1. Mathematics Before Euclid


1.1 The Empirical Nature of preHellenic 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. NonEuclidean 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 NonEuclidean Geometry 

3.5 The Consistency and the Significance of NonEuclidean 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 SetsEquivalence and Consistency 

6.4 Properties of Postulate SetsIndependence, 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 TwentyEight 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 

