Foundations and Fundamental Concepts of Mathematics

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