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AuthorTrudeau, Richard J. author
TitleThe Non-Euclidean Revolution [electronic resource] : With an Introduction by H.S.M Coxeter / by Richard J. Trudeau
ImprintBoston, MA : Birkhรคuser Boston, 2001
Connect tohttp://dx.doi.org/10.1007/978-1-4612-2102-9
Descript XIV, 270 p. online resource

SUMMARY

How unique and definitive is Euclidean geometry in describing the "real" space in which we live? Richard Trudeau confronts the fundamental question of truth and its representation through mathematical models in The Non-Euclidean Revolution. First, the author analyzes geometry in its historical and philosophical setting; second, he examines a revolution every bit as significant as the Copernican revolution in astronomy and the Darwinian revolution in biology; third, on the most speculative level, he questions the possibility of absolute knowledge of the world. Trudeau writes in a lively, entertaining, and highly accessible style. His book provides one of the most stimulating and personal presentations of a struggle with the nature of truth in mathematics and the physical world. A portion of the book won the Pรณlya Prize, a distinguished award from the Mathematical Association of America


CONTENT

1 First Things -- The Origin of Deductive Geometry -- ?aterial Axiomatic Systems -- Logic -- Proofs -- A Simple Example of a ?aterial Axiomatic System -- Exercises -- Notes -- 2 Euclidean Geometry -- ?ow ?ig Is a Point? -- Euclidโ{128}{153}s Primitive Terms -- Euclidโ{128}{153}s Defined Terms (Part 1) -- โ{128}{156}Sufficient for Each Day Is the Rigor Thereofโ{128}{157} -- Euclidโ{128}{153}s Defined Terms (Part 2) -- Euclidโ{128}{153}s Axiorns -- Theorems Proven Without Postulate 5 -- Theorems Proven With Postulate 5 -- Index to Euclidean Geometry -- Exercises -- Notes -- 3 Geometry and the Diamond Theory of Truth -- ?antโ{128}{153}s Distinctions -- Synthetic A Priori Statements -- Geometry as Synthetic A Priori -- ?antโ{128}{153}s Doctrine of Space -- The Diamond Theory of Truth -- Notes -- 4 The Problem With Postulate 5 -- Poseidonios -- Proof of Postulate 5, After Poseidonios -- Metageometry -- Evaluation of Poseidoniosโ{128}{153} Reorganization -- Overview of Later Attempts -- So Near -- An Experimental Test of Postulate 5 -- Exercises -- Notes -- 5 The Possibility of Non-Euclidean Geometry -- The Logical Possibility of Non-Euclidean Geometry -- The Founders of Non-Euclidean Geometry -- The Psychological Impossibility of Non-Euclidean Geometry -- Formal Axiomatic Systems -- A Simple Example of a Formal Axiomatic System -- How to Not Let the Pictures Bother You -- Exercise -- Notes -- 6 Hyperbolic Geometry -- Hyperbolic Geometry (Part 1) -- Reconciliation With Common Sense -- Hyperbolic Geometry (Part 2) -- Glimpses -- Exercises -- Notes -- 7 Consistency -- Models -- Poincarรฉโ{128}{153}s Model -- Can We Be Sure Euclidean Geometry Is Consistent? -- Notes -- 8 Geometry and the Story Theory of Truth -- Kant Revisited -- The Luneburgโ{128}{148}Blank Theory of Visual Space -- The Diamond Theory in Decline -- The Story Theory of Truth -- Notes


Mathematics Geometry History Mathematics Geometry Mathematics general History of Mathematical Sciences



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