philosophy of physics

The Oxford Handbook of Philosophy of Physics REdited by obert Batterman Oxford University Press is a department of the University of Oxford. It furthers the University's objec tive of exc ellenc e in researc h, sc holarship, and education by publishing worldwide. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Braz il Chile Cz ec h Republic  Franc e Greec e Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switz erland Thailand Turkey Ukraine Vietnam Oxford is a registered trademark of Oxford University Press in the UK and c ertain other c ountries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016 © Oxford University Press 2013 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by lic ense, or under terms agreed with the appropriate reproduc tion rights organiz ation. Inquiries c onc erning reproduc tion outside the sc ope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not c irc ulate this work in any other form and you must impose this same c ondition on any ac quirer. Library of Congress Cataloging-in-Public ation Data Page 1 of 2

The Oxford handbook of philosophy of physic s / edited by Robert Batterman. p. c m. ISBN 978-0-19-539204-3 (alk. paper) 1. Physic s– Philosophy. I. Batterman, Robert W. II. Title: Handbook of philosophy of physic s. QC6.O925 2012 530.1– dc 23  2012010291 1 3 5 7 9 8 6 4 2

CONTENTS                   Contributors Introduction Robert Batterman 1. For a Philosophy of Hydrodynamics Olivier Darrigol 2. What Is \"Classical Mechanics\" Anyway? Mark Wilson 3. Causation in Classical Mechanics Sheldon R. Smith 4. Theories of Matter: Infinities and Renormalization Leo P. Kaodanoff 5. Turn and Face the Strange …Ch-ch-changes: Philosophical Questions Raised by Phase Transitions Tarun Menon and Craig Callender 6. Effective Field Theories ]onathan Bain 7. The Tyranny of Scales Robert Batterman 8. Symmetry Sorin Bangu 9. Symmetry and Equivalence Gordon Belot 10. lndistinguishability Simon Saunders

11. Unification in Physics Margaret Morrison 12. Measurement and Classical Regime in Quantum Mechanics Guido Bacciagaluppi 13. The Everett Interpretation David Wallace 14. Unitary Equivalence and Physical Equivalence Laura Ruetsche 15. Substantivalist and Relationalist Approaches to Spacetime Oliver Pooley 16. Global Spacetime Structure John Byron Manchak 17. Philosophy of Cosmology Chris Smeenk Index

Contributors The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Co ntributo rs Guido Bacciagaluppi is Reader in Philosophy at the University of Aberdeen. His field of researc h is the philosophy of physic s, in partic ular the philosophy of quantum theory. He also works on the history of quantum theory and has published a book on the 1927 Solvay c onferenc e (together with A. Valentini). He also has interests in the foundations of probability and in issues of time symmetry and asymmetry. Jo nathan Bain is Assoc iate Professor of Philosophy of Sc ienc e at the Polytec hnic Institute of New York University. His researc h interests inc lude philosophy of spac e-time, sc ientific realism, and philosophy of quantum field theory. Page 1 of 5

Contributors So rin Bangu is Associate Professor of Philosophy at the University of Bergen, Norway. He received his Ph.D. from the University of Toronto and has previously been a postdoctoral fellow at the University of Western Ontario and a fixed-term lecturer at the University of Cambridge, Department of History and Philosophy of Science. His main interests are in philosophy of sc ienc e (espec ially philosophy of physic s, mathematic s, and probability) and later Wittgenstein. He has published extensively in these areas and has recently c ompleted a book manusc ript on the metaphysic al and epistemologic al issues arising from the applicability of mathematics to science. Ro bert Batterman is Professor of Philosophy at the University of Pittsburgh. He is a Fellow of the Royal Soc iety of Canada. He is the author of The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence (Oxford, 2002). His work in philosophy of physics focuses primarily upon the area of condensed matter broadly construed. His researc h interests inc lude the foundations of statistic al physic s, dynamic al systems and c haos, asymptotic reasoning, mathematic al idealiz ations, the philosophy of applied mathematic s, explanation, reduc tion, and emergenc e. Go rdo n Belo t is Professor of Philosophy at the University of Michigan. He has published a number of artic les on philosophy of physic s and related areas—and one small book, Geometric possibility (Oxford, 2011). Craig Callender is Professor of Philosophy at the University of California, San Diego. He has written widely in philosophy of science, metaphysics, and philosophy of physics. He is the editor of Physics meets philosophy at the Planck length (with Huggett) and the Oxford handbook of the philosophy of time. He is currently working on a book monograph on the relationship between physic al time and time as we experienc e it. Page 2 of 5

