The Non-Euclidean Revolution

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?.- Euclid's Primitive Terms.- Euclid's Defined Terms (Part 1).- "Sufficient for Each Day Is the Rigor Thereof".- Euclid's Defined Terms (Part 2).- Euclid'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.- ?ant's Distinctions.- Synthetic A Priori Statements.- Geometry as Synthetic A Priori.- ?ant'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 Poseidonios' 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.- Poincaré's Model.- Can We Be Sure Euclidean Geometry Is Consistent?.- Notes.- 8 Geometry and the Story Theory of Truth.- Kant Revisited.- The Luneburg-Blank Theory of Visual Space.- The Diamond Theory in Decline.- The Story Theory of Truth.- Notes.

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.

"Trudeau meets the challenge of reaching a broad audience in clever ways...(The book) is a good addition to our literature on non-Euclidean geometry and it is recommended for the undergraduate library."

--Choice, February 1988

"...the author, in this remarkable book, describes in an incomparable way the fascinating path taken by the geometry of the plane in its historical evolution from antiquity up to the discovery of non-Euclidean geometry. This 'non-Euclidean revolution', in all its aspects, is described very strikingly here...Many illustrations and some amusing sketches complement the very vividly written text."

--Mathematical Reviews