Foundations and Fundamental Concepts of Mathematics

ISBN-10: 048669609X

ISBN-13: 9780486696096

Edition: 3rd 1997 (Revised)

Authors: Howard W. Eves

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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, more. Emphasis on axiomatic procedures. Problems. Solution Suggestions for Selected Problems. Bibliography.
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Book details

List price: $16.95
Edition: 3rd
Copyright year: 1997
Publisher: Dover Publications, Incorporated
Publication date: 5/20/1997
Binding: Paperback
Pages: 368
Size: 6.50" wide x 9.25" long x 0.75" tall
Weight: 0.990
Language: English

Mathematics Before Euclid
The Empirical Nature of pre-Hellenic Mathematics
Induction Versus Deduction
Early Greek Mathematics and the Introduction of Deductive Procedures
Material Axiomatics
The Origin of the Axiomatic Method Problems
Euclid's Elements
The Importance and Formal Nature of Euclid's Elements
Aristotle and Proclus on the Axiomatic Method
Euclid's Definitions, Axioms, and Postulates
Some Logical Shortcomings of Euclid's Elements
The End of the Greek Period and the Transition to Modern Times Problems
Non-Euclidean Geometry
Euclid's Fifth Postulate
Saccheri and the Reductio ad Absurdum Method
The Work of Lambert and Legendre
The Discovery of Non-Euclidean Geometry
The Consistency and the Significance of Non-Euclidean Geometry Problems
Hilbert's Grundlagen
The Work of Pasch, Peano, and Pieri
Hilbert's Grundlagen der Geometrie
Poincar�'s Model and the Consistency of Lobachevskian Geometry
Analytic Geometry
Projective Geometry and the Principle of Duality Problems
Algebraic Structure
Emergence of Algebraic Structure
The Liberation of Algebra
The Significance of Groups in Algebra and Geometry
Relations Problems
Formal Axiomatics
Statement of the Modern Axiomatic Method
A Simple Example of a Branch of Pure Mathematics
Properties of Postulate Sets--Equivalence and Consistency
Properties of Postulate Sets--Independence, Completeness, and Categoricalness
Miscellaneous Comments Problems
The Real Number System
Significance of the Real Number System for the Foundations of Analysis
The Postulational Approach to the Real Number System
The Natural Numbers and the Principle of Mathematical Induction
The Integers and the Rational Numbers
The Real Numbers and the Complex Numbers Problems
Sets and Their Basic Relations and Operations
Boolean Algebra
Sets and the Foundations of Mathematics
Infinite Sets and Transfinite Numbers
Sets and the Fundamental Concepts of Mathematics Problems
Logic and Philosophy
Symbolic Logic
The Calculus of Propositions
Other Logics
Crises in the Foundations of Mathematics
Philosophies of Mathematics Problems
The First Twenty-Eight Propositions of Euclid
Euclidean Constructions
Removal of Some Redundancies
Membership Tables
A Constructive Proof of the Existence of Transcendental Numbers
The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics
Nonstandard Analysis
The Axiom of Choice
A Note on G�del's Incompleteness Theorem Bibliography
Solution Suggestions for Selected Problems
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