Contributors Olivier Darrigo l is a CNRS researc h direc tor in the SPHERE/Rehseis researc h team in Paris. He investigates the history of physic s, mostly nineteenth and twentieth c entury, with a strong interest in related philosophic al questions. He is the author of several books inc luding From c-numbers to q-numbers: The classical analogy in the history of quantum theory (Berkeley: University of California Press, 1992), Electrodynamics from Ampère to Einstein (Oxford: Oxford University Press, 2000), Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl (Oxford: Oxford University Press, 2005), and A history of optics from Greek antiquity to the nineteenth century (Oxford: Oxford University Press, 2012). Leo P. Kadano ff is a theoretic al physic ist and applied mathematic ian who has c ontributed widely to researc h in the properties of matter, the development of urban areas, statistic al models of physic al systems, and the development of c haos in simple mec hanic al and fluid systems. His best-known c ontribution was in the development of the c onc epts of “sc ale invarianc e” and “universality” as they are applied to phase transitions. More rec ently, he has been involved in the understanding of singularities in fluid flow. Jo hn Byro n Manchak is an Assistant Professor of Philosophy at the University of Washington. His primary researc h interests are in philosophy of physic s and philosophy of sc ienc e. His researc h has foc used on foundational issues in general relativity. Tarun Meno n is a graduate student in Philosophy at the University of California, San Diego. His researc h interests are in the philosophy of physic s and metaphysic s, partic ularly time, probability, and the foundations of statistic al mec hanic s. He is also interested in formal epistemology and the c ognitive struc ture of sc ienc e. Page 3 of 5

Contributors Margaret Mo rriso n is Professor of Philosophy at the University of Toronto. She is the author of several artic les on various aspec ts of philosophy of sc ienc e inc luding physic s and biology. She is also the author of Unifying scientific theories: Physical concepts and mathematical structures (Cambridge, 2000) and the editor (with Mary Morgan) of Models as mediators: Essays on the philosophy of natural and social science (Cambridge, 1999). Oliver Po o ley is University Lec turer in the Fac ulty of Philosophy at the University of Oxford and a Fellow and Tutor at Oriel College, Oxford. He works in the philosophy of physic s and in metaphysic s. Muc h of his researc h foc uses on the nature of spac e, time, and spac etime. Laura Ruetsche is Professor of Philosophy at the University of Mic higan. Her Interpreting quantum theories: The art of the possible (Oxford, 2011) aims to artic ulate questions about the foundations of quantum field theories whose answers might hold interest for philosophy more broadly c onstrued. Simo n Saunders is Professor in the Philosophy of Physic s and Fellow of Linac re College at the University of Oxford. He has worked in the foundations of quantum field theory, quantum mec hanic s, symmetries, thermodynamic s, and statistic al mec hanic s and in the philosophy of time and spac etime. He was an early proponent of the view of branc hing in the Everett interpretation as an “effec tive” proc ess based on dec oherenc e. He is c o-editor (with Jonathan Barrett, Adrian Kent, and David Wallac e) of Many worlds? Everett, quantum theory, and reality (OUP 2010). Chris Smeenk is Assoc iate Professor of Philosophy at the University of Western Ontario. His researc h interests are history and philosophy of physic s, and seventeenth-c entury natural philosophy. Page 4 of 5

Contributors Sheldo n R. Smith is Professor of Philosophy at UCLA. He has written artic les on the philosophy of c lassic al mec hanic s, the relationship between c ausation and laws, the philosophy of applied mathematic s, and Kant's philosophy of sc ienc e. David Wallace studied physic s at Oxford University before moving into philosophy of physic s. He is now Tutorial Fellow in Philosophy of Sc ienc e at Balliol College, Oxford, and university lec turer in Philosophy at Oxford University. His researc h interests inc lude the interpretation of quantum mec hanic s and the philosophic al and c onc eptual problems of quantum field theory, symmetry, and statistic al physic s. Mark Wilso n is Professor of Philosophy at the University of Pittsburgh, a Fellow of the Center for Philosophy of Sc ienc e, and a Fellow at the Americ an Ac ademy of Arts and Sc ienc es. His main researc h investigates the manner in whic h physic al and mathematic al c onc erns bec ome entangled with issues c harac teristic of metaphysic s and philosophy of language. He is the author of Wandering significance: An essay on conceptual behavior (Oxford, 2006). He is c urrently writing a book on explanatory struc ture. He is also interested in the historic al dimensions of this interc hange; in this vein, he has written on Desc artes, Frege, Duhem, and Wittgenstein. He also supervises the North Americ an Traditions Series for Rounder Rec ords.

