Mathematical Reasoning Mathematical Reasoning Patterns, Problems, Conjectures, and Proofs Raymond S. Nickerson Psychology Press ¥ Taylor & Francis Croup New York London Psychology Press Psychology Press Taylor & Francis Group Taylor & Francis Group 270 Madison Avenue 27 Church Road New York, NY 10016 Hove, East Sussex BN3 2FA © 2010 by Taylor and Francis Group, LLC This edition published in the Taylor & Francis e-Library, 2011. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. Psychology Press is an imprint of Taylor & Francis Group, an Informa business International Standard Book Number: 978-1-84872-827-1 (Hardback) For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organiza- tion that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Nickerson, Raymond S. Mathematical reasoning : patterns, problems, conjectures, and proofs/ Raymond Nickerson. p. cm. Includes bibliographical references and index. ISBN 978-1-84872-827-1 1. Mathematical analysis. 2. Reasoning. 3. Logic, Symbolic and mathematical. 4. Problem solving. I. Title. QA300.N468 2010 510.1’9--dc22 2009021567 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the Psychology Press Web site at http://www.psypress.com ISBN 0-203-84802-0 Master e-book ISBN CONTENTS The Author vii Preface ix 1 What Is Mathematics? 1 2 Counting 17 3 Numbers 41 4 Deduction and Abstraction 79 5 Proofs 97 6 Informal Reasoning in Mathematics 137 7 Representation in Mathematics 161 v vi Contents 8 Infinity 191 9 Infinitesimals 217 10 Predilections, Presumptions, and Personalities 249 11 Esthetics and the Joys of Mathematics 277 12 The Usefulness of Mathematics 319 13 Foundations and the “Stuff” of Mathematics 343 14 Preschool Development of Numerical and Mathematical Skills 373 15 Mathematics in School 391 16 Mathematical Problem Solving 423 17 Final Thoughts 461 References 471 Appendix: Notable (Deceased) Mathematicians, Logicians, Philosophers, and Scientists Mentioned in the Text 555 Author Index 563 Subject Index 575 THE AUTHOR Raymond S. Nickerson is a research professor at Tufts University, from which he received a PhD in experimental psychology, and is retired from Bolt Beranek and Newman Inc. (BBN), where he was a senior vice presi- dent. He is a fellow of the American Association for the Advancement of Science, the American Psychological Association, the Association for Psychological Science, the Human Factors and Ergonomics Society, and the Society of Experimental Psychologists. Dr. Nickerson was the founding editor of the Journal of Experimental Psychology: Applied (American Psychological Association), the founding and first series editor of Reviews of Human Factors and Ergonomics (Human Factors and Ergonomics Society), and is the author of several books. Published titles include: The Teaching of Thinking (with David N. Perkins and Edward E. Smith) Using Computers: Human Factors in Information Systems Reflections on Reasoning Looking Ahead: Human Factors Challenges in a Changing World Psychology and Environmental Change Cognition and Chance: The Psychology of Probabilistic Reasoning Aspects of Rationality: Reflections on What It Means to Be Rational and Whether We Are vii PREFACE What does it means to reason well? Do the characteristics of good rea- soning differ from one context to another? Do engineers reason differ- ently, when they reason well, than do lawyers when they reason well? Do physicians use qualitatively different reasoning principles and skills when attempting to diagnose a medical problem than do auto mechanics when attempting to figure out an automotive malfunction? Is there any- thing about mathematics that makes mathematical reasoning unique, or at least different in principle from the reasoning that, say, nonmath- ematical biologists do? As a psychologist, I find such questions intriguing. I do not address them directly in this book, but mention them to note the context from which my interest in reasoning in mathematics stems. Inasmuch as I am not a mathematician, attempting to write a book on this subject might appear presumptuous—and undoubtedly it is. My excuse is that I wished to learn something about mathematical reasoning and I believe that one way to learn about anything—an especially good one in my view—is to attempt to explain to others, in writing, what one thinks one is learning. But can a nonmathematician hope to understand mathematical reason- ing in a more than superficial way? This is a good question, and the writ- ing of this book represents an attempt to find out if one can. I beg the indulgence of my mathematically sophisticated friends and colleagues if specific attempts at exposition reveal only mathematical naiveté. I count on their kindness to give me some credit for trying. Although much