Bigger than Chaos Copyright © 2003 The President and Fellows of Harvard College1 Copyright © 2003 The President and Fellows of Harvard College2 Bigger than Chaos Understanding Complexity through Probability z michael strevens Harvard University Press Cambridge, Massachusetts London, England 2003 Copyright © 2003 The President and Fellows of Harvard College3 Copyright © 2003 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress Cataloging-in-Publication Data Strevens, Michael. Bigger than chaos : understanding complexity through probability / Michael Strevens. p. cm. Includes bibliographical references and index. ISBN 0-674-01042-6 (alk. paper) 1. Probabilities. I. Title. QC174.85.P76 S77 2003 003—dc21 2002192237 Copyright © 2003 The President and Fellows of Harvard College4 To Joy Copyright © 2003 The President and Fellows of Harvard College5 Copyright © 2003 The President and Fellows of Harvard College6 Acknowledgments It has been eleven years since Barry Loewer, in response to my very first gradu- ate school paper on probabilistic explanation in biological and social systems, said, “Yes, but where do the probabilities come from?” Thanks to Barry for raising the question and for much subsequent aid and encouragement. Thanks also to the other people who have provided helpful comments in the course of this project, in particular David Albert, Joy Connolly, Persi Diaconis, Pe- ter Godfrey-Smith, Alan H ´ ajek, Tim Maudlin, Ken Reisman, and anonymous readers for Harvard University Press. Copyright © 2003 The President and Fellows of Harvard College7 Copyright © 2003 The President and Fellows of Harvard College8 Contents Note to the Reader xiii 1 The Simple Behavior of Complex Systems 1 1.1 Simplicity in Complex Systems 2 1.2 Enion Probability Analysis 12 1.3 Towards an Understanding of Enion Probabilities 27 2 The Physics of Complex Probability 38 2.1 Complex Probability Quantified 39 2.2 Microconstant Probability 47 2.3 The Interpretation of IC-Variable Distributions 70 2.4 Probabilistic Networks 73 2.5 Standard IC-Variables 81 2.6 Complex Probability and Probabilistic Laws 96 2.7 Effective and Critical IC-Values 101 2.A The Method of Arbitrary Functions 118 2.B More on the Tossed Coin 122 2.C Proofs 127 3 The Independence of Complex Probabilities 139 3.1 Stochastic Independence and Selection Rules 140 3.2 Probabilities of Composite Events 141 3.3 Causal Independence 145 3.4 Microconstancy and Independence 150 3.5 The Probabilistic Patterns Explained 161 3.6 Causally Coupled Experiments 163 3.7 Chains of Linked IC-Values 178 3.A Conditional Probability 213 3.B Proofs 214 Copyright © 2003 The President and Fellows of Harvard College9 x Contents 4 The Simple Behavior of Complex Systems Explained 249 4.1 Representing Complex Systems 250 4.2 Enion Probabilities and Their Experiments 251 4.3 The Structure of Microdynamics 253 4.4 Microconstancy and Independence of Enion Probabilities 263 4.5 Independence of Microdynamic Probabilities 275 4.6 Aggregation of Enion Probabilities 286 4.7 Grand Conditions for Simple Macrolevel Behavior 292 4.8 Statistical Physics 293 4.9 Population Ecology 319 5 Implications for the Philosophy of the Higher-Level Sciences 333 5.1 Reduction 333 5.2 Higher-Level Laws 339 5.3 Causal Relevance 346 5.4 The Social Sciences 351 5.5 The Mathematics of Complex Systems 355 5.6 Are There Simple Probabilities? 357 Notes 363 Glossary 387 References 397 Index 403 Copyright © 2003 The President and Fellows of Harvard College10 [...]