Introduction Introduction Robert Batterman The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter disc usses the theme of this book, whic h is about the philosophy of physic s. The book provides an overview of the topic s being studied by philosophers of physic s and identifies theories that would not have been c onsidered fundamental during the 1980s. It desc ribes new problems and issues that bec ame the foc us of the philosophy of physic s in rec ent years, whic h inc lude the philosophy of hydrodynamic s, c lassic al mec hanic s, effec tive field theories, and measurement in quantum mec hanic s. K ey words: ph i l osoph y of ph y si cs, h y drody n ami cs, cl assi cal mech an i cs, effecti v e fi el d th eori es, qu an tu m mech an i cs When I was in graduate sc hool in the 1980s, philosophy of physic s was foc used primarily on two dominant reasonably self-c ontained theories: Orthodox nonrel-ativisitic quantum mec hanic s and relativistic spac etime theories. Of c ourse, there were a few papers published on c ertain questions in other fields of physic s suc h as statistic al mec hanic s and its relation to thermodynamic s. These latter, however, primarily targeted the extent to whic h the reduc tive relations between the two theories c ould be c onsidered a straightforward implementation of the orthodox strategy outlined by Ernest Nagel. Philosophical questions about the measurement problem, the question of the possibility of hidden variables, and the nature of quantum loc ality dominated the philosophy of physic s literature on the quantum side. Questions about relationalism vs. substantivalism, the c ausal and temporal struc ture of the world, as well as issues about underdetermination of theories dominated the literature on the spac etime side. Some worries about determinism vs. indeterminism c rossed the divide between these theories and played a signific ant role in shaping the development of the field. (Here I am thinking of Earman's A Primer on Determinism (1986) as a partic ular driving forc e.) These issues still rec eive c onsiderable attention from philosophers of physic s. But many philosophers have shifted their attention to other questions related to quantum mec hanic s and to spac etime theories. In partic ular, there has been c onsiderable work on understanding quantum field theory, partic ularly from the point of view of algebraic or axiomatic formulations. New attention has also been given to philosophic al issues surrounding quantum information theory and quantum c omputing. And there has, naturally, been c onsiderable interest in understanding the relations between quantum theory and relativity theory. Questions about the possibility of unifying these two fundamental theories arise. Relatedly, there has been a foc us on understanding gauge invarianc e and symmetries. However, I believe philosophy of physic s has evolved even further, and this belief prompts the public ation of this volume. Rec ently, many philosophers have foc used their attentions on theories that, for the most part, were largely ignored in the past. As noted above, the relationship between thermodynamic s and statistic al mec hanic s—onc e thought to be a paradigm instanc e of unproblematic theory reduc tion—is now a hotly debated topic . Philosophers and physic ists have long implic itly or explic itly adopted a reduc tionist methodologic al bent. Yet, over the years this methodologic al slant has been questioned dramatic ally. Attention has been foc used on the explanatory and desc riptive roles of “non-fundamental,” phenomenological theories. In large part bec ause of this shift of foc us, Page 1 of 8

Introduction “old” theories suc h as c lassic al mec hanic s, onc e deemed to be of little philosophic al interest, have inc reasingly bec ome the foc us of deep methodologic al investigations. Furthermore, some philosophers have bec ome more interested in less “fundamental” c ontemporary physic s. For instanc e, there are deep questions that arise in c ondensed matter theory. These questions have interesting and important implic ations for the nature of models, idealiz ations, and explanation in physic s. For example, model systems, suc h as the Ising model, play important c omputational and c onc eptual roles in understanding how there c an be phase transitions with spec ific c harac teristic s. And, the use of the thermodynamic limit is an idealiz ation that (some have argued) plays an essential, ineliminable role in understanding and explaining the observed universality of c ritic al phenomena. These spec ific issues are disc ussed in several of the c hapters in this volume. In the United States during the 1970s and 1980s, there was a great debate between partic le physic ists who pushed for funding of high-energy partic le ac c elerators and solid-state or c ondensed-matter theorists for whom the siphoning off of so muc h government funding to “fundamental” physic s was unac c eptable. A famous paper championing the latter position is Philip Anderson's “More Is Different” (1972). Not only was this a debate over funding, but it raised issues about exac tly what should c ount as “fundamental” physic s. While historians of physic s have foc used c onsiderable attention on this public debate, philosophers of physic s have really only rec ently begun to engage with the c onc eptual implic ations of the possibility that c ondensed matter theory is in some sense just as fundamental as high-energy partic le physic s. This c ollec tion aims to do two things. First, it tries to provide an overview of many of the topic s that c urrently engage philosophers of physic s. And sec ond, it foc uses attention on some theories that by orthodox 1980s standards would not have been c onsidered fundamental. It strives to survey some of these new issues and the problems that have bec ome a foc us of attention in rec ent years. Additionally, it aims to provide up-to-date discussions of the deep problems that dominated the field in the past. In the first c hapter, “For a Philosophy of Hydrodynamic s,” Olivier Darrigol foc uses attention on lessons that c an be learned from the historic al development of fluid mec hanic s. He notes that hydrodynamic s has probably rec eived the least attention of any physic al theory from philosophers of physic s. Hydrodynamic s is not a “fundamental” theory along the lines of quantum mec hanic s and relativity theory, and its basic formulation has not evolved muc h for two c enturies. These fac ts, together with a lac k of detailed historic al studies of hydrodynamic s, have kept the theory off the radar.1 Darrigol provides an ac c ount of the development of hydrodynamic s as a c omplex theory— one that is not fully c aptured by the basic Navier-Stokes equations. For the theory to be applic able, partic ularly for it to play an explanatory role, a host of tec hniques—idealiz ations, modeling strategies, and empiric ally determined data must c ome into play. This disc ussion shows c learly how intric ate, sophistic ated, and modern the theory of hydrodynamic s ac tually is. Darrigol draws a number of lessons about the struc tures of phenomenologic al theories from his detailed disc ussion, foc using partic ularly on what he c alls the “modular struc ture” of hydrodynamic s. Continuing the disc ussion of “old”—but by no means dead or eliminated— theories, Mark Wilson takes on the formidable task of trying to say exac tly what is the nature of c lassic al mec hanic s. A c ommon initial reac tion to this topic is to dismiss it: “Surely we all know what c lassic al mec hanic s is! Just look at any textbook.” But as Wilson shows in “What Is ‘Classic al Mec hanic s’ Anyway?”, this dismissive attitude is misleading on a number of important levels. Classic al mec hanic s is like a five-legged stool on a very uneven floor. It shifts dramatic ally from one founda- tional perspec tive to another depending upon the problem at hand, whic h in turn is often a func tion of the sc ale length at whic h the phenomenon is investigated. In the c ontext of planetary motions, billiards, and simplified ideal gases in boxes, the point-partic le interpretation of c lassic al mec hanic s will most likely provide an appropriate theoretic al setting. However, as soon as one tries to provide a more realistic desc ription of what goes on inside ac tual billiard ball c ollisions, one must c onsider the fac t that the balls will deform and build up internal stresses upon c ollision. In suc h situations, the point-partic le foundation will fail and one will need to shift to an alternative foundation, provided by c lassic al c ontinuum mec hanic s. Yet a third potential foundation for c lassic al mec hanic s c an be found within so-c alled analytic mec hanic s, in whic h the notion of a rigid body bec omes c entral. Here c onstraint forc es (suc h as the c onnec tions that allow a ball to roll, rather than skid, down an inc lined plane) play a c ruc ial role. Forc es of this type are not wholly c onsistent with the suppositions c entral to either the point-partic le or c ontinuum points of view. A major lesson from Wilson's disc ussion is that c lassic al mec hanic s should best be thought of as c onstituted by various foundational methodologies that do not fit partic ularly well with one another. This goes against c urrent orthodoxy that a theory must be seen as a formally axiomatiz able c onsistent struc ture. Page 2 of 8

