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New Foundations for Physical Geometry: The Theory of Linear Structures
Author
Publisher
Oxford University Press
Publication Date
2014
Language
English
Description
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Table of Contents
From the Book - 1st ed.
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.
From the eBook
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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More Details
ISBN
9780198701309
9780191771613
9780191004551
9780191771613
9780191004551
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