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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 Poincaré), 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.
List of contents
1 At the Origins of Geometrical Knowledge.- 1. Conceptual space, mental spatial models, latent geometry.- 2. Figures, symbols, and the Greek origins of geometry.- 3. The ritual origin of geometry.- 2 Geometry: the Model of Knowledge.- 1. Sensibility.- 2. Imagination.- 3. Understanding.- 4. Pure apprehension and geometry.- 5. Empirical apprehension and empirical schematism.- 6. Geometrical schemata and constructions: models of philosophy.- 7. Space as the object of geometry.- 3 Constructions, Logic, Categories.- 1. Space and logic.- 2. Intuition, construction, and the logic of singular terms.- 3. Pure and applied geometry.- 4. Why is geometry synthetic?.- 5. Categories and Axioms of Intuition.- 6. Mathematical schematism.- 4 The ???????? in Ancient Geometrical Knowledge.- 1. Geometry, drawing, and writing.- 2. Mathematical objects.- 3. Geometrical reasoning.- 4. The science that studies shapes: geometry.- 5. "History of geometry" and "elements" of geometry.- 5 Geometry and Convention.- 1. Crude facts, relations, conventions.- 2. Pure and applied geometry.- 3. Sensible, geometric, and physical space.- 4. Geometrical intuition.- 5. Geometrical apriorism and empiricism.- 6. The genesis of geometry.- 7. The interchangeability of geometries.- 8. Withdrawing conventions.- 9. Withdrawing principles of coordination.- 6 Geometry, Problem Solving, Abduction.- 1. Geometrical constructions and problem solving.- 2. Model-based and manipulative abduction.- 3. Geometrical construction is a kind of manipulative abduction.- 4. Diagrams, abduction, and deductive reasoning.- 7 Geometry and Cognition.- 1. Geometry of visibles, protogeometry, manipulations.- 2. At the origins of geometrical knowledge II.- 3. Non-conceptual and spatial abilities.- 4. Computational geometricalconstructions.- 5. Spatial imagery.- 6. Logical models of diagrammatic reasoning.- References.- Author Index.
Summary
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 Poincaré), 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.
