Ernest W. Adams

with a foreword by Patrick Suppes

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.

*“This is a remarkable piece of work of great conceptual interest. The literature on geometry is, in almost every direction to be named, enormous. I think that Adams has done a remarkable job giving a new conceptual outlook on surfaces, and perhaps even more, on the longstanding problem of how to think about superposition in geometry.”*

–Patrick Suppes, Stanford University

Read an excerpt from this book

Ernest W. Adams is Professor Emeritus of Philosophy at the University of California, Berkley, and fellow of the American Academy of Arts and Sciences.

- Forward

Patrick Suppes
- Preface
## I Preliminaries

- 1 Characteristics of the Approach
- 1.1 Introduction
- 1.2 Characteristics of the Approach
- 1.3 Illustrations in the Case of Points on the Surface
- 1.4 Relevance to Geometry
- 1.5 An Empiricists-Operationalist Program
- 1.6 The Problem of Appearances and Reality
- 1.7 Summary of Themes of Following Chapters

- 2 The Concrete Superficial
- 2.1 Introduction
- 2.2 Immateriality and Two-Dimensionality
- 2.3 Incidence and Identity
- 2.4 Asides on Dependent Surface Features
- 2.5 Multi-Modal Incidence Judgements
- 2.6 Standard Surface Features
- 2.7 The Substantiality of Surfaces
- 2.8 Ontological and Epistemological Remarks

- 3 The Logic of Constructability
- 3.1 Introduction
- 3.2 The Logic of Constructability

- 4 Remarks on Physical Abstraction
- 4.1 Introduction
- 4.2 Instantiation, Individuation of Abstracta and the Dual Interpretation of Coincidence
- 4.3 Processes of Individuation
- 4.4 Principles of Physical Abstraction I: ‘Principal Principles’ and Their Grounds
- 4.5 Principles of Physical Abstraction II: Identity
- 4.6 Principles of Physical Abstraction III: Other Abstraction Principles
- 4.7 Identity over Time: Standards of Constancy
- 4.8 Summary

## II Surface Topologies

- 5 Overview
- 5.1 Introduction
- 5.2 Theory of Points on the Surface
- 5.3 Basic Surface Topologies
- 5.4 Boundaries
- 5.5 Dimensionality
- 5.6 Linearity

- 6 Points on Surfaces
- 6.1 Introduction
- 6.2 Basic Concepts
- 6.3 The Separation Test and Its Theory
- 6.4 Intersective Systems
- 6.5 Indivisibility
- 6.6 Abstract Points and a Problem
- 6.7 Other Views and Nature of the Points

- 7 Towards a Topology of Physical
- 7.1 Introduction: The Problem of Physical Topology
- 7.2 The Basic Topology
- 7.3 Finite Coverability and the Hausdorff Property
- 7.4 Metrizability: A Hypothesis
- 7.5 Topological Connectedness

- 8 Boundaries
- 8.1 Introductions
- 8.2 Theory of U-boundary Covers
- 8.3 Interiors
- 8.4 Remarks on Boundary Topologies
- 8.5 Boundaries of Spaces
- 8.6 Remarks on Representing Boundaries

- 9 Surface Dimensionality
- 9.1 Introduction
- 9.2 Summary of Concepts and Results of Modern Dimension Theory
- 9.3 Fractal Possibilities: Methodological Remarks

- 10 Aspects of a Platonic Account of Linearity
- 10.1 Introduction
- 10.2 Abstract Chacterization and its Applications to Surface Spaces
- 10.3 Operational Characterization of Linearity in the Case of Boundary Segments
- 10.4 Linear Ordering
- 10.5 Representation Lines
- 10.6 Open Problems

## III Superposition

- 11 The Method of Superposition and Its Problems
- 11.1 Historical Background
- 11.2 Logical Problems on the Surface Superposition
- 11.3 Suggested Resolutions
- 11.4 Looking Ahead

- 12 Phenomena and Topology of Superposition
- 12.1 Introduction: Empirical Difficulties
- 12.2 Fundamentals of Composite Surface Spaces: Points of the Spaces
- 12.3 The Paradoxes of Superposition
- 12.4 The Justification of Superposition Claims
- 12.5 Composite Surface Topologies
- 12.6 On Countable Composite Surfaces
- 12.7 On Orientability

- 13 Possible Superpositions
- 13.1 Introduction
- 13.2 Speculative Remarks on Superpositionality Assumptions in
*The Elements*
- 13.3 A Special Law of Superpositionability
- 13.4 Decompositions and Their Spaces

- 14 Rigidity
- 14.1 Aspects of Rigidity
- 14.2 An Atemporal Rigidity Presupposition of
*The Elements*: Constructive Reference and Abstraction
- 14.3 Rigid Motion
- 14.4 Length, Distance, and Rigidity, and Their Relation to Congruence

- 15 Rigid Frames and Their Spaces
- 15.1 Introduction
- 15.2 Euclidean Plane Geometry
- 15.3 Rigid Frames and the Application of Geometry to Objects in Them
- 15.4 Remarks on the Topologies of Spaces of Rigid-Frames
- 15.5 Relations Between Space
- 15.6 Comments on Measuring-Tape Geometry

## IV Miscellaneous Topics

- 16 Connections with Physical Theory
- 16.1 Introduction
- 16.2 The Rôle of Non-GeometricalConsiderations in Defining Spatial Relations in Physical Applications of Geometry
- 16.3 Marks in the Application of Physical Theory
- 16.4 Liquids and Matter

- 17 Surface, Feature, Sense Datum, and Psychology
- 17.1 Introduction
- 17.2 Similarities between Surface Features and Sense Data
- 17.3 Appearance, Reality, Superposition, and Construction
- 17.4 Toward a Positive Account of Appearances
- 17.5 Physical and Mental Pictures
- 17.6 Visual Geometry I: Two Philosophical Theories
- 17.7 Visual Geometry II: Marrâ€™s Theory
- 17.8 Concluding Philosophical Reflections

- 8 Objectives, Theses, and Objections
- 18.1 Summary of Aims and Claims of This Essay
- 18.2 Objections Formulated and Discussed

- References
- Index

11/1/2001