Introduction On the c ontrary, to properly employ c lassic al mec hanic s for desc riptive and explanatory purposes, one pushes a foundational methodology appropriate at one sc ale of investigation to its limiting utility, after whic h one shifts to a different set of c lassic al modeling tools in order to c apture the physic s ac tive at a lower siz e sc ale. Wilson argues that a good deal of philosophic al c onfusion has arisen from failing to rec ogniz e the c omplic ated sc ale-dependent struc tures of c lassic al physic s. Sheldon Smith's c ontribution adds to our understanding of a partic ular aspec t of c lassic al physic s. In “Causation in Classic al Mec hanic s,” he addresses skeptic al arguments initiated by Bertrand Russell to the effec t that c ausation is not a fundamental feature of the world. In the c ontext of c lassic al physic s, one way of making this c laim more prec ise is to argue that there is no reason to privilege retarded over advanc ed Green's func tions for a system. Green's func tions, c rudely, desc ribe the effec t of an instantaneous, loc aliz ed disturbanc e that ac ts upon the system. It seems that the laws of motion for elec tromagnetism or for the behavior of a harmonic osc illator do not distinguish between retarded (presumably “c ausal”) and advanc ed (presumably “ac ausal”) solutions. If there is to be room for a princ iple of c ausality in c lassic al physic s, then it looks like we need to find extra-nomologic al reasons to privilege the retarded solutions. Smith surveys a wide range of attempts to answer the c ausal skeptic in the c ontexts of the use of Green's func tions and the imposition of (Sommerfeld) radiation c onditions, among other attempts. The upshot is that it is remarkably diffic ult to find justific ation within physic al theory for the maxim that c auses prec ede their effec ts. The next c hapter, by Leo Kadanoff, foc uses on c ondensed matter physic s. In partic ular, Kadanoff disc usses progress in physic ally understanding the fac t that matter c an abruptly c hange its qualitative state as it undergoes a phase transition. An everyday example oc c urs with the boiling water in a teakettle. As the temperature inc reases, the water c hanges from its liquid phase to its vapor phase in the form of steam. Mathematic ally, suc h transitions are desc ribed by an important c onc ept c alled an order parameter. In a first-order phase transition, suc h as the liquid vapor transition, the order parameter c hanges disc ontinuously. Certain phase transitions, however, are c ontinuous in the sense that the disc ontinuity in the behavior of the order parameter approac hes z ero at some spec ific c ritic al value of the relevant parameters suc h as temperature and pressure. For a long time there were theoretic al attempts to understand the physic s involved in suc h c ontinuous transitions that failed to adequately represent the ac tual behavior of the order parameter as it approac hed its c ritic al value. The development of the renormalization group in the 1970s remedied this situation. Kadanoff played a pivotal role in the c onc eptual development of renormaliz ation group theory. In this c hapter, he foc uses on these developments (partic ularly, the improvement upon early mean field theories) and on a deeply interesting feature he c alls the “extended singularity theorem.” This is the idea that sharp, qualitatively distinc t, c hanges in phase involve the presenc e of a mathematic al singularity. This singularity typic ally emerges in the limit in whic h the system siz e bec omes infinite. The understanding of the behavior of systems at and near phase transitions requires radic ally different c onc eptual apparatuses. It involves a synthesis between standard statistic al mec hanic al uses of probabilities and c onc epts from dynamic al systems theory—partic ularly, the topologic al c onc eptions of basins of attrac tion and fixed points of a dynamical transformation. The disc ussion of the renormaliz ation group and phase transitions c ontinues as Tarun Menon and Craig Callender examine several philosophic al questions raised by phase transitions. Their c hapter, “Turn and Fac e the Strange … Ch-c h-c hanges,” foc uses on the question of whether phase transitions are to be understood as genuinely emergent phenomena. The term “emergent” is muc h abused and c onfused in both the philosophic al and physic s literatures and so Menon and Callender provide a kind of road map to several c onc epts that have been invoked in the inc reasing number of papers on emergenc e and phase transitions. In partic ular, they disc uss c onc eptions of reduc tion and c orresponding notions of emergenc e: c onc eptual novelty, explanatory irreduc ibility, and ontologic al irreduc ibility. Their goal is to establish that for any reasonable senses of reduc ibility and emergenc e, phase transitions are not emergent phenomena, and they do not present problems for those of a reduc tionist explanatory bent. In a sense, their disc ussion c an be seen as c hallenging the importanc e of the extended singularity theorem mentioned above. Menon and Callender also c onsider some rec ent work in physic s that attempts to provide well- defined notions of phase transition for finite systems. Their c ontribution serves to highlight the c ontroversial and evolving nature of our philosophic al understanding of phase transitions, emergenc e, and reduc tionism. Jonathan Bain's c ontribution on “Effec tive Field Theories” looks at several physic al and methodologic al c onsequenc es of the fac t that some theories at low-energy sc ales are effec tively independent of, or dec oupled from, theories desc ribing systems at higher energies. Sometimes we know what the high-energy theory looks like Page 3 of 8

