Mathematical Proof and the Principles of Mathematics/Introduction/Objectives and prerequisites

This is a more detailed look at what a prospective reader will need to understand this book and what topics will be covered.

Prerequisites

The material in this book represents a break from the kind of computational mathematics you may have been exposed to previously. So there aren't many mathematical facts that will be required. But this knowledge will be helpful in that it will be used in some of the examples given. You will need to be familiar with unknowns and varables as used in algebra and calculus, and be familiar with the concept of functions. Perhaps most importantly, you should not be daunted by mathematical symbols and expressions; they are the vocabulary of the language of mathematics.

Objectives

After reading this book and completing exercises the reader should:

• Appreciate the reasons for rigor in mathematics and understand the advantages of proof over intuition and inductive reasoning.
• Understand the axiomatic method and how if differs with the scientific method.
• Understand and be able to work with commonly used constructions in mathematics.
• Be proficient at constructing simple proofs of mathematical statements.
• Be able to construct more complex proofs given some hints.
• Be proficient at understanding mathematical texts.
• Be familiar with problem solving techniques as applied to mathematics.
• Appreciate the scope of modern mathematics.
• Be familiar with the fundamental concepts of the different areas of mathematics.
• Appreciate the aesthetics of mathematics.
• Be familiar with some of the lore of mathematics: famous names, unsolved problems, historical anecdotes, famous quotes etc.