Surfaces and Superposition: Field Notes on Some Geometrical Excavations

by Ernest W. Adams

Overview

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.

Product Details

ISBN-13:	9781575862804
Publisher:	Center for the Study of Language and Inf
Publication date:	02/28/2001
Series:	Center for the Study of Language and Information - Lecture Notes Series
Pages:	200
Product dimensions:	6.00(w) x 9.00(h) x 0.70(d)

Table of Contents

	Foreword	xiii
	Preface	xv
I	Preliminaries	1
1	Characteristics of the Approach	3
1.1	Introduction	3
1.2	Characteristics of the Approach	4
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
1.7	Summary of Themes of Following Chapters	19
2	The Concrete Superficial	21
2.1	Introduction	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
3.1	Introduction	37
3.2	The Logic of Constructability	39
4	Remarks on Physical Abstraction	43
4.1	Introduction	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
4.8	Summary	67
II	Surface Topologies	69
5	Overview	71
5.1	Introduction	71
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
6.1	Introduction	85
6.2	Basic Concepts	86
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
8.1	Introduction	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
9.1	Introduction	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
10.1	Introduction	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
11.1	Historical Background	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
13.1	Introduction	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
15.1	Introduction	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
16.1	Introduction	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
17.1	Introduction	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
	References	295
	Index	301