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The Mathematics of Behavior

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  • 14 tables
  • Page extent: 356 pages
  • Size: 228 x 152 mm
  • Weight: 0.491 kg

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 (ISBN-13: 9780521615228 | ISBN-10: 0521615224)

The Mathematics of Behavior Cambridge University Press
978-0-521-85012-4 - THE MATHEMATICS OF BEHAVIOR - by Earl Hunt
Frontmatter/Prelims



The Mathematics of Behavior

Mathematical thinking provides a clear, crisp way of defining problems. Our whole technology is based on it. What is less appreciated is that mathematical thinking can also be applied to problems in the social and behavioral sciences. This book illustrates how mathematics can be employed for understanding human and animal behavior, using examples in psychology, sociology, economics, ecology, and even marriage counseling.

Earl Hunt is Professor Emeritus of Psychology at the University of Washington in Seattle. He has written many articles and chapters in contributed volumes and is the past editor of Cognitive Psychology and Journal of Experimental Psychology. His books include Concept Learning: An Information Processing Problem, Experiments in Induction, Artificial Intelligence, and Will We Be Smart Enough?, which won the William James Book Award from the American Psychological Association in 1996. His most recent book is Thoughts on Thought.



The Mathematics of Behavior

EARL HUNT

University of Washington, Seattle




CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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Information on this title: www.cambridge.org/9780521850124

© Cambridge University Press 2007

This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 2007

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Hunt, Earl B.
The mathematics of behavior/Earl Hunt
p. cm.
Includes bibliographical references and index.
ISBN 0 521 85012 6 (hardcover) – ISBN 0 521 61522 4 (pbk.)
1. Psychology – Mathematical models. 2. Social sciences – Mathematical models. I. Title.
BF39.H86 2006
150.1′51–dc22  2005030591

ISBN-13 978-0-521-85012-4 hardback
ISBN-10 0-521-85012-6 hardback

ISBN-13 978-0-521-61522-8 paperback
ISBN-10 0-521-61522-4 paperback


Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.




Contents

Prefacepage ix
1INTRODUCTION1
1.1. What’s in the Book?1
1.2. Some Examples of Formal and Informal Thinking2
1.3. A Bit of History4
1.4. How Big Is the Earth? Eratosthenes’ Solution5
1.5. A Critique of Eratosthenes12
1.6. Applications of Mathematics to Social and Behavioral Issues14
1.7. Statistics16
2APPLYING PROBABILITY THEORY TO PROBLEMS IN SOCIOLOGY AND PSYCHOLOGY18
2.1. Introduction18
2.2. Defining Probability and Probability Measures19
2.3. How Closely Connected Are We?23
2.4. Conscious and Unconscious Memories27
2.5. Some Final Comments32
Appendix 2A. The Basis for Kolmogorov’s Axioms32
Appendix 2B. Some Important Properties of Probability Measures33
3FROM PHYSICS TO PERCEPTION42
3.1. The Psychophysical Problem42
3.2. Weber’s Law44
3.3. Fechner’s Law47
3.4. Stevens’s Scaling Technique: Deriving the Psychophysical Function from Magnitude Estimation53
3.5. Judging Complex Objects61
3.6. A Comment on Measurement65
4WHEN SYSTEMS EVOLVE OVER TIME67
4.1. Systems of Variables67
4.2. Differences and Differentiation68
4.3. Exponential Growth and Decay70
4.4. Numerical Analysis: The Transmission of Jokes and Colds76
4.5. Questions about Modeling81
4.6. Graphical Analysis: The Evolution of War and Peace86
4.7. Making Love, Not War: The Gottman-Murray Model of Marital Interactions96
4.8. Concluding Comments on Modeling Simple Systems101
Appendix 4A. A Proof of the Exponential Growth Equation103
5NON-LINEAR AND CHAOTIC SYSTEMS104
5.1. Continuous Change and Sudden Jumps104
5.2. The Lotka-Volterra Model of Predator and Prey Interactions106
5.3. The Logistic Equation: Introduction and Behavior When k < 1111
5.4. Non-zero Asymptotes and Cycles as k Increases116
5.5. Chaos121
5.6. Chaos and Network Models123
5.7. Closing Comments on Chaos130
6DEFINING RATIONALITY132
6.1. Axiomatic Reasoning132
6.2. Decision Making under Risk133
6.3. The Concept of Utility135
6.4. Von Neumann and Morgenstern’s Axiomatic Approach to Decision Making139
6.5. The Utility of Money143
6.6. A Summary of the Argument148
6.7. Psychological Research on Decision Making151
6.8. The Problem of Voting158
6.9. Definition and Notation161
6.10. Arrow’s Axioms: The Restrictions on Social Welfare Functions162
6.11. Illustration of the Definitions and Concepts for the Three-Person Society164
6.12. A Proof of Arrow’s Theorem166
6.13. Commentary on the Implications of Arrow’s Theorem173
6.14. Summary Comments and Questions About Axiomatic Reasoning174
7HOW TO EVALUATE EVIDENCE176
7.1. The Legacy of Reverend Bayes176
7.2. Bayes’ Theorem178
7.3. Some Numerical Examples180
7.4. Calculating the Odds184
7.5. Some Examples of Signal Detection185
7.6. A Mathematical Formulation of the Signal Detection Problem187
7.7. The Decision Analyst’s Problem191
7.8. A Numerical Example of ROC Analysis199
7.9. Establishing a Criterion203
7.10. Examples207
7.11. Four Challenge Problems213
8MULTIDIMENSIONAL SCALING216
8.1. The Basic Idea216
8.2. Steps and Technique219
8.3. Extensions to Non-geometric Data222
8.4. Extending the Idea to Conceptual Classes223
8.5. Generalizations of Semantic Space Models227
8.6. Qualifications on the Semantic Space Model229
9THE MATHEMATICAL MODELS BEHIND PSYCHOLOGICAL TESTING231
9.1. Introduction231
9.2. A Brief Review of Correlation and Covariance234
9.3. Predicting One Variable from Another: Linear Regression240
9.4. The Single Factor Model: The Case of General Intelligence244
9.5. Multifactor Theories of Intelligence and Personality249
9.6. Geometric and Graphic Interpretations254
9.7. What Sort of Results Are Obtained?255
Appendix 9A. A Matrix Algebra Presentation of Factor Analysis256
10HOW TO KNOW YOU ASKED A GOOD QUESTION259
10.1. The Problem259
10.2. An Illustrative Case: Vocabulary Testing260
10.3. The Basics of Item Response Theory262
10.4. Standardization: Estimating Item and Person Parameters Simultaneously265
10.5. The Application Phase: Adaptive Testing267
10.6. More Complicated IRT Models269
10.7. Mathematics Meets the Social World: Mathematical Issues and Social Relevance272
Appendix 10A. The Adaptive Testing Algorithm274
Appendix 10B. An Exercise in Adaptive Testing275
11THE CONSTRUCTION OF COMPLEXITY277
11.1. Some Grand Themes277
11.2. The Problem of Complexity278
11.3. Cellular Automata Can Create Complicated Constructions281
11.4. Is Capitalism Inherently Unfair? Reconstructing a Simple Market Economy283
11.5. Residential Segregation, Genocide, and the Usefulness of the Police289
11.6. Is This a New Kind of Science?294
12CONNECTIONISM297
12.1. The Brain and the Mind297
12.2. Computation at the Neural Level299
12.3. Computations at the Network Level303
12.4. A Philosophical Aside307
12.5. Connectionist Architectures309
12.6. Simulating a Phenomenon in Visual Recognition: The Interactive Activation Model311
12.7. An Artificial Intelligence Approach to Learning313
12.8. A Biological Approach to Learning: The Hebbian Algorithm319
12.9. The Auto-associator321
12.10. A Final Word324
13L’ENVOI325
References328
Index of Names333
Index of Subjects337



Preface

Many, many years ago, when I was a graduate student at Yale University, I attended Professor Robert Abelson’s seminar on mathematical psychology. This was in the late 1950s, just as mathematical techniques were beginning to hit psychology. Subsequently I met Professor Jacob Marschak, an economist whose work on the economics of information was seminal in the field. After I received my doctorate in 1960 I had the great opportunity to work with Marschak’s group at the University of California, Los Angeles. Marschak set a gold standard for the use of mathematics to support clear, precise thinking. It is now almost 50 years later, near the end of my own career, and I have yet to meet someone whose logic was so clear. I have had the opportunity to see some people come close to Marschak’s standard, both in my own discipline of psychology and in other fields. This book is an attempt to let future students see how our understanding of behaviors, by both humans and non-humans, can be enhanced by mathematical analysis.

Is such a goal realistic today? Many people have deplored the alleged decline in mathematical training among today’s college students. I do not think that that is fair. On an absolute level, students at the major universities arrive far better trained than they were 50 years ago. High school courses in the calculus are common today; they were rare even 25 years ago. It is true that on a comparative basis American students have slipped compared to their peers abroad, but on an absolute basis the better students in all countries are simply better prepared than they used to be. I have set my sights accordingly. This book should be easily accessible to anyone who has a basic understanding of the calculus, and most of the book will not even require that. It will require the ability (the willingness?) to follow a mathematical argument. I hope that the effort will be rewarded. Curious about the mathematics of love? Or how unprejudiced people can produce a segregated society? Read on!

And to those of you on college and university faculties, consider teaching a course that covers topics like this; mathematics used to analyze important issues in our day, or important issues in the history of science. Don’t restrict it to your own discipline; think broadly. I hope you find this book helpful, but if you don’t, get some readings and teach the course anyway. I have been fortunate to teach such a course in the University of Washington Honors Program for the past several years, and the discussions among students pursuing majors from philosophy to biology and engineering have been informative and enjoyable.

No one prepares a book without a great deal of support. I have had it. I thank the Honors Program and, most especially, the students in my classes, for letting me lead the course. I also thank the Psychology Department for letting me lead a predecessor of this course, focusing somewhat more on psychology. Cambridge University Press provided assistance in book preparation that was far superior to that of any other press with which I have ever worked. I thank Regina Paleski for production editing assistance, and I particularly thank Phyllis Berk for a superb job of copyediting a difficult manuscript. I also thank the editor, Philip Laughlin, for his assistance, and in particular for his obtaining very high-quality editorial reviews. Naturally, the people who wrote them are thanked, too! The final review, by Professor Jerome Busemeyer of the University of Indiana, was a model of constructive criticism.

Every author closes with thanks to family … or at least, he should. My wife, Mary Lou Hunt, has supported me in this and all my scholarly work. I could not accomplish any efforts without her loving aid and assistance.

Earl Hunt
Bellevue, Washington, and
Hood Canal, Washington
February 2006



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