New Foundations for Physical Geometry: The Theory of Linear Structures

Hardcover | March 7, 2014

byTim Maudlin

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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 ascontinuity, 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 andRelativistic 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 foundationalnotion 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, aredefined 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) andmetrical structure.

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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 ascontinuity, connectedness, boundaries of set...

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 professo...

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Format:HardcoverDimensions:376 pages, 9.21 × 6.14 × 0.01 inPublished:March 7, 2014Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198701306

ISBN - 13:9780198701309

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Table of Contents

AcknowledgmentsIntroductionMathematical Foundations1. Topology and Its Shortcomings2. Linear Structures, Neighborhoods, Open SetsAppendix to Chapter 23. Closed Sets, Open Sets (Again), Connected Spaces4. Separation Properties, Convergence, and Extensions5. Properties of Functions6. Subspaces and Substructures; Straightness and Differentiability7. Metrical StructureAppendix: A Remark about Minimal Regular Metric Spaces8. Product Spaces and Fiber Bundles9. Beyond ContinuaAxioms and DefinitionsBibliography