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Surfaces and Superposition

Field Notes on some Geometrical Excavations

By: Ernest W. Adams

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Paperback | 1 February 2001 | Edition Number 1

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200 Pages

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2.28 x 1.52 x 0.18

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Buildings appear to rest on top of the earth's surface, yet the surface is actually permeated by the buildings' foundations-out of view. If a foundation's blueprints are unavailable, as in archaeology, excavation would be needed to discover what actually supports a specific building. Analogously, the fields of geometry and topology have easily observable concepts resting on the surface of theoretical underpinnings that have not been completely discovered, unearthed or understood. Moreover, geometrical and topological principles of superposition provide insight into probing the connections between accessible superstructures and their hidden underpinnings. This book develops and applies these insights broadly, from physics to mathematics to philosophy. Even analogies and abstractions can now be seen as foundational superpositions.

This book examines the dimensionality of surfaces, how superpositions can make stable frameworks, and gives a quasi-Leibnizian account of the relative `spaces' that are defined by these frameworks. Concluding chapters deal with problems concerning the spatio-temporal frameworks of physical theories and implications for theories of visual geometry. The numerous illustrations, while surprisingly simple, are satisfyingly clear.

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Non-Fiction Mathematics Geometry Philosophy Science Science in General Philosophy of Science Engineering & Technology Civil Engineering Structural Engineering Miscellaneous Items

Foreword	p. xiii
Preface	p. xv
Preliminaries	p. 1
Characteristics of the Approach	p. 3
Introduction	p. 3
Characteristics of the Approach	p. 4
Illustrations in the Case of Points on Surfaces	p. 9
Relevance to Geometry	p. 13
An Empiricist-Operationalist Program	p. 16
The Problem of Appearance and Reality	p. 17
Summary of Themes of Following Chapters	p. 19
The Concrete Superficial	p. 21
Introduction	p. 21
Immateriality and Two-Dimensionality	p. 21
Incidence and Identity	p. 23
Asides on Dependent Surface Features	p. 27
Multi-Modal Incidence Judgments	p. 29
Standard Surface Features	p. 29
The Substantiality of Surfaces	p. 32
Ontological and Epistemological Remarks	p. 33
The Logic of Constructability	p. 37
Introduction	p. 37
The Logic of Constructability	p. 39
Remarks on Physical Abstraction	p. 43
Introduction	p. 43
Instantiation, Individuation of Abstracta and the Dual Interpretation of Coincidence	p. 44
Processes of Individuation	p. 47
Principles of Physical Abstraction I: 'Principal Principles' and Their Grounds	p. 56
Principles of Physical Abstraction II: Identity	p. 59
Principles of Physical Abstraction III: Other Abstraction Principles	p. 63
Identity over Time: Standards of Constancy	p. 65
Summary	p. 67
Surface Topologies	p. 69
Overview	p. 71
Introduction	p. 71
Theory of Points on Surfaces	p. 72
Basic Surface Topologies	p. 74
Boundaries	p. 76
Dimensionality	p. 79
Linearity	p. 81
Points on Surfaces	p. 85
Introduction	p. 85
Basic Concepts	p. 86
The Separation Test and Its Theory	p. 88
Intersective Systems	p. 96
Indivisibility	p. 102
Abstract Points and a Problem	p. 106
Other Views on the Nature of Points	p. 112
Towards a Topology of Physical Surfaces	p. 115
Introduction: The Problem of Physical Topology	p. 115
The Basic Topology	p. 119
Finite Coverability and the Hausdorff Property	p. 121
Metrizability: A Hypothesis	p. 124
Topological Connectedness	p. 127
Boundaries	p. 131
Introduction	p. 131
Theory of U-boundary Covers	p. 134
Interiors	p. 138
Remarks on Boundary Topologies	p. 140
Boundaries of Spaces	p. 143
Remarks on Representing Boundaries	p. 145
Surface Dimensionality	p. 147
Introduction	p. 147
Summary of Concepts and Results of Modern Dimension Theory	p. 149
Operationalizations	p. 152
Fractal Possibilities: Methodological Remarks	p. 157
Aspects of a Platonic Account of Linearity	p. 163
Introduction	p. 163
Abstract Characterization and its Application to Surface Spaces	p. 165
Operational Characterization of Linearity in the Case of Boundary Segments	p. 168
Linear Ordering	p. 170
Representing Lines	p. 172
Open Problems	p. 174
Superposition	p. 177
The Method of Superposition and Its Problems	p. 179
Historical Background	p. 179
Logical Problems of Surface Superposition	p. 182
Suggested Resolutions	p. 183
Looking Ahead	p. 184
Phenomena and Topology of Superposition	p. 187
Introduction: Empirical Difficulties	p. 187
Fundamentals of Composite Surface Spaces: Points of the Spaces	p. 194
The Paradoxes of Superposition	p. 196
The Justification of Superposition Claims	p. 197
Composite Surface Topologies	p. 201
On Countable Composite Surfaces	p. 202
On Orientability	p. 206
Possible Superpositions	p. 211
Introduction	p. 211
Speculative Remarks on Superpositionality Assumptions in The Elements	p. 213
A Special Law of Superposability	p. 218
Decompositions and Their Spaces	p. 220
Rigidity	p. 231
Aspects of Rigidity	p. 231
An Atemporal Rigidity Presupposition of The Elements: Constructive Reference and Abstraction	p. 233
Rigid Motion	p. 235
Length, Distance, and Rigidity, and Their Relation to Congruence	p. 236
Rigid Frames and Their Spaces	p. 241
Introduction	p. 241
Euclidean Plane Geometry	p. 242
Rigid Frames and the Application of Geometry to Objects in Them	p. 244
Remarks on the Topologies of Spaces of Rigid-Frames	p. 248
Relations Between Spaces	p. 249
Comments on Measuring-Tape Geometry	p. 251
Miscellaneous Topics	p. 255
Connections with Physical Theory	p. 257
Introduction	p. 257
The Role of Non-Geometrical Considerations in Defining Spatial Relations in Physical Applications of Geometry	p. 257
Marks in the Application of Physical Theory	p. 258
Liquids and Matter	p. 262
Surface Feature, Sense Datum, and Psychology	p. 265
Introduction	p. 265
Similarities between Surface Features and Sense Data	p. 267
Appearance, Reality, Superposition, and Construction	p. 269
Towards a Positive Account of Appearances	p. 269
Physical and Mental Pictures	p. 271
Visual Geometry I: Two Philosophical Theories	p. 273
Visual Geometry II: Marr's Theory	p. 276
Concluding Philosophical Reflections	p. 279
Objectives, Theses, and Objections	p. 283
Summary of Aims and Claims of This Essay	p. 283
Objections Formulated and Discussed	p. 285
References	p. 295
Index	p. 301
Table of Contents provided by Ingram. All Rights Reserved.

Surfaces and Superposition

Field Notes on some Geometrical Excavations

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Paperback

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