... explaining large-scale regularities, probability is relegated to a subsidiary role The stability of statistics is put down to some non-probabilistic cause; the role of probability is only to describe fluctuations from the ordained rate of occurrence of a given event Probability governs short-term disorder then, but does not—by contrast with Poisson’s view—play a positive role in producing long-term order... be found on the Bigger than Chaos website, at www.stanford.edu/ ~strevens/ bigger References to the website are of this form: see website section 3.6B For the most part, the book is designed to be read from beginning to end Several notions, however, are introduced some time before they are put to use Examples include degrees of microconstancy (section 2.23) and effective and critical ic-values (section... often behave in simple ways It is almost as if there is something about low-level complexity and chaos itself that is responsible for high-level simplicity What could that something be? That is the subject of this book The key to understanding the simplicity of the behavior of many, perhaps all, complex systems, I will propose, is probability More exactly, the key is to understand the foundations of a... colliding coins: Non-linear coupling 172 3.7 Effective ic-values for chained trials on straight wheel 182 3.8 Multi-spin experiments on a straight wheel I 191 3.9 Multi-spin experiments on a straight wheel II 192 Copyright © 2003 The President and Fellows of Harvard College11 xii Figures 3.10 Restricted ic-evolution function 198 3.11 Creating macroperiodicity from uniformity 203 3.12 Multi-spin experiments... enion probability analysis, or epa, and that is exemplified by, among other theories, the kinetic theory of gases and population genetics It is not enough simply to master epa itself, as epa makes probabilistic assumptions about the dynamics of complex systems that beg the most important questions about the ways in which low-level complexity gives rise to high-level simplicity What is required is an understanding. ..Figures 2.1 ic-density function 41 2.2 Complex probability 44 2.3 Evolution function for wheel of fortune 50 2.4 Wheel of fortune with two different croupiers 51 2.5 Evolution function for tossed coin 63 2.6 Asymmetric eliminability 69 2.7 Probability flow in traffic 74 2.8 Probabilistic network 75 2.9 Loop founded by raw ic-density 78 2.10 True probability 80 2.11 Gerrymandered evolution... fact of the simplicity itself I want to begin by creating, or re-creating, a sense of wonder at the phenomenon of simplicity emerging from complexity Along the way, I pose, and try to answer, a number of questions: How widespread is simple behavior? What is simple behavior? What is a complex system? Why should probability play a role in understanding the behavior of complex systems? Most important of... somewhat simple behaviors: quasi-periodic behavior, in which there is an irregular cycle, as in the case of El Ni˜ o’s three- to ten-year cycle; and general trends, such as insular pygmyism/ n gigantism in mammals, or the linguistic rule that peripheries conserve while centers innovate.2 (For more on the relation between particular laws and general trends, see section 5.24.) Rather than cataloguing various... inventing a name for a distinction that I do not, from this point on, discuss, I simply reserve the term complex for the particular kinds of systems with which this study is concerned 1.14 Understanding Complexity through Probability: Early Approaches The notion inspiring this book, that laws governing complex systems might owe their simplicity to some probabilistic element of the systems’ underlying dynamics,... variable 93 2.14 Effective and critical parts of an ic-value 112 2.15 Ordinal scheme for assigning critical ic-values 113 2.16 Teleological scheme for assigning critical ic-values 115 3.1 Independence in two simple networks 146 3.2 Independence in two slightly less simple networks 146 3.3 Microconstant composite evolution function 151 3.4 Outcome map for non-colliding coins 166 3.5 Outcome map for colliding . Cataloging-in-Publication Data Strevens, Michael. Bigger than chaos : understanding complexity through probability / Michael Strevens. p. cm. Includes bibliographical references and index. ISBN 0-6 7 4-0 104 2-6 . Bigger than Chaos Copyright © 2003 The President and Fellows of Harvard College1 Copyright © 2003 The President and Fellows of Harvard College2 Bigger than Chaos Understanding Complexity through. indicated by underlin- ing, like so. 1 Some extended discussions of points raised in this book can be found on the Bigger than Chaos website, at www.stanford.edu/ ~strevens/ bigger. References to