Introduction and c an follow a rec ipe for c onstruc ting low-energy effec tive theories by systematic ally eliminating high-energy interactions that are essentially “unobservable” at the lower energies. But, at other times, we simply do not know the c orrec t high-energy theory, yet nonetheless, we still c an have effec tive low-energy theories. Broadly c onstrued, hydrodynamic s is an example of the latter type of effec tive theory, if we c onsider it as a nineteenth c entury theory c onstruc ted before we knew about the atomic c onstitution of matter. Bain's foc us is on effec tive theories in quantum field theory and c ondensed matter physic s. His disc ussion c onc entrates on the intertheoretic relations between low-energy effec tive theories and their high-energy c ounterparts. Given the effec tive independence of the former from the latter, should one think of this relation as autonomous or emergent? Bain c ontends that an answer to this question is quite subtle and depends upon the type of renormaliz ation sc heme employed in c onstruc ting the effec tive theory. My own c ontribution to the volume c onc erns a general problem in physic al theoriz ing. This is the problem of relating theories or models of systems that appear at widely separated sc ales. Of c ourse, the renormaliz ation group theory (disc ussed by Kadanoff, Menon and Callender, and Bain in this volume) is one instanc e of bridging ac ross sc ales. But more generally, we may try to address the relations between finite statistic al theories at atomic and nanosc ales and c ontinuum theories that apply at sc ales 10+ orders of magnitude higher. One c an ask, for example, why the Navier-Cauc hy equations for isotropic elastic solids work so well to desc ribe the bending behavior of steel beams at the mac rosc ale. At the mic rosc ale the lattic e struc ture of iron and c arbon atoms looks nothing like the homogeneous mac rosc ale theory. Nevertheless, the latter theory is remarkably robust and safe. The c hapter disc usses strategies for upsc aling from theories or models at small sc ales to those at higher sc ales. It examines the philosophic al c onsequenc es of having to c onsider, in one's modeling prac tic e, struc tures that appear at sc ales intermediate between the mic ro and the mac ro. There has been c onsiderable debate about the nature of symmetries in physic al theories. Rec ent foc us on gauge symmetries has led philosophers to a deeper understanding of the role of loc al invarianc es in elec tromagnetism, partic le physic s, and the hunt for the Higgs’ partic le. Sorin Bangu provides a broad and c omprehensive survey of c onc epts of symmetry and invarianc e in his c ontribution to this volume. One of the most seduc tive features of symmetry c onsiderations c omes out of Wigner's suggestion that one might be able to understand, explain, or ground laws of nature by appeal to a kind of superprinc iple expressing symmetries and invarianc es that c onstrain laws to have the forms that they do. On this c onc eption symmetries are, perhaps, ontologic ally and epistemic ally prior to laws of nature. This raises deep questions for further researc h on the relationship between formal mathematic al struc tures and our physic al understanding of the world. Gordon Belot also c onsiders issues of symmetry and invarianc e. His c ontribution explores the c onnec tions between being a symmetry of a theory—a map that leaves invariant c ertain struc tures that enc ode the laws of the theory—and what it is for solutions to a theory to be physically equivalent. It is fairly c ommonplac e for philosophers to adopt the idea that, in effec t, these two notions c oinc ide. And if they do, then we have tight c onnec tion between a purely formal c onc eption of the symmetries of a theory and a methodologic al/interpretive c onc eption of what it is for two solutions to represent the same physic al state of affairs. Belot notes that in the c ontext of spac etime theories there seem to be well-established arguments supporting this tight c onnec tion between symmetries and physic al equivalenc e. However, he explores the diffic ulties in attempting to generaliz e this c onnec tion in c ontexts that inc lude c lassic al dynamic al theories. Belot examines different ways one might make prec ise the notion of the symmetries of a c lassic al theory and shows that they do not c omport well with reasonable c onc eptions of physic al equivalenc e. The c hallenge to the reader is then to find appropriate, nontrivial notions of symmetries for c lassic al theories that will respec t reasonable notions of physic al equivalenc e. Yet another type of symmetry, permutation symmetry, is the subjec t of the c hapter by Simon Saunders, entitled “Indistinguishability.” He foc uses on the proper understanding of partic le indistinguishability in c lassic al statistic al mec hanic s and in quantum theory. In the c lassic al c ase, Gibbs had already (prior to quantum mec hanic s) rec ogniz ed a need to treat partic les, at least sometimes, as indistinguishable. This is related to the infamous Gibbs paradox that Saunders disc usses in detail. The c onc ept of “indistinguishability” had meanwhile entered physic s in a c ompletely new way, involving a new kind of statistic s. This c ame with the derivation of Planc k's spec tral distribution, in whic h Planc k's quantum of ac tion h first entered physic s. Common wisdom has long held that partic le indistinguisha-bility is stric tly a quantum c onc ept, inapplic able to the c lassic al realm; and that c lassic al statistic al mec hanic s is anyway only the c lassic al limit of a quantum theory. This fits with the standard view of the explanation of quantum statistic s (Bose-Einstein or Fermi-Dirac statistic s): departures from c lassic al (Maxwell- Page 4 of 8

