Category Theory in Physics, Mathematics, and Philosophy (Springer Proceedings in Physics)

Description

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science.  Introduction Contents Contributors 1 Why Categories? 1.1 Introduction 1.2 The Beginnings of Category Theory 1.3 Category Theory and Philosophy 1.3.1 Structuralism Versus Object Realism 1.3.2 The Foundations of Mathematics 1.3.3 The Problem with Unifying Mathematics 1.3.4 Category Theory in Metaphysics 1.4 Physics: Category Theory as an Ontology Rescuing Tool 1.5 Conclusion References 2 Category Theory and Philosophy 2.1 General Remarks Concerning Category Theory (CT) 2.2 CT and Some Classical Problems in Philosophy 2.2.1 CT and Ontology 2.2.2 CT and Epistemology 2.2.3 CT and Epistemology Contd.: Theories of the Development of Mathematics and Science 2.3 Conclusion References 3 Are There Category-Theoretical Explanations of Physical Phenomena? 3.1 Introduction 3.2 Mathematical Explanations of Physical Phenomena 3.3 Category Theory in Physics 3.4 Conclusion References 4 The Application of Category Theory to Epistemic and Poietic Processes 4.1 Introduction 4.2 Epistemic and Poietic Processes 4.3 The Potential of Category Theory References 5 Asymmetry of Cantorian Mathematics from a Categorial Standpoint: Is It Related to the Direction of Time? 5.1 Introduction 5.2 Products and Coproducts 5.2.1 Products in Certain Categories of Sets with Structures 5.2.2 Coproducts in the Same Categories 5.2.3 Recapitulation of Main Points 5.3 A Philosophical Discussion References 6 Extending List's Levels 6.1 List's Descriptive, Explanatory, and Ontological Levels 6.2 Supervenience and Reduction 6.3 Extensions: Partial and Non-surjective Maps 6.4 Levels as Categories 6.5 Conclusions and Prospects References 7 From Quantum-Mechanical Lattice of Projections to Smooth Structure of mathbbR4 7.1 Introduction 7.2 Quantum Mechanics—Preliminaries 7.3 Boolean Subalgebras of mathbbL 7.4 The Smooth Structure of a Spacetime 7.5 Correspondence: From mathbbL to Smooth Structures 7.6 Discussion References 8 Beyond the Space-Time Boundary 8.1 Introduction 8.2 Preliminaries 8.3 Manifold down to the Smallest Scale 8.4 A Model 8.1 Appendix: Through the Boundary 8.2 Appendix: Non-holonomous Monads References 9 Aspects of Perturbative Quantum Gravity on Synthetic Spacetimes 9.1 Introduction 9.2 Synthetic Differential Geometry, Spacetime and Gravity 9.2.1 SDG and the Perturbative QG 9.2.2 Classical gµν as a Quantized Field 9.3 Can Perturbative QG Be Any Fundamental Theory? References 10 Category Theory as a Foundation for the Concept Analysis of Complex Systems and Time Series 10.1 Introduction 10.2 Polarities and Complete Lattices 10.2.1 Poset Categories 10.2.2 Contexts 10.2.3 Galois Connections and Galois Correspondences 10.2.4 Concepts 10.2.5 Two-Element Posets 10.3 Categories of Concepts 10.3.1 The Category of Bonds 10.3.2 The Category of Bond Pairs 10.3.3 The Category of Bondings 10.3.4 A Complete Lattice Homomorphism 10.4 Quasilattices 10.4.1 Lattices and Semilattices as Quasilattices 10.4.2 The Structure of Quasilattices 10.4.3 The Diagonal Embedding of the Two-Element Lattice 10.5 Concept Analysis of a Complex System 10.5.1 Complex Polarities 10.5.2 Concept Quasilattices 10.5.3 Set Representations: The Layers of History 10.6 Conclusion References