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Three Papers on How Physics Bears on Philosophy, and How Philosophy Bears on Physics

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Three Papers on How Physics Bears on Philosophy, and How Philosophy Bears on Physics Three Papers on How Physics Bears on Philosophy, and How Philosophy Bears on Physics Erik Curiel Doctor of Philosophy Department of Philosophy Division of the Humanities JUNE 2005 2 I respectfully and affectionately dedicate this work to Prof. Howard Stein. It is a pleasure to acknowledge and thank the following people. Bob Wald|for giving me a desk in the Relativity Group and kindly treating me like one of his graduate students when he had absolutely no reason to do so other than his interest in supporting all sorts of intellectual investigation, and especially for often proving Bob Geroch wrong in our weekly informal seminars over lunch. Ian Mueller|for exemplifying the spirit of careful scholarship, and for making me realize that sometimes (not often, but sometimes) studying the secondary literature can be almost as rewarding as reading the original text. Arthur Adkins|for being a scholar, a gentleman and an inspiration nonpareil, displaying a stoicism so natural as to seem no burden or affect, but merely the least he could do in living his εὐδαιμόnikoc life, all the time with that twinkle in his eye and not-quite-hidden smile. All the Erics|for never letting me get away with any bullshit while we were plastered at Jimmy's, or anywhere else for that matter. Michelle and Hisham|for that rarest of comradeships, intellectual, social and just plain damn fun. And for having taken me to see \Celine and Julie Go Boating". The whole Taylor clan, and George and Meg in particular|for being the most wonderful second family anyone could ever dare hope for. JDH|for, well, for more than I can well say. Mom|for giving me a well furnished mind. Elpis|for supporting me, believing in me, and reminding me that appreciation of all things physical, spiritual and aesthetic is not an alternative to intellectual contemplation, but its most necessary and inextricable companion. It almost seems silly to, the debt is so great, but, finally, I thank David Malament for his unwavering, supererogatory support and for teaching by example and inspiration what it means to think clearly, carefully and precisely about all matters, both theoretical and practical, all while being the paragon of a mensch; Bob Geroch for betting me beers about the answers to physics problems, and for teaching by example and inspiration what it means to think like a philosopher (though he will not thank me for saying so) while being a mathematician and a physicist without peer; and Howard Stein for never having given me an `A' on any paper I wrote for him, and for teaching by example and inspiration what it means to think like a mathematician and a physicist while being a philosopher without peer. I do not claim to have absorbed any of these lessons; I claim only their attempt and their appreciation. 3 A student repays his teacher very poorly by never disproving him. { Nietzsche, Beyond Good and Evil 4 To make a beginning out of particulars To roll up the sum by defective means. Rigor of beauty is the quest But how will you find it when it is locked away in the mind beyond all remonstrance? William Carlos Williams, Paterson When science starts to be interpretive it is more unscientific even than mysticism. \Self-Protection", D. H. Lawrence Ahhh, it's so hard, ya' know, it's so hard to believe in anything anymore, ya' know what I mean? It's like religion, you can't really take it seriously, because it seems so mythological and it seems so arbitrary, and then on the other hand, science is just pure empiricism, and by virtue of its method it excludes metaphysics. And I guess I wouldn't believe in anything if it weren't for my Lucky Astrology Mood-Watch. Steve Martin, \A Wild and Crazy Guy" How can we lose when we're so sincere? Charlie Brown Contents 1 Pr´ecisof Dissertation1 1.1 The Spirit of the Thing..................................1 1.2 The Constraints General Relativity Places on Physicalist Accounts of Causality...3 1.3 The Analysis of Singular Spacetimes...........................4 1.4 On the Formal Consistency of Experiment and Theory in Physics...........4 2 The Constraints General Relativity Places on Physicalist Accounts of Causality9 2.1 Introduction.........................................9 2.2 Causality and Energy................................... 13 2.3 Causal Relations and Energetic Processes........................ 14 2.4 The Propagation and Transfer of Energy......................... 17 2.5 Energy in General Relativity............................... 25 2.6 Causality after General Relativity............................. 30 3 The Analysis of Singular Spacetimes 33 3.1 Introduction......................................... 