Surfaces and superposition / Ernest W. Adams ; foreword by Patrick Suppes.

Format:

Book

Author/Creator:

Adams, Ernest W. (Ernest Wilcox), 1926-

Language:

English

Subjects (All):

Geometry.

Physical Description:

xxi, 310 pages : illustrations ; 24 cm

Place of Publication:

Stanford, Calif. : CSLI Publications, [2001]

Summary:

Buildings appear to rest on top of the earth's surface, yet the buildings' foundations, which are embedded in it, are 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.

Contents:

1 Characteristics of the Approach 3

1.3 Illustrations in the Case of Points on Surfaces 9

1.4 Relevance to Geometry 13

1.5 An Empiricist-Operationalist Program 16

1.6 The Problem of Appearance and Reality 17

2 The Concrete Superficial 21

2.2 Immateriality and Two-Dimensionality 21

2.3 Incidence and Identity 23

2.4 Asides on Dependent Surface Features 27

2.5 Multi-Modal Incidence Judgments 29

2.6 Standard Surface Features 29

2.7 The Substantiality of Surfaces 32

2.8 Ontological and Epistemological Remarks 33

3 The Logic of Constructability 37

4 Remarks on Physical Abstraction 43

4.2 Instantiation, Individuation of Abstracta and the Dual Interpretation of Coincidence 44

4.3 Processes of Individuation 47

4.4 Principles of Physical Abstraction I: 'Principal Principles' and Their Grounds 56

4.5 Principles of Physical Abstraction II: Identity 59

4.6 Principles of Physical Abstraction III: Other Abstraction Principles 63

4.7 Identity over Time: Standards of Constancy 65

II Surface Topologies 69

5.2 Theory of Points on Surfaces 72

5.3 Basic Surface Topologies 74

5.4 Boundaries 76

5.5 Dimensionality 79

5.6 Linearity 81

6 Points on Surfaces 85

6.3 The Separation Test and Its Theory 88

6.4 Intersective Systems 96

6.5 Indivisibility 102

6.6 Abstract Points and a Problem 106

6.7 Other Views on the Nature of Points 112

7 Towards a Topology of Physical Surfaces 115

7.1 Introduction: The Problem of Physical Topology 115

7.2 The Basic Topology 119

7.3 Finite Coverability and the Hausdorff Property 121

7.4 Metrizability: A Hypothesis 124

7.5 Topological Connectedness 127

8 Boundaries 131

8.2 Theory of U-boundary Covers 134

8.3 Interiors 138

8.4 Remarks on Boundary Topologies 140

8.5 Boundaries of Spaces 143

8.6 Remarks on Representing Boundaries 145

9 Surface Dimensionality 147

9.2 Summary of Concepts and Results of Modern Dimension Theory 149

9.3 Operationalizations 152

9.4 Fractal Possibilities: Methodological Remarks 157

10 Aspects of a Platonic Account of Linearity 163

10.2 Abstract Characterization and its Application to Surface Spaces 165

10.3 Operational Characterization of Linearity in the Case of Boundary Segments 168

10.4 Linear Ordering 170

10.5 Representing Lines 172

10.6 Open Problems 174

III Superposition 177

11 The Method of Superposition and Its Problems 179

11.2 Logical Problems of Surface Superposition 182

11.3 Suggested Resolutions 183

11.4 Looking Ahead 184

12 Phenomena and Topology of Superposition 187

12.1 Introduction: Empirical Difficulties 187

12.2 Fundamentals of Composite Surface Spaces: Points of the Spaces 194

12.3 The Paradoxes of Superposition 196

12.4 The Justification of Superposition Claims 197

12.5 Composite Surface Topologies 201

12.6 On Countable Composite Surfaces 202

12.7 On Orientability 206

13 Possible Superpositions 211

13.2 Speculative Remarks on Superpositionality Assumptions in The Elements 213

13.3 A Special Law of Superposability 218

13.4 Decompositions and Their Spaces 220

14 Rigidity 231

14.1 Aspects of Rigidity 231

14.2 An Atemporal Rigidity Presupposition of The Elements: Constructive Reference and Abstraction 233

14.3 Rigid Motion 235

14.4 Length, Distance, and Rigidity, and Their Relation to Congruence 236

15 Rigid Frames and Their Spaces 241

15.2 Euclidean Plane Geometry 242

15.3 Rigid Frames and the Application of Geometry to Objects in Them 244

15.4 Remarks on the Topologies of Spaces of Rigid-Frames 248

15.5 Relations Between Spaces 249

15.6 Comments on Measuring-Tape Geometry 251

IV Miscellaneous Topics 255

16 Connections with Physical Theory 257

16.2 The Role of Non-Geometrical Considerations in Defining Spatial Relations in Physical Applications of Geometry 257

16.3 Marks in the Application of Physical Theory 258

16.4 Liquids and Matter 262

17 Surface Feature, Sense Datum, and Psychology 265

17.2 Similarities between Surface Features and Sense Data 267

17.3 Appearance, Reality, Superposition, and Construction 269

17.4 Towards a Positive Account of Appearances 269

17.5 Physical and Mental Pictures 271

17.6 Visual Geometry I: Two Philosophical Theories 273

17.7 Visual Geometry II: Marr's Theory 276

17.8 Concluding Philosophical Reflections 279

18 Objectives, Theses, and Objections 283

18.1 Summary of Aims and Claims of This Essay 283

18.2 Objections Formulated and Discussed 285.

Notes:

Includes bibliographical references (pages 295-300) and index.

ISBN:

1575862808

OCLC:

46240354