Beginning Rigorous Mathematics
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This book is intended to be an introduction to proofs and mathematical reasoning. Many universities offer a course that serves as a transition from calculus to courses which involve proving more general statements and more abstract concepts, 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.
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.
In a sense, no book or instructor can teach you how to do the kind of mathematics we're talking about. You must teach yourself and the only way to learn is by doing. A book can get you started on the right path and provide sign posts on the way, but no one can make the journey for you. For this reason, only a few proofs will be given here as examples, the rest will be left for the reader to fill in.
It is often said that mathematics is a language; some go as far as saying it's the language through which the universe speaks to us. This book attempts to describe the grammar and vocabulary of this language, but no language book would be be complete without some description of the culture of its speakers. So this book will also include something of the history and lore of mathematics: information which is common knowledge among mathematicians but may not be covered in a typical book on mathematics.
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.
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.