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...

Ausführliche Beschreibung

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Weitere Titel:
Theory of linear structures
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 --
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) --
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 --
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 --
Separation Properties, Convergence, and Extensions --
Separation Properties --
Convergence and Unpleasantness.
Sequences and Convergence --
Extensions --
The Topologist's Sine Curve --
Physical Interlude: Thomson's Lamp --
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 --
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.
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 --
Appendix: A Remark about Minimal Regular Metric Spaces --
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 --
Beyond Continua --
How Can Continua and Non-Continua Approximate Each Other? --
Continuous Functions --
Homotopy --
Compactness --
Summary of Mathematical Results and Some Open Questions --
1. Verfasser:
Maudlin, Tim
Format:
Elektronisch E-Book
Sprache:
Englisch
Veröffentlicht:
Oxford : Oxford University Press, 2014.
Zusammenfassung:
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 structures.
Umfang:
1 online resource (ix, 363 pages) : illustrations
Anmerkungen:
English
Bibliografie:
Includes bibliographical references and index.
ISBN:
0-19-102269-1
0-19-100455-3
Schlagworte:
Line geometry.
Space and time.
Geometry.
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