| 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 |
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