Introduction Boltz mann) statistic s are explained by partic le indistinguishability. With this Saunders takes issue. He shows how it is possible to treat the statistic al mec hanic al statistic s for c lassic al partic les as invariant under permutation symmetry in exac tly the same way that it is treated in the quantum c ase. He argues that the c onc eption of permutation symmetry deserves a plac e alongside all the other symmetries and invarianc es of physic al theories. Spec ific ally, he argues that the c onc ept of indistinguishable, permutation invariant, classical partic les is c oherent and reasonable c ontrary to many c laims found in the literature. Margaret Morrison's topic is “Unific ation in Physic s.” She argues that there are a number of distinc t senses of unific ation in physic s, eac h of whic h has different implic ations for how we view unified theories and phenomena. On the one hand, there is a type of unific ation that is ac hieved via reduc tionist programs. Here a paradigm example is the unific ation provided by Maxwellian elec trodynamic s. Maxwell's emphasis on mec hanic al models in his early work involved the introduc tion of the displac ement c urrent, whic h was nec essary for a field theoretic representation of the phenomena. These models also enabled him to identify the luminiferous aether with the medium of transmission of elec tromagnetic phenomena. Two aethers were essentially reduc ed to one. When these models were abandoned in his later derivation of the field equations, the displac ement c urrent provided the unifying parameter or theoretic al quantity that allowed for the identific ation of elec tromagnetic and optic al phenomena within the framework of a single field theoretic ac c ount. This type of unific ation was analogous to Newton's unific ation of the motions of the planets and terrestrial trajec tories under the same (gravitational) theoretic al framework. However, not all c ases of unific ation are of this type. Morrison disc usses the example of the elec troweak theory in some detail, arguing that this unific atory suc c ess represents a kind of synthetic , rather than reduc tive, unity. The elec troweak theory also involves a unifying parameter, namely, the “Weinberg angle.” However, the unity ac hieved through gauge symmetry is a synthesis of struc ture, rather than of substanc e, as exemplified by the reduc tive c ases. Finally, in c alling attention to the diffic ulties with the Standard Model more generally, Morrison notes that yet a different kind of unific ation is ac hieved in the framework of effec tive field theory. This provides another vantage point from whic h to understand the importanc e of the renormaliz ation group. Morrison argues for a third type of unific ation in terms of the universality c lasses, one that foc uses on unific ation of phenomena but should be understood independently of the type of mic ro-reduc tion c harac teristic of unified field theory approaches. As noted earlier, there c ontinues to be signific ant researc h on foundational problems in quantum mec hanic s. Guido Bac c iagaluppi's c hapter provides an up-to-date disc ussion of work on two distinc t problems in the foundations of quantum mec hanic s that are typic ally c onflated in the literature. These are the problem of the c lassic al regime and the measurement problem. Both problems arise from deep issues involving entanglement and the failure of an ignoranc e interpretation of reduc ed quantum states. Bac c iagaluppi provides a c ontemporary and thorough introduc tion to these issues. The problem of the c lassic al regime is that of providing a quantum mec hanic al explanation or ac c ount of the suc c ess of c lassic al physic s at the mac rosc ale. It is, in essenc e, a problem of intertheoretic relations. Contemporary work has c onc entrated on the role of environmental dec oherenc e in the emergenc e of c lassic al kinetic s and dynamic s. Bac c iagaluppi argues that the suc c ess of appeals to dec oherenc e to solve this problem will depend upon one's interpretation of quantum mec hanic s. He surveys an ontologic ally minimalist instrumental interpretation and a standard, ontologic ally more robust or realistic interpretation. The measurement problem is the distinc t problem of deriving the c ollapse postulate and the Born rule from the first princ iples (Sc hrödinger evolution) of the quantum theory. In examining the measurement problem, Bac c iagaluppi provides a detailed presentation of a modern, realistic theory of measurement that goes beyond the usual idealiz ed disc ussions of spin measurements using Stern-Gerlac h magnets. This disc ussion generaliz es the usual c ollapse postulate and the Born rule to take into ac c ount the fac t that real measurements are unsharp. It does so by employing the apparatus of positive operator value (POV) measures and observables. The upshot is that the measurement problem remains a real worry for someone who wants to maintain a standard, reasonably orthodox interpretation of quantum theory. Perhaps Everett theories, GRW-like spontaneous c ollapse theories, and so on are required for a solution. The Everett, or Many Worlds, interpretation of quantum mec hanic s is the subjec t of David Wallac e's c hapter. It is well known that the linearity of quantum mec hanic s leads, via the princ iple of superposition, to the possibility that mac rosc opic objec ts suc h as c ats c an be found in biz arre states—superpositions of being alive and being dead. Wallac e argues that a proper understanding of what quantum mec hanic s ac tually says will enable us to understand suc h biz arre situations in a way that does not involve c hanging the physic s (e.g., as in Bohmian hidden Page 5 of 8

