INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter free, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Infonnadon Company 300 North Zeeb Road, Ann Aibor MI 48106-1346 USA 313/761-4700 800/521-0600 DEFENDING CHAOS: AN EXAMINATION AND DEFENSE OF THE MODELS USED IN CHAOS THEORY DISSERTATION Presented in Partial Fulfillment o f the Requirements for the Degree of Doctor o f Philosophy in the Graduate School of The Ohio State University by Jeffrey David Koperski, B.E.E., M.A. The Ohio State University 1997 Dissertation Committee: Approved by Professor Robert Batterman, advisor Professor Mark Wilson Professor Diana Raffman Advisor Department of Philosophy UMI Number: 9731654 Copyright 19 9 7 by Koperski, Jeffrey David All rights reserved. UMI Microform 9731654 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Copyright by Jeffrey D. Koperski 1997 ABSTRACT The indispensable role of models in science has long been recognized by philosophers. In contemporary dynamics, the models are often simply sets of equations. Bridging the gap between pure mathematics and real-world phenomenon is especially difficult when the model is chaotic. I address the charge that this bridge has not, in fact, been built and that chaos remains “just math.” Although the problems discussed have become acute with the rise of modem chaos theory, their roots were recognized nearly a century ago by Pierre Duhem. The skeptical attacks are both foundational and epistemic. Targeting the foundations of chaos theory, the skeptic claims that the (fractal) geometry instantiated by some chaotic models cannot be representative of real-world processes. Specifically, mathematical fractals are said to have more complexity than can be captured in a material world o f discrete atoms. Furthermore, there is what physicists call “the problem of quantum chaos:” quantum mechanics appears to forbid full-blown chaotic evolutions. I argue that the counterfactual nature o f chaotic models can be used to resolve these tensions. Moreover, I show that the skeptic cannot attack the foundations of chaos theory without calling into question unproblematic areas of mathematical science. The epistemological problems are driven by an essential property of chaotic dynamics, viz., sensitive dependence on Initial conditions. This sensitivity confounds the confirmation schemes familiar to philosophers by precluding future predictions of the state o f a system. The skeptic argues that without such predictions, the models ii cannot be tested. I show how researchers in chaos theory can overcome these skeptical objections via a recently developed method of model construction. This new method not only yields a decisive answer to the skeptic, but reveals a lacuna in the philosophical literature on models in science. I present a new taxonomy o f models that incorporates these recent discoveries and conforms more closely to contemporary scientific practice than do previous classifications. HI To my wife, Marie IV ACKNOWLEDGMENTS My primary indebtedness is to my teachers, starting with J.P. Moreland and ending with those who directed me through my doctoral studies: Robert Batterman, Mark Wilson, and Ronald Laymon. Special thanks to Robert for his guidance and support. Thanks also to Diana Raffman, Robert Kraut, Stuart Sh^iro, and Philip West. Thanks to Peter Smith for being a gracious target and for his many helpful suggestions. I am also deeply grateful to my family. My great grandmother, Dorothy Miller, did not live to see this monograph, but helped make graduate school possible. My mother, Karen, was also an important source of support and encouragement. To my wife, Marie, thank you for your love and endurance. Your presence and patience have been far more important to this work than anyone will ever know. To my little boy, Andrew, thanks for letting me off the hook so often. Go find your ball, son, your papa’s ready to play. VITA February 25, 1965 ................................. Born - Dayton, Ohio 1987 ....................................................... B.E.E. Electrical Engineering, University of Dayton 1991.......................................................M.A. Christian Thought (Philosophy), Liberty University 1991.......................................................M.A. Philosophy, Ohio State University 1991 - present .....................................Graduate Teaching Associate, Ohio State University FIELDS OF STUDY Major Field: Philosophy VI TABLE OF CONTENTS Page Abstract................................................................................................................................. ii Dedication ............................................................................................................................. iv Acknowledgments.................................................................................................................v V ita ........................................................................................................................................ vi List of Figures ...................................................................................................................... x Chapters: CHAPTER 1 INTRODUCTION......................................................................................... 1 1. Characteristics of Chaos ...............................................................................................2 2. The Skeptic’s Case: Foundational and Epistemological .............................................6 3. Some Historical Perspective: Poincaré and Duhem .................................................. 10 4. Overview ...................................................................................................................... 12 CHAPTER 2 MODELING CHAOS.................................................................................... 15 1. Physical Models, Mathematical Models, and State Spaces ...................................... 16 A. Physical Models ......................................................................................................... 16 B. Mathematical M odels ...............................................................................................21 C. State Spaces and Phase Portraits ............................................................................ 25 2. Models of Data ............................................................................................................ 30 3. Artefacts...................................................................................................................... 36 4. Catastrophe Theory ..................................................................................................... 38 A. Applied Catastrophe Theory (ACT) .......................................................................39 B. The Critics................................................................................................................ 43 C. Ad hocness ............................................................................................................... 44 Vll CHAPTER 3 FRACTALS, ATTRACTORS, AND ARTEFACTS.................................. 48 1. Chaos, Fractals, and Fine Structure ........................................................................... 54 A. Fractals and Strange Attractors ...............................................................................57 B. Fractal or Fractal-like? ............................................................................................ 63 C. A More Careful Skeptic ...........................................................................................70 D. Problems with the Paradigm Cases ........................................................................ 74 2. Lessons from Continuum M echanics ........................................................................ 76 A. Underlying physics and conflict .............................................................................78 B. Surplus structure added vs. detail ignored ............................................................. 83 C. Straw man?..............................................................................................................