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.
By:
Tim Maudlin (New York University)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Dimensions:
Height: 240mm,
Width: 162mm,
Spine: 31mm
Weight: 1g
ISBN: 9780198701309
ISBN 10: 0198701306
Pages: 374
Publication Date: 13 March 2014
Audience:
College/higher education
,
Professional and scholarly
,
Further / Higher Education
,
Undergraduate
Format: Hardback
Publisher's Status: Active
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
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.
Reviews for New Foundations for Physical Geometry: The Theory of Linear Structures
Maudlin's New Foundations for Physical Geometry is a fascinating and remarkable book Carolyn Brighouse, Mind It is impressive because Maudlin develops his own, new and independent, foundational scheme for geometry. Not only does he propose new principles, he elaborates them to such a degree that a serious alternative to traditional approaches begins to take shape. I think that this result, and the enthusiasm, energy and persistence displayed in the enterprise, deserve admiration. Dennis Dieks, Notre Dame Philosophical Reviews Online
