Philosophy and Geometry


Philosophy and Geometry
Theoretical and Historical Issues

Lorenzo Magnani
University of Pavia

Pavia, Italy

Georgia Institute of Technology

Atlanta, Georgia, USA


Contents



Chapter 1 At the Origins of Geometrical Knowledge		1

1. Conceptual space, mental spatial models, latent geometry		1
1.1 Conceptual space and geometrical shapes		1
1.2 Mental spatial models and spatial descriptions		9
1.3 Latent geometry		11
2. Figures, symbols, and the Greek origins of geometry		15
2.1 Sefirot		15
2.2 Yantra		18
2.3 Numbers, points, geometrical diagrams		19
3. The ritual origin of geometry		22

Chapter 2 Geometry: the Model of Knowledge		27

1. Sensibility		27
2. Imagination		29
3. Understanding		30
4. Pure apprehension and geometry		32
5. Pure apprehension and empirical schematism		39
6. Geometrical schemata and constructions: models of philosophy		47
7. Space as the object of geometry		54

Chapter 3 Constructions, Logic, Categories		57

1. Space and logic		57
2. Intuition, construction, and the logic of singular terms		59
3. Pure and applied geometry		66
4. Why is geometry synthetic?		69
5. Categories and Axioms of Intuition		70
5.1 Transcendental categories and schemes		70
5.2 The Axioms of Intuition: why we can apply geometry to
experience		73
5.3 The foundation of geometry and objectivity		78
6. Mathematical schematism		84

Chapter 4 The F antasi¢a in Ancient Geometrical Knowledge		91

1. Geometry, drawing, and writing		91
2. Mathematical objects		96
3. Geometrical reasoning		97
4. The science that studies shapes: geometry		99
5. "History of geometry" and "elements" of geometry		100

Chapter 5 Geometry and Convention		105

1. Crude facts, relations, conventions		105
2. Pure and applied geometry		107
3. Sensible, geometric, and physical space		108
4. Geometrical intuition		112
5. Geometrical apriorism and empiricism		114
6. The genesis of geometry		118
7. The interchangeability of geometries		119
8. Withdrawing conventions		123
9. Withdrawing principles of coordination		132

Chapter 6 Geometry, Problem Solving, Abduction		139

1. Geometrical constructions and problem solving		139
1.1 Generate and test		148
2. Model-based and manipulative abduction		151
2.1 Abductive reasoning		151
2.2 Thinking through drawing: model-based abduction		157
2.3 Thinking through doing: manipulative abduction		160
3. Geometrical construction is a kind of manipulative abduction		171
4. Diagrams, abduction, and deductive reasoning		172

Chapter 7 Geometry and Cognition		175

1. Geometry of visibles, protogeometry, manipulations		175
2. At the origin of geometrical knowledge II		178
2.1 Adumbrations		178
2.2 The genesis of space		179
2.3 Anticipations as abductions		182
2.4 The genesis of geometrical idealities		183
3. Non-conceptual and spatial abilities		189
4. Computational geometrical constructions		192
4.1 Automatic constructions as epistemic mediators		192
4.2 Automatic "thinking through doing"		193
5. Spatial imagery		196
5. Spatial thinking and the discovery of non-Euclidean
geometries		198
6. Logical models of diagrammatic reasoning		204
6.1 Diagrams, heuristics, abduction		204
6.2 Diagrams and geometrical constructions as deductions		207

References		211

Author Index		231

Subject Index		237