Introduction To Mathematical Thinking

Introduction to Mathematical Thinking Keith Devlin © Keith Devlin, 2012 Publisher: Keith Devlin 331 Poe St, Unit 4 Palo Alto, CA 94301 USA http://profkeithdevlin.com © Keith Devlin, 2012 All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission of the copyright owner. Publication data Devlin, Keith, Introduction to Mathematical Thinking First published, July 2012 ISBN-13: 978-0615653631 ISBN-10: 0615653634 Contents Preface What this book is about 1 What is mathematics? 1.1 More than arithmetic 1.2 Mathematical notation 1.3 Modern college-level mathematics 1.4 Why do you have to learn this stuff? 2 Getting precise about language 2.1 Mathematical statements 2.2 The logical combinators and, or, and not 2.3 Implication 2.4 Quantifiers 3 Proofs 3.1 What is a proof? 3.2 Proof by contradiction 3.3 Proving conditionals 3.4 Proving quantified statements 3.5 Induction proofs 4 Proving results about numbers 4.1 The integers 4.2 The real numbers 4.3 Completeness 4.4 Sequences APPENDIX: Set theory Index Preface Many students encounter difficulty going from high school math to collegelevel mathematics. Even if they do well at math in school, most students are knocked off course for a while by the shift in emphasis from the K-12 focus on mastering procedures to the “mathematical thinking” characteristic of much university mathematics. Though the majority survive the transition, many do not, and leave mathematics for some other major (possibly outside the sciences or other mathematically-dependent subjects). To help incoming students make the shift, colleges and universities often have a “transition course.” This short book is written to accompany such a course, but it is not a traditional “transition textbook.” Rather than give beginning college students (and advanced high school seniors) a crash course in mathematical logic, formal proofs, some set theory, and a bit of elementary number theory and elementary real analysis, as is commonly done, I attempt to help students develop that crucial but elusive ability: mathematical thinking. This is not the same as “doing math,” which usually involves the application of procedures and some heavy-duty symbolic manipulations. Mathematical thinking, by contrast, is a specific way of thinking about things in the world. It does not have to be about mathematics at all, though I would argue that certain parts of mathematics provide the ideal contexts for learning how to think that way, and in this book I will concentrate my attention on those areas. Mathematicians, scientists, and engineers need to “do math.” But for life in the twenty-first century, everyone benefits from being able to think mathematically to some extent. (Mathematical thinking includes logical and analytic thinking as well as quantitative reasoning, all crucial abilities.) This is why I have tried to make this book accessible to anyone who wants or needs to extend and improve their analytic thinking skills. For the student who goes beyond a basic grasp of logical and analytic thinking, and truly masters mathematical thinking, there is a payoff at least equal to those advantages incidental to twenty-first century citizenship: mathematics goes from being confusing, frustrating, and at times seemingly impossible, to making sense and being hard but doable. I developed one of the first college transition courses in the late 1970s, when I was teaching at the University of Lancaster in England, and I wrote one of the first transition textbooks, Sets, Functions and Logic, which was p