1. Mathematics Before Euclid
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1.1 The Empirical Nature of pre-Hellenic Mathematics |
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1.2 Induction Versus Deduction |
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1.3 Early Greek Mathematics and the Introduction of Deductive Procedures |
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1.4 Material Axiomatics |
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1.5 The Origin of the Axiomatic Method |
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Problems |
2. Euclid's Elements |
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2.1 The Importance and Formal Nature of Euclid's Elements |
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2.2 Aristotle and Proclus on the Axiomatic Method |
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2.3 Euclid's Definitions, Axioms, and Postulates |
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2.4 Some Logical Shortcomings of Euclid's Elements |
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2.5 The End of the Greek Period and the Transition to Modern Times |
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Problems |
3. Non-Euclidean Geometry |
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3.1 Euclid's Fifth Postulate |
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3.2 Saccheri and the Reductio ad Absurdum Method |
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3.3 The Work of Lambert and Legendre |
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3.4 The Discovery of Non-Euclidean Geometry |
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3.5 The Consistency and the Significance of Non-Euclidean Geometry |
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Problems |
4. Hilbert's Grundlagen |
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4.1 The Work of Pasch, Peano, and Pieri |
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4.2 Hilbert's Grundlagen der Geometrie |
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4.3 Poincaré's Model and the Consistency of Lobachevskian Geometry |
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4.4 Analytic Geometry |
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4.5 Projective Geometry and the Principle of Duality |
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Problems |
5. Algebraic Structure |
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5.1 Emergence of Algebraic Structure |
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5.2 The Liberation of Algebra |
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5.3 Groups |
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5.4 The Significance of Groups in Algebra and Geometry |
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5.5 Relations |
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Problems |
6. Formal Axiomatics |
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6.1 Statement of the Modern Axiomatic Method |
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6.2 A Simple Example of a Branch of Pure Mathematics |
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6.3 Properties of Postulate Sets--Equivalence and Consistency |
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6.4 Properties of Postulate Sets--Independence, Completeness, and Categoricalness |
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6.5 |
Miscellaneous Comments |
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Problems |
7. The Real Number System |
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7.1 Significance of the Real Number System for the Foundations of Analysis |
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7.2 The Postulational Approach to the Real Number System |
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7.3 The Natural Numbers and the Principle of Mathematical Induction |
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7.4 The Integers and the Rational Numbers |
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7.5 The Real Numbers and the Complex Numbers |
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Problems |
8. Sets |
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8.1 Sets and Their Basic Relations and Operations |
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8.2 Boolean Algebra |
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8.3 Sets and the Foundations of Mathematics |
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8.4 Infinite Sets and Transfinite Numbers |
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8.5 Sets and the Fundamental Concepts of Mathematics |
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Problems |
9. Logic and Philosophy |
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9.1 Symbolic Logic |
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9.2 The Calculus of Propositions |
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9.3 Other Logics |
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9.4 Crises in the Foundations of Mathematics |
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9.5 Philosophies of Mathematics |
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Problems |
Appendix 1. The First Twenty-Eight Propositions of Euclid |
Appendix 2. Euclidean Constructions |
Appendix 3. Removal of Some Redundancies |
Appendix 4. Membership Tables |
Appendix 5. A Constructive Proof of the Existence of Transcendental Numbers |
Appendix 6. The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics |
Appendix 7. Nonstandard Analysis |
Appendix 8. The Axiom of Choice |
Appendix 9. A Note on Gödel's Incompleteness Theorem |
Bibliography; Solution Suggestions for Selected Problems; Index |
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