Mathematical Proof and the Principles of Mathematics

This wikibook aims to teach you the elements of pure mathematics in a self-contained, accessible style. The main objective is to introduce the reader to material usually found in an undergraduate course intended for mathematics majors at a university. In fact, many universities offer a course that serves as a transition from calculus to courses which involve more abstract concepts and writing proofs, and this book might serve as a text for such a course. It's also intended for people who are considering mathematics, especially pure mathematics, as a career or serious avocation, and wish to know what to expect as they advance to higher levels. We also intend to introduce the reader to style of proofs and rigor needed to read and write mathematical literature. The material is covered in greater detail and more rigorously than you may be used to. In fact much of the material will already be familiar, though the approach to it may not.

In the experience of most people, mathematics consists mostly of the mechanical application of rules of computation at various levels: arithmetic, solving equations, finding derivatives and integrals. But for a mathematician, mathematics is a process for discovering and establishing truths. It requires an analytical mind, but also a certain amount of creativity and intuition. It can also be, as we hope you'll discover through this book, very rewarding.

Table of Contents

 * Introduction
 * The reason this book was written
 * Objectives and prerequisites
 * A history of mathematical rigor
 * Euclid
 * After Euclid, the next two thousand years
 * The problem of parallels
 * Infrastructure
 * Four schools of thought
 * Preliminaries
 * What is mathematics?
 * Mathematical Statements
 * Mathematical proof
 * Proof anatomy
 * /Logic/
 * Logical connectives
 * Direct proofs for implication
 * Indirect proofs
 * Proofs with conjunction and disjunction
 * Logical equivalence
 * Non-classical logic (optional)
 * Quantifiers and predicates
 * The universal quantifier
 * The existential quantifier
 * Rules of inference summary
 * Axioms and equality
 * /Sets/
 * History
 * Elements and subsets
 * Classes and collectivizing predicates
 * Pairs
 * Union and intersection
 * Classes and foundation
 * Power sets
 * Natural numbers
 * Replacement
 * Axioms for sets
 * Operations on sets
 * Numbers
 * Natural numbers
 * Cardinality and counting
 * Integers
 * Rational numbers
 * Real numbers
 * Complex numbers
 * Abstract number systems
 * The scope of mathematics
 * Algebra
 * Analysis
 * Geometry
 * Number theory
 * Combinatorics