NEW FOUNDATIONS FOR PHYSICAL GEOMETRY C

Erscheinungsjahr 2014
ISBN: 978-0-19-870130-9
Verlag: ACADEMIC

Buch, Englisch, 374 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 725 g

Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. This is the initial volume in a two-volume
set, the first of which develops the mathematical structure and the second of which applies it to classical and Relativistic physics.

The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed in
those terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed up
through homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.

Zielgruppe

Mathematicians, physicists, and philosophers

Autoren/Hrsg.

Maudlin

Maudlin, Tim

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Philosophie der Mathematik
• Mathematik | Informatik Mathematik Geometrie Geometrie der modernen Physik
• Geisteswissenschaften Philosophie Philosophie der Mathematik, Philosophie der Physik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Topologie
• Naturwissenschaften Physik Quantenphysik Relativität, Gravitation

Inhaltsverzeichnis

Acknowledgments
Introduction
Mathematical Foundations
1: Topology and Its Shortcomings
2: Linear Structures, Neighborhoods, Open Sets
Appendix to Chapter 2
3: Closed Sets, Open Sets (Again), Connected Spaces
4: Separation Properties, Convergence, and Extensions
5: Properties of Functions
6: Subspaces and Substructures; Straightness and Differentiability
7: Metrical Structure
Appendix: A Remark about Minimal Regular Metric Spaces
8: Product Spaces and Fiber Bundles
9: Beyond Continua
Axioms and Definitions
Bibliography

Über Autor(innen)

Tim Maudlin was born and raised in Washington DC. Having received his BA in Physics and Philosophy from Yale and his PhD in History and Philosophy of Science from the University of Pittsburgh, he was a faculty member at Rutgers University for a quarter century before joining New York University in 2011. He has been a visiting professor of philosophy at Harvard.