New foundations for physical geometry : the theory of linear structures /

Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original theory of linear struct...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Maudlin, Tim (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Oxford : Oxford University Press, 2014.
Temas:
Line geometry.
Space and time.
Géométrie de la ligne.
MATHEMATICS > Topology.
Line geometry
Space and time
Raumstruktur
Topologischer Raum
Raum-Zeit
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Machine generated contents note: Metaphorical and Geometrical Spaces
  • A Light Dance on the Dust of the Ages
  • The Proliferation of Numbers
  • Descartes and Coordinate Geometry
  • John Wallis and the Number Line
  • Dedekind and the Construction of Irrational Numbers
  • Overview and Terminological Conventions
  • 1. Topology and Its Shortcomings
  • Standard Topology
  • Closed Sets, Neighborhoods, Boundary Points, and Connected Spaces
  • The Hausdorff Property
  • Why Discrete Spaces Matter
  • The Relational Nature of Open Sets
  • The Bill of Indictment (So Far)
  • 2. Linear Structures, Neighborhoods, Open Sets
  • Methodological Morals
  • The Essence of the Line
  • The (First) Theory of Linear Structures
  • Proto-Linear Structures
  • Discrete Spaces, Mr Bush's Wild Line, the Woven Plane, and the Affine Plane
  • A Taxonomy of Linear Structures
  • Neighborhoods in a Linear Structure
  • Open Sets.
  • Finite-Point Spaces
  • Return to Intuition
  • Directed Linear Structures
  • Linear Structures and Directed Linear Structures
  • Neighborhoods, Open Sets, and Topologies Again
  • Finite-Point Spaces and Geometrical Interpretability
  • A Geometrically Uninterpretable Topological Space
  • Segment-Spliced Linear Structures
  • Looking Ahead
  • Exercises
  • Appendix: Neighborhoods and Linear Structures
  • 3. Closed Sets, Open Sets (Again), Connected Spaces
  • Closed Sets: Preliminary Observations
  • Open and Closed Intervals
  • JP-closed and JP-open Sets
  • Jp-open Sets and Open Sets, JP-closed Sets and Closed Sets
  • Zeno's Combs
  • Closed Sets, Open Sets, and Complements
  • Interiors, Boundary Points, and Boundaries
  • Formal Properties of Boundary Points
  • Connected Spaces
  • Chains and Connectedness
  • Directedness and Connectedness
  • Exercises
  • 4. Separation Properties, Convergence, and Extensions
  • Separation Properties
  • Convergence and Unpleasantness.
  • Sequences and Convergence
  • Extensions
  • The Topologist's Sine Curve
  • Physical Interlude: Thomson's Lamp
  • Exercises
  • 5. Properties of Functions
  • Continuity: an Overview
  • The Intuitive Explication of Continuity and Its Shortcomings
  • The Standard Definition and Its Shortcomings
  • What the Standard Definition of Continuity Defines
  • The Essence of Continuity
  • Continuity at a Point and in a Direction
  • An Historical Interlude
  • Remarks on the Architecture of Definitions; Lineal Functions
  • Lines and Continuity in Standard Topology
  • Exercises
  • 6. Subspaces and Substructures; Straightness and Differentiability
  • The Geometrical Structure of a Subspace: Desiderata
  • Subspaces in Standard Topology
  • Subspaces in the Theory of Linear Structures
  • Substructures
  • One Way Forward
  • Euclid's Postulates and the Nature of Straightness
  • Convex Affine Spaces
  • Example: Some Conical Spaces
  • Tangents
  • Upper and Lower Tangents, Differentiability
  • Summation
  • Exercises.
  • 7. Metrical Structure
  • Approaches to Metrical Structure
  • Ratios Between What?
  • The Additive Properties of Straight Lines
  • Congruence and Comparability
  • Eudoxan and Anthyphairetic Ratios
  • The Compass
  • Metric Linear Structures and Metric Functions
  • Open Lines, Curved Lines, and Rectification
  • Continuity of the Metric
  • Exercises
  • Appendix: A Remark about Minimal Regular Metric Spaces
  • 8. Product Spaces and Fiber Bundles
  • New Spaces from Old
  • Constructing Product Linear Structures
  • Examples of Product Linear Structures
  • Neighborhoods and Open Sets in Product Linear Structures
  • Fiber Bundles
  • Sections
  • Additional Structure
  • Exercises
  • 9. Beyond Continua
  • How Can Continua and Non-Continua Approximate Each Other?
  • Continuous Functions
  • Homotopy
  • Compactness
  • Summary of Mathematical Results and Some Open Questions
  • Exercises.

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