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 regardingaspects is illustrated with an application to social choice and group decision theory. Structural dependence is connected with the idea of concomitant variations and the mathematical notion of invariance. A distinction between dependence in the sense of determination and in the sense of relevance is drawn. Structural dependence both in the sense of determination and of relevance can be found in degrees from complete determination and relevance to complete undetermination and irrelevance. The possibility of different "scales" of determination and relevance is considered. One chapter of the book is devoted to a study of the dependence relation between the social ordering and the individual orderings in social choice and group decision theory.
Content:
Front Matter....Pages I-XII
Front Matter....Pages 1-1
The Concept of Dependence in Applied Mathematics; a First Account....Pages 3-39
Basic Formal Concepts and Terminology....Pages 40-70
Front Matter....Pages 71-71
Relationals....Pages 73-95
Subordination, Uncorrelation and Derivation....Pages 96-134
An Example: Social Choice....Pages 135-151
Conformity and Measures....Pages 152-172
Front Matter....Pages 173-173
Transitions Between Systems of Relationals....Pages 175-200
The Structure of Subordination....Pages 201-218
Isomorphic Mappings and Invariance....Pages 219-234
Back Matter....Pages 235-248
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