Foundations and Fundamental Concepts of Mathematics

Häftad, Engelska, 2003

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Third edition of popular undergraduate-level text offers overview of historical roots and evolution of several areas of mathematics. Topics include mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, and more. Emphasis on axiomatic procedures. Problems. Solution Suggestions for Selected Problems. Bibliography.

Produktinformation

Utgivningsdatum2003-03-28
Mått155 x 234 x 22 mm
Vikt535 g
FormatHäftad
SpråkEngelska
SerieDover Books on MaTHEMA 1.4tics
Antal sidor368
FörlagDover Publications Inc.
ISBN9780486696096

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Matematik inom Naturvetenskap och teknik

1. Mathematics Before Euclid1.1 The Empirical Nature of pre-Hellenic Mathematics1.2 Induction Versus Deduction1.3 Early Greek Mathematics and the Introduction of Deductive Procedures1.4 Material Axiomatics1.5 The Origin of the Axiomatic MethodProblems2. Euclid's Elements2.1 The Importance and Formal Nature of Euclid's Elements2.2 Aristotle and Proclus on the Axiomatic Method2.3 Euclid's Definitions, Axioms, and Postulates2.4 Some Logical Shortcomings of Euclid's Elements2.5 The End of the Greek Period and the Transition to Modern TimesProblems3. Non-Euclidean Geometry3.1 Euclid's Fifth Postulate3.2 Saccheri and the Reductio ad Absurdum Method3.3 The Work of Lambert and Legendre3.4 The Discovery of Non-Euclidean Geometry3.5 The Consistency and the Significance of Non-Euclidean GeometryProblems4. Hilbert's Grundlagen4.1 The Work of Pasch, Peano, and Pieri4.2 Hilbert's Grundlagen der Geometrie4.3 Poincaré's Model and the Consistency of Lobachevskian Geometry4.4 Analytic Geometry4.5 Projective Geometry and the Principle of DualityProblems5. Algebraic Structure5.1 Emergence of Algebraic Structure5.2 The Liberation of Algebra5.3 Groups5.4 The Significance of Groups in Algebra and Geometry5.5 RelationsProblems6. Formal Axiomatics6.1 Statement of the Modern Axiomatic Method6.2 A Simple Example of a Branch of Pure Mathematics6.3 Properties of Postulate Sets--Equivalence and Consistency6.4 Properties of Postulate Sets--Independence, Completeness, and Categoricalness6.5 Miscellaneous CommentsProblems7. The Real Number System7.1 Significance of the Real Number System for the Foundations of Analysis7.2 The Postulational Approach to the Real Number System7.3 The Natural Numbers and the Principle of Mathematical Induction7.4 The Integers and the Rational Numbers7.5 The Real Numbers and the Complex NumbersProblems8. Sets8.1 Sets and Their Basic Relations and Operations8.2 Boolean Algebra8.3 Sets and the Foundations of Mathematics8.4 Infinite Sets and Transfinite Numbers8.5 Sets and the Fundamental Concepts of MathematicsProblems9. Logic and Philosophy9.1 Symbolic Logic9.2 The Calculus of Propositions9.3 Other Logics9.4 Crises in the Foundations of Mathematics9.5 Philosophies of MathematicsProblemsAppendix 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