Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

A successor relation between theories T und T₁ express that T₁, the successor of T, has justifiably superseded T. In physics, for instance, Newton's theory of gravitation has superseded Kepler's laws and, in turn, Einstein's theory has become the successor of Newton's. By now there is no agreement on how a general concept of successor relation would have to be construed. In the present paper attention is drawn to two types of such relations, one deductive the other confirmatory. It seems (...) that what most people are looking for is the deductive type of relation. However, a deductive successor relation has to be justified by showing that if it holds then another, confirmatory relation holds, saying roughly that the actual empirical success of the deducing theory is at least as good as the corresponding success of the deduced theory. Under certain restrictions such a justification is given in the paper. (shrink)

Mit der vorliegenden Arbeit verfolgen wir drei Ziele. Erstens exemplifizieren wir einige wissenschaftstheoretische Fragen und die zugehörigen Antworten am sehr einfachen Beispiel der klassischen Stoßmechanik. Zweitens läßt sich an diesem Beispiel besonders klar der Begriff des Meßmodells darstellen; insbesondere erhalten wir eine vollständige Übersicht über alle Meßmodelle. Und drittens erhalten wir ein schönes Beispiel für den Begriff der Reduktion einer Theorie auf eine andere, denn die Stoßmechanik läßt sich auf einfache Weise auf eine Spezialisierung der Klassischen Partikelmechanik reduzieren.

Als eine wesentliche Schwäche des strukturalistischen Theorienkonzepts musste die Vagheit erscheinen, mit der die Unterscheidung zwischen bezüglich einer Theorie theoretischen und nichttheoretischen Funktionen bisher belastet war. In der vorliegenden Untersuchung wird ein formal präzises, pragmatisch nichtrelativiertes Kriterium für T-Theoretizität vorgestellt. Erste Schritte zur Klärung seiner Bedeutung für die holistische Sichtweise erfahrungswissenschaftlicher Theorien werden unternommen.

This is the basis of the first part of the book.