Foundations and Fundamental Concepts of Mathematics

This third edition of a popular, well-received text offers undergraduates an opportunity to obtain an overview of the historical roots and the evolution of several areas of mathematics.The selection of topics conveys not only their role in this historical development of mathematics but also their value as bases for understanding the changing nature of mathematics. Among the topics covered in this wide-ranging text are: mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, the real numbers system, sets, logic and philosophy and more. The emphasis on axiomatic procedures provides important background for studying and applying more advanced topics, while the inclusion of the historical roots of both algebra and geometry provides essential information for prospective teachers of school mathematics.The readable style and sets of challenging exercises from the popular earlier editions have been continued and extended in the present edition, making this a very welcome and useful version of a classic treatment of the foundations of mathematics. "A truly satisfying book." ? Dr. Bruce E. Meserve, Professor Emeritus, University of Vermont.

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 Problems2. 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 Problems3. 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 Problems4. 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 Problems5. 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 Problems6. 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 Problems7. 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 Problems8. 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 Problems9. 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 ProblemsAppendix 1. The First Twenty-Eight Propositions of EuclidAppendix 2. Euclidean ConstructionsAppendix 3. Removal of Some RedundanciesAppendix 4. Membership TablesAppendix 5. A Constructive Proof of the Existence of Transcendental NumbersAppendix 6. The Eudoxian Resolution of the First Crisis in the Foundations of MathematicsAppendix 7. Nonstandard AnalysisAppendix 8. The Axiom of ChoiceAppendix 9. A Note on Gödel's Incompleteness TheoremBibliography; Solution Suggestions for Selected Problems; Index