Mathematical reasoning : patterns, problems, conjectures, and proofs

596 Pages·2010·3.25 MB·English

by  Raymond S Nickerson

Mathematical
Reasoning

Mathematical
Reasoning
Patterns,
Problems,
Conjectures,
and Proofs
Raymond S. Nickerson
¥
Psychology Press
Taylor & Francis Croup
New York London
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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
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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