has been written about the importance of the teaching and learning of mathematics at all levels of formal education, and much angst has been expressed about the relatively poor job that is being done in American schools in this regard, especially at the primary and secondary levels, the fact is that mathematics is not of great interest to most people. Hammond (1978) refers to mathematics as an “invisible culture” and raises the question as to what it is in the nature of “this ix x Preface unique human activity that renders it so remote and its practitioners so isolated from popular culture” (p. 15). One conceivable answer is that the fundamental ideas of mathemat- ics are inherently difficult to grasp. Certainly the world of mathematics is populated with objects that are not part of everyday parlance: vectors, tensors, twisters, manifolds, geodesics, and so forth. Even concepts that are relatively familiar can quickly become complex when one begins to explore them; geometry, for example, which we know from high school math deals with properties of such mundane objects as points, lines, and angles, encompasses a host of more esoteric subdisciplines: differential geometry, projective geometry, complex geometry, Lobachevskian geom- etry, Riemannian geometry, and Minkowskian geometry, to name a few. But, even allowing that there are areas of mathematics that are arcane, and will remain so, to most of us, mathematics also offers countless delights and uses for people with only modest mathematical training. I hope I am able to convey in this book some sense of the fascination and pleasure that is to be found even in the exploration of only limited parts of the mathematical domain. Although there are a few significant exceptions, mathematicians generally are notoriously bad about communicating their subject matter to the general public. Why is that the case? Undoubtedly, many math- ematicians are sufficiently busy doing mathematics that they would find an attempt to explain to nonmathematicians what they are doing to be an unwelcome, time-consuming distraction. There is also the possibility that the abstract nature of much of mathematics is extraordinarily dif- ficult to communicate in lay terms. Steen (1978) makes this point and contrasts the “otherworldly vocabulary” of mathematics with the some- what more concrete terms (“molecules, DNA, and even black holes”) that provide chemists, biologists, and physicists with links to material reality with which they can communicate their interests. “In contrast, not even analogy and metaphor are capable of bringing the remote vocabulary of mathematics into the range of normal human experience” (p. 2). Whatever the cause of the paucity of books about the doing of math- ematics or about the nature of mathematical reasoning written by mathema- ticians, we should be especially grateful to those mathematicians who have proved to be exceptions to the rule: G. H. Hardy, Mark Kac, Imre Lakatos, George Polya, and Stanislav Ulam, along with a few contem- porary writers, come quickly to mind. (Hardy wrote about the doing of mathematics only when he considered himself too old to be able to do mathematics effectively, and expressed his disdain for the former activ- ity; nevertheless, his Apology provides many insights into the latter.) I have found the writings of these expositors of mathematical reasoning to be not only especially illuminating, but easy and pleasurable to read. Preface xi I hope in this book to convey a sense of the enriching experience that reflection on mathematics can provide even to those whose mathemati- cal knowledge is not great. I owe thanks to several people who generously read drafts of sections of the book and gave me the benefit of much insightful and helpful feedback. These include Jeffrey Birk, Susan Chipman, Russell Church, Carol DeBold, Francis Durso, Ruma Falk, Carl Feehrer, Samuel Glucksberg, Earl Hunt, Peter Killeen, Thomas Landauer, Duncan Luce, Joseph Psotka, Judah Schwartz, Thomas Sheridan, Richard Shiffrin, Robert Siegler, and William Uttal. I am especially grateful to Neville Moray, who read and commented on the entire manuscript. Stimulating and enlightening conversations on matters of psychology and math with son, Nathan Nickerson, and colleagues at Tufts, especially Susan Butler, Richard Chechile, and Robert Cook, have been most helpful and enjoy- able. Special thanks go also to granddaughters Amara Nickerson for criti- cally reading the chapters on learning math and problem solving from the perspective of a first-year teacher with Teach America, and Laura Traverse for pulling the cited references out of a cumbersome master reference file and catching various grammatical blunders in the manu- script in the process.