Invariance and Structural Dependence

This book deals with the notices of structural dependenceand independence, which are used in many applications ofmathematics to science. For instance, a physical law statesthat one physical aspect is structurally dependent on one ormore other aspects. Structural dependence is closely relatedto the mathematical idea of functional dependence. However, functional dependence expresses a kind of completedependence while structural dependence includes partialdependence. Further, structural dependence is primarilythought of as a relation holding between aspects rather thanbetween their measures. In this book, the traditional way oftreating aspects within measurement theory is modified. Anaspect is not viewed as a set-theoretical structure but as afunction which has sets as arguments and set-theoreticalstructures as values. This way of regardingaspects isillustrated with an application to social choice and groupdecision theory. Structural dependence is connected with theidea of concomitant variations and the mathematical notionof invariance. A distinction between dependence in the senseof determination and in the sense of relevance is drawn. Structural dependence both in the sense of determination andof relevance can be found in degrees from completedetermination and relevance to complete undetermination andirrelevance. The possibility of different "scales" ofdetermination and relevance is considered. One chapter ofthe book is devoted to a study of the dependence relationbetween the social ordering and the individual orderings insocial choice and group decision theory.