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Mathematics Before Euclid | |
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The Empirical Nature of pre-Hellenic Mathematics | |
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Induction Versus Deduction | |
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Early Greek Mathematics and the Introduction of Deductive Procedures | |
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Material Axiomatics | |
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The Origin of the Axiomatic Method Problems | |
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Euclid's Elements | |
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The Importance and Formal Nature of Euclid's Elements | |
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Aristotle and Proclus on the Axiomatic Method | |
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Euclid's Definitions, Axioms, and Postulates | |
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Some Logical Shortcomings of Euclid's Elements | |
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The End of the Greek Period and the Transition to Modern Times Problems | |
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Non-Euclidean Geometry | |
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Euclid's Fifth Postulate | |
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Saccheri and the Reductio ad Absurdum Method | |
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The Work of Lambert and Legendre | |
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The Discovery of Non-Euclidean Geometry | |
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The Consistency and the Significance of Non-Euclidean Geometry Problems | |
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Hilbert's Grundlagen | |
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The Work of Pasch, Peano, and Pieri | |
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Hilbert's Grundlagen der Geometrie | |
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Poincar�'s Model and the Consistency of Lobachevskian Geometry | |
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Analytic Geometry | |
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Projective Geometry and the Principle of Duality Problems | |
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Algebraic Structure | |
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Emergence of Algebraic Structure | |
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The Liberation of Algebra | |
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Groups | |
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The Significance of Groups in Algebra and Geometry | |
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Relations Problems | |
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Formal Axiomatics | |
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Statement of the Modern Axiomatic Method | |
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A Simple Example of a Branch of Pure Mathematics | |
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Properties of Postulate Sets--Equivalence and Consistency | |
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Properties of Postulate Sets--Independence, Completeness, and Categoricalness | |
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Miscellaneous Comments Problems | |
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The Real Number System | |
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Significance of the Real Number System for the Foundations of Analysis | |
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The Postulational Approach to the Real Number System | |
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The Natural Numbers and the Principle of Mathematical Induction | |
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The Integers and the Rational Numbers | |
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The Real Numbers and the Complex Numbers Problems | |
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Sets | |
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Sets and Their Basic Relations and Operations | |
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Boolean Algebra | |
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Sets and the Foundations of Mathematics | |
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Infinite Sets and Transfinite Numbers | |
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Sets and the Fundamental Concepts of Mathematics Problems | |
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Logic and Philosophy | |
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Symbolic Logic | |
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The Calculus of Propositions | |
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Other Logics | |
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Crises in the Foundations of Mathematics | |
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Philosophies of Mathematics Problems | |
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The First Twenty-Eight Propositions of Euclid | |
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Euclidean Constructions | |
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Removal of Some Redundancies | |
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Membership Tables | |
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A Constructive Proof of the Existence of Transcendental Numbers | |
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The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics | |
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Nonstandard Analysis | |
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The Axiom of Choice | |
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A Note on G�del's Incompleteness Theorem Bibliography | |
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Solution Suggestions for Selected Problems | |
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Index | |