Mathematics: Powerful Patterns in Nature and Society

189 Pages·2007·3.7 MB·English

by  Harry Henderson

Milestones in Discovery and Invention
c
M
ATHEMATICS
POWERFUL PATTERNS IN
NATURE AND SOCIETY
Harry Henderson
MATHEMATICS: Powerful Patterns in Nature and Society
Copyright © 2007 by Harry Henderson
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Chelsea House
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New York NY 10001
ISBN-10: 0-8160-5750-8
ISBN-13: 978-0-8160-5750-4
Library of Congress Cataloging-in-Publication Data
Henderson, Harry, 1951–
Mathematics : powerful patterns in nature and society / Harry Henderson.
p. cm. — (Milestones in discovery and invention)
Includes bibliographical reference and index.
ISBN 0-8160-5750-8
1. Mathematics. 2. Mathematics—History. 3. Mathematics—Social aspects. I.
Title.
QA21.H46 2007
510—dc22 2006024680
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Text design by James Scotto-Lavino
Cover design by Dorothy M. Preston
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Printed in the United States of America
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This book is printed on acid-free paper.
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CONTENTS
PREFACE ix
ACKNOWLEDGMENTS xiii
INTRODUCTION xv
1
HOW NATURE COUNTS: LEONARDO OF
PISA DISCOVERS FIBONACCI NUMBERS 1
Those Useful Arabic Numerals 2
Practical Mathematics 3
Reviving Mathematics in Europe 4
Number Theory 4
Other Mathematicians: Arab Mathematicians 5
A Problem with Rabbits 7
Fibonacci Numbers in Nature 8
The Golden Ratio 9
An Inner Harmony? 10
Parallels: Mathematics and Greek Philosophy 11
Leonardo’s Legacy 12
Chronology 12
Further Reading 13
2
TOOLS FOR PATTERN-FINDERS:
KARL PEARSON AND STATISTICS 15
A Roving Mind 15
The Grammar of Science 17
Other Scientists: Francis Galton 18
Statistical Tools 19
Eugenics and the Galton Laboratory 21
Issues: Evaluating Experiments 22
Issues: The Political Use of Science 23
Later Life 24
Chronology 26
Further Reading 26
3
SURMISES AND SIMULATIONS: JOHN VON
NEUMANN PUTS THE COMPUTER IN PLAY 29
Young Genius in a Brilliant City 29
Finding a Career 31
Contributions to Physics and Mathematics 31
Economics and Game Theory 32
Issues: Von Neumann and “The Bomb” 33
Need for Speed 34
Designing the Digital Computer 35
The Self-Reproducing Automaton 37
A Difficult Fate 40
Chronology 40
Other Scientists: Stanisław Ulam 41
Further Reading 42
4
A DELICATE EQUILIBRIUM:
JOHN NASH AND GAME THEORY 44
A “Different” Child 44
A Wayward Path to Math 46
Life at Princeton 47
Game Theory and the Nash Equilibrium 48
Falling Short 50
I Was There: Infuriating but So Brilliant 51
Breakdown 51
In the Shadowlands 52
Connections: Math and Madness? 53
The Long Climb Back 54
Nobel Triumph 54
Chronology 56
Further Reading 57
5
ENDLESS STRUCTURE: BENOÎT
MANDELBROT OPENS THE FRACTAL PORTAL 58
Learning in the Shadow of War 58
A Different Mathematical Path 60
Mysterious Clumps 61
Chaos and Fractals 64
The Mandelbrot Set 64
Finding Applications 67
Connections: Generating and Applying Fractals 67
Other Scientists: Christopher Scholz 68
Popularizing Fractals 69
I Was There: Ego or Necessity? 70
Achievements 71
Chronology 71
Further Reading 72
6
ON BUTTERFLY WINGS:
EDWARD LORENZ AND CHAOS THEORY 74
Clouds and Calculations 74
Forecasting and Meteorology 75
A New Approach to Weather 76
A Berserk Computer? 77
The Limits of Forecasting 78
Trends: Weather Forecasting Today 79
Demonstrations of Chaos 79
Strange Attractors 82
Other Scientists: Mitchell Feigenbaum 82
Chaos: A New Paradigm? 83
Achieving Recognition 86
Issues: Theories and Fads 87
Chronology 88
Further Reading 88
7
GAMES OF EMERGENCE: JOHN H.
CONWAY, “LIFE,” AND OTHER PASTIMES 90
“I Want to Be a Mathematician” 90
Cambridge and “Surreal Numbers” 92
Packed in 24 Dimensions 92
Puzzles and Pastimes 93
The Game of Life 94
Life without End? 97
Seriously Weird? 100
Other Mathematicians: Martin Gardner 100
Other Mathematicians: William Gosper 101
Chronology 103
Further Reading 103
8
FROM COSMOS TO MIND: ROGER PENROSE
SUGGESTS HIDDEN CONNECTIONS 105
A Talented Family 105
Turning to Mathematics 107
Mathematical Physics 107
Black Holes and Hawking 108
Twisted Space and Tricky Tiles 109
Other Scientists: Stephen Hawking 109
The Physics of Consciousness 110
Connections: Unifying Relativity
and Quantum Mechanics 112
Is the Mind a Quantum Computer? 113
Issues: Penrose and His Critics 114
Major Achievements 115
Chronology 116
Further Reading 117
9
ARTIFICIAL EVOLUTION: CHRISTOPHER
LANGTON CREATES VIRTUAL LIFE 119
A Dilettante Gets a Computer 120
Discovering “Life” 121
Genetic Programming 123
In Von Neumann’s Footsteps 124
The “Langton Loop” 124
A Field without a Name 126
A Virtual Ant Colony 127
Parallels: Artificial Life and Computer Animation 127
Information Is Life 128
The Los Alamos Conference 129
Connections: Artificial Intelligence and Artificial Life 131
Artificial Life and Consciousness 131
Chronology 133
Further Reading 133
10
A NEW KIND OF SCIENCE? STEPHEN
WOLFRAM AND THE UNIVERSAL AUTOMATON 135
Teenage Physicist 136
Computers at Caltech 137
Studying Cellular Automata 139
Mathematica 139
A Shortcut to Complexity? 140
Connections: Applying Wolfram’s Ideas 142
Universal Automaton 142
A New Kind of Science? 143
Issues: Is Wolfram’s Work Pseudoscience? 146
Assessing Wolfram’s Science 147
Chronology 148
Further Reading 150
CHRONOLOGY 152
GLOSSARY 154
FURTHER RESOURCES 159
INDEX 163