Introduction variable mec hanic s or GRW spontaneous c ollapse theories). Neither, he c laims, does it involve c hanging one's philosophy by, for example, providing an operationalist interpretation that imposes some spec ial status to the observer or to what c ounts as measurement, along the lines of Bohr. Suc h interpretations are at odds with our understanding of, say, the role of the observer in the rest of sc ienc e. Wallac e argues for a straightforward, fully realist interpretation of the bare mathematic al formalism of quantum mec hanic s and c laims that this interpretation will make sense of superposed c ats, and so on, without c hanging the theory and without c hanging our overall view of science. The straightforward realist interpretation that is to do all of this work is the Everett interpretation. Prima fac ie, this c laim is itself biz arre: after all, the Everett interpretation has us multiplying worlds or universes upon measurements. Nevertheless, Wallac e makes a strong c ase that an understanding of superposition as a desc ription of multiplic ity, rather than of the indefiniteness of states, is exac tly what is needed. Furthermore, that is exac tly what the Everett interpretation (and no other) provides. The bulk of Wallac e's c ontribution examines various problems that have been raised for the Everett interpretation. In partic ular, he foc uses on (1) the problem of providing a preferred basis—what ac tually justifies our understanding of superposition in terms of multiplic ity of worlds, and (2) the probability problem—how to understand the probabilistic nature of quantum mec hanic s if one has only the fully deterministic dynamic s provided by the Sc hrödinger equation. He argues that the c ontemporary understanding of the Everett interpretation has the resourc es to address these issues. Laura Ruetsc he's c hapter “Unitary Equivalenc e and Physic al Equivalenc e” inves-tigates a question of deep physic al and philosophic al importanc e: The demand for c riteria establishing the physic al equivalenc e of two formulations of a physic al theory. In “ordinary” quantum mec hanic s the rec eived view is that two quantum theories are physic ally equivalent just in c ase they are unitarily equivalent. Any pair of theories purporting, say, to desc ribe two entangled spin 1/2 systems are really just one and the same bec ause of the Jordan and Wigner theorem showing that a theory that represents the c anonic al antic ommutation relations for a system of n spins is unique up to unitary equivalenc e. A similar theorem due to Stone and von Neumann guarantees an analogous result for any Hilbert spac e representation of the c anonic al c ommutation relations for a Hamiltonian system. What are the c onsequenc es of the breakdown of unitary equivalenc e for those quantum systems for whic h these theorems fail to hold? Suc h systems inc lude the infinite systems studied in quantum field theory, quantum statistic al mec hanic s, and even simpler infinite systems like an infinite one-dimensional c hain of quantum spins. She c alls these theories c ollec tively QM∞. The plethora of unitarily inequivalent representations in these infinite c ases demands that we revisit our assumptions about physic al equivalenc e and the nature of quantum theories. Ruetsc he examines various c ompeting suggestions, or c ompeting princ iples that may guide the investigation into this problem. The next c hapter, by Oliver Pooley, provides an up-to-date, c omprehensive disc ussion of substantivalist and relationalist approac hes to spac etime. Crudely, this is a debate about the ontology of our theories of spac e and spac etime. The substantivalists hold that among the fundamental objec ts of the world is spac e-time itself. Relationists, to the c ontrary, deny that propositions about spac etime are ultimately to be understood in terms of c laims about material objec ts and possible spatiotemporal relations that may obtain between them. Pooley presents a historic al introduc tion, as well as a detailed disc ussion of the c urrent landsc ape in the literature. Spec ific ally, he c onsiders rec ent relationist, neo-Mac hian proposals by Barbour, as well as dynamic al approac hes favored by Brown, and Pooley and Brown, that aim to provide a reduc tive ac c ount of the spac etime symmetries in terms of the dynamic al symmetries of laws governing the behavior of matter. In addition, Pooley provides a c urrent assessment of the impac t of the so-c alled Hole Argument against substantivalism. In “Global Spac etime Struc ture” John Manc hak examines the qualitative, primarily topologic al and c ausal, aspec ts of general relativity. He provides an abstrac t c lassific ation of various loc al and global spac etime properties. In the global c ausal c ontext he explic itly defines a set of c ausal c onditions that form a stric t hierarc hy of possible c asual properties of spac etime. The strongest is the c ondition of global hyperbolic ity, whic h implies others inc luding c ausality and c hronology. Another set of global properties of spac etime c onc erns in what sense a spac etime c an be said to possess singularities. Here he foc uses on the notion of geodesic inc ompleteness. Manc hak then takes up philosophic al questions c onc erning the physic al reasonableness of these various spac etime properties. In a loc al c ontext, being a solution to Einstein's Field Equation is typic ally taken to be physic ally reasonable. But, global properties c onc erning the existenc e and nature of singularities and the possibility of time travel lead to open questions of philosophic al interest that are c urrently being investigated. Last, but not least, Chris Smeenk's c ontribution c onc erns philosophic al issues raised in c ontemporary work on c osmology. A c ommon view is that c osmology requires a distinc tive methodology bec ause the universe-as-a- Page 6 of 8

