Foundations and Fundamental... | Buy

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
Poincaré'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
Groups
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
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 Gödel's Incompleteness Theorem Bibliography
Solution Suggestions for Selected Problems
Index
Table of Contents provided by Publisher. All Rights Reserved.

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.