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Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and PoincarA(c)), and discussing conventionalist and neopositivist perspectives and the problem of the origins of geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field.
From the reviews:
"In this book different points of view with respect to geometry are combined. This mix of approaches from philosophy, history, mathematic, psychology and cognitive science with references to logic and computational models of geometry, represents a ] the strength of the book. a ] the authora (TM)s erudition is impressive. a ] Undoubtedly the book is good medicine against one-sided views of what geometry is." (Teun Koetsier, Zentralblatt MATH, Vol. 1045 (20), 2004)
| At the Origins of Geometrical Knowledge | p. 1 |
| Conceptual space, mental spatial models, latent geometry | p. 1 |
| Conceptual space and geometrical shapes | p. 1 |
| Mental spatial models and spatial descriptions | p. 9 |
| Latent geometry | p. 11 |
| Figures, symbols, and the Greek origins of geometry | p. 15 |
| Sefirot | p. 15 |
| Yantra | p. 18 |
| Numbers, points, geometrical diagrams | p. 19 |
| The ritual origin of geometry | p. 22 |
| Geometry: the Model of Knowledge | p. 27 |
| Sensibility | p. 27 |
| Imagination | p. 29 |
| Understanding | p. 30 |
| Pure apprehension and geometry | p. 32 |
| Empirical apprehension and empirical schematism | p. 39 |
| Geometrical schemata and constructions: models of philosophy | p. 47 |
| Space as the object of geometry | p. 54 |
| Constructions, Logic, Categories | p. 57 |
| Space and logic | p. 57 |
| Intuition, construction, and the logic of singular terms | p. 59 |
| Pure and applied geometry | p. 66 |
| Why is geometry synthetic? | p. 69 |
| Categories and Axioms of Intuition | p. 70 |
| Transcendental categories and schemata | p. 70 |
| The Axioms of Intuition: why we can apply geometry to experience | p. 73 |
| The foundation of geometry and objectivity | p. 78 |
| Mathematical schematism | p. 84 |
| The [Phi alpha nu tau alpha sigma iota alpha] in Ancient Geometrical Knowledge | p. 91 |
| Geometry, drawing, and writing | p. 91 |
| Mathematical objects | p. 96 |
| Geometrical reasoning | p. 97 |
| The science that studies shapes: geometry | p. 99 |
| "History of geometry" and "elements" of geometry | p. 100 |
| Geometry and Convention | p. 105 |
| Crude facts, relations, conventions | p. 105 |
| Pure and applied geometry | p. 107 |
| Sensible, geometric, and physical space | p. 108 |
| Geometrical intuition | p. 112 |
| Geometrical apriorism and empiricism | p. 114 |
| The genesis of geometry | p. 118 |
| The interchangeability of geometries | p. 119 |
| Withdrawing conventions | p. 123 |
| Withdrawing principles of coordination | p. 132 |
| Geometry, Problem Solving, Abduction | p. 139 |
| Geometrical constructions and problem solving | p. 139 |
| Generate and test | p. 148 |
| Model-based and manipulative abduction | p. 151 |
| Abductive reasoning | p. 151 |
| Thinking through drawing: model-based abduction | p. 157 |
| Thinking through doing: manipulative abduction | p. 160 |
| Geometrical construction is a kind of manipulative abduction | p. 171 |
| Diagrams, abduction, and deductive reasoning | p. 172 |
| Geometry and Cognition | p. 175 |
| Geometry of visibles, protogeometry, manipulations | p. 175 |
| At the origins of geometrical knowledge II | p. 178 |
| Adumbrations | p. 178 |
| The genesis of space | p. 179 |
| Anticipations as abductions | p. 182 |
| The genesis of geometrical idealities | p. 183 |
| Non-conceptual and spatial abilities | p. 189 |
| Computational geometrical constructions | p. 192 |
| Automatic constructions as epistemic mediators | p. 192 |
| Automatic "thinking through doing" | p. 193 |
| Spatial imagery | p. 196 |
| Spatial thinking and the discovery of non-Euclidean geometry | p. 198 |
| Logical models of diagrammatic reasoning | p. 204 |
| Diagrams, heuristics, abduction | p. 204 |
| Diagrams and geometrical constructions as deductions | p. 207 |
| References | p. 211 |
| Author Index | p. 231 |
| Subject Index | p. 237 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780792369332
ISBN-10: 0792369335
Series: Western Ontario Series in Philosophy of Science
Audience:
Professional
Format:
Hardcover
Language:
English
Published: 31st May 2001
Dimensions (cm): 24.0 x 16.0
x 1.7
Weight (kg): 1.26
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