The Non-Euclidean Revolution

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.

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.

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.