Introduction whole is a unique objec t. Restric tions on observational ac c ess to the universe due to the finite speed of light pose severe c hallenges to establishing global properties of the universe. How c an we know that the loc al generaliz ations we take to be lawful in our limited region c an be extended in a global fashion? Here, of c ourse, there is overlap with the disc ussions of the previous c hapter. Suc c esses of the so-c alled Standard Model for c osmology inc lude big- bang nuc leosynthesis and the understanding of the c osmic bac kground radiation, among others. Challenges to the Standard Model result from growing evidenc e that if it is c orrec t, then most of the matter and energy present in the universe is not what we would c onsider ordinary. Instead, there apparently needs to be dark matter and dark energy. Smeenk provides an overview of rec ent hypotheses about dark matter and energy, and relates these disc ussions to philosophic al debates about underdetermination. A different kind of problem arises in assessing theories regarding the very early universe. These theories are often motivated by the idea that the initial state required by the Standard Model is highly improbable. This defic ienc y c an be addressed by introduc ing a dynamic al phase of evolution, suc h as inflationary c osmology, that alleviates this need for a spec ial initial state. Smeenk notes that assessing this response to fine-tuning is c onnec ted with debates about explanation and foundational disc ussions regarding time's arrow. One very important aspec t of rec ent work in c osmology is the appeal to anthropic reasoning to help explain features of the early universe. A sec ond rec ent development, often related to anthropic c onsiderations, is the multiverse hypothesis—the existenc e of c ausally isolated poc ket universes. This c hapter brings these fasc inating issues to the fore and raises a number of philosophic al questions about the nature of explanation and c onfirmation appropriate for c osmology. It is my hope that readers of this volume will gain a sense of the wide variety of issues that c onstitute the general field of philosophy of physic s. The foc us of the field has expanded tremendously over the last thirty years. New problems have c ome up, and old problems have been refoc used and refined. It is indeed my pleasure to thank all of the authors for their c ontributions. In addition, I would like to thank Peter Ohlin from Oxford University Press. A number of others c ontributed to this projec t in various ways. I am partic ularly indebted to Gordon Belot, Julia Bursten, Nic olas Fillion, Laura Ruetsc he, Chris Smeenk, and Mark Wilson for invaluable advic e and support. References P. W. Anderson. More is different. Science, 177(4047):393–396, 1972. Olivier Darrigol. Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford, 2005. John Earman. A Primer on Determinism. Reidel, Dordrec ht, 1986. Notes: (1) Darrigol's rec ent Worlds of Flow fills this lac una