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PREFACE
The Milestones in Discovery and Invention set is based on a simple
but powerful idea—that science and technology are not separate
from people’s daily lives. Rather, they are part of seeking to under-
stand and reshape the world, an activity that virtually defines being
human.
More than a million years ago, the ancestors of modern humans
began to shape stones into tools that helped them compete with the
specialized predators around them. Starting about 35,000 years
ago, the modern type of human, Homo sapiens, also created elabo-
rate cave paintings and finely crafted art objects, showing that tech-
nology had been joined with imagination and language to compose
a new and vibrant world of culture. Humans were not only shaping
their world but representing it in art and thinking about its nature
and meaning.
Technology is a basic part of that culture. The mythologies of
many peoples include a trickster figure, who upsets the settled
order of things and brings forth new creative and destructive pos-
sibilities. In many myths, for instance, a trickster such as the Native
Americans’ Coyote or Raven steals fire from the gods and gives it
to human beings. All technology, whether it harnesses fire, electric-
ity, or the energy locked in the heart of atoms or genes, partakes of
the double-edged gift of the trickster, providing power to both hurt
and heal.
An inventor of technology is often inspired by the discoveries of
scientists. Science as we know it today is younger than technology,
dating back about 500 years to a period called the Renaissance.
During the Renaissance, artists and thinkers began to explore
nature systematically, and the first modern scientists, such as
Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642),
ix