33 3.2 Curve Incompleteness................................... 36 3.3 Explosive Curvature Growth along Incomplete Curves................. 38 3.4 Missing Points....................................... 43 3.5 Global vs. Local Properties of a Manifold........................ 49 3.6 The Finitude of Existence................................. 51 4 On the Formal Consistency of Experiment and Theory in Physics 57 4.1 Introduction......................................... 58 4.2 Relativistic Formulations of the Navier-Stokes Equations................ 62 4.2.1 Parabolic Theories and Their Problems..................... 62 4.2.2 Hyperbolic Theories................................ 63 4.2.3 The Breakdown of Partial-Differential Equations as Models in Physics.... 64 4.3 The Regime of a Physical Theory............................. 66 4.3.1 Constraints on the Measure of Spatiotemporal Intervals............ 69 4.3.2 Infimal Decoupages................................. 72 4.3.3 The Kinematical Regime............................. 74 4.4 Physical Fields....................................... 78 4.4.1 Algebraic Operations on the Values of Quantities Treated by a Physical Theory 78 4.4.2 Inaccurate Scalars................................. 83 4.4.3 Algebraic Operations on Inaccurate Scalars................... 87 4.4.4 Inaccurate Scalar Fields and Their Derivations................. 95 4.4.5 Inaccurate Tensorial Fields and Their Derivations............... 100 5 6 CONTENTS 4.4.6 Inaccurately Linear Operators.......................... 107 4.4.7 Integrals of Inaccurate Fields, and Topologies on Their Spaces........ 108 4.4.8 Colorings...................................... 112 4.5 Physical Theories...................................... 114 4.5.1 Exact Theories with Regimes and Inaccurate, Colored, Kinematically Con- strained Theories.................................. 115 4.5.2 Idealization and Approximation.......................... 121 4.5.3 An Inaccurate, Well Set Initial-Value Formulation............... 129 4.5.4 A Physically Well Set Initial-Value Formulation................. 133 4.5.5 Maxwell-Boltzmann Theories........................... 136 4.5.6 The Consistency of Theory and Experiment.................. 139 4.6 The Soundness of Physical Theory............................ 140 4.6.1 The Comparison of Predicted and Observed Values............... 140 4.6.2 Consistent Maxwell-Boltzmann Theories..................... 143 4.6.3 The Dynamical Soundness of a Physical Theory................. 145 4.6.4 Theoretical Under-Determination......................... 147 4.7 The Theory Is and Is Not the Equations......................... 148 References 151 Chapter 1 Pr´ecisof Dissertation 1.1 The Spirit of the Thing The dissertation consists of three separate papers. The three could not be more different from each other, as determined using a legion of different measures. The most important is this: the earliest one, on physicalist interpretations of causality, though I stand by the substance of its claims, is jejune in style and presentation; the later two, on, respectively, singular structure in relativistic spacetimes and the regime of applicability of theories of mathematical physics, are relatively more mature. If I had had the opportunity, I would have liked to have substituted another paper for the first, or to have rewritten it; but time did not allow this. The three do possess two related features of great importance, gestured at by the title of the dissertation. On the one hand, all three could be said to manifest a spirit of mitigated skepticism with regard to the interest and indeed the coherence of many questions dealt with traditionally by philosophers, both in their content and in their framing. On the other hand, they also all manifest a spirit that may be thought at first contrary to the first, though it is not, one of guarded optimism for the advancement of real understanding and comprehension of the issues these traditionally philo- sophical questions have purported to describe, albeit an optimism qualified by a demand intimated as well by the title of the dissertation: that many questions traditionally dealt with by philosophers require knowledge both detailed and comprehensive of our best physical theories, and that of both their formal and empirical content, if one is to make any substantive progress on them. The time is long since past when anyone, philosopher or physicist, can hope to address in any serious way questions deeply rooted in the physical states of affairs of the world by sequestering himself in his study and requiring of the physical world that it conform, in nature and substance, to the limitations of his imagination and his \powers" of pure reasoning. 1 This is perhaps recognized by most professional philosophers practicing today, and indeed many workers in many fields at least pay lip-service to the idea that, in investigating certain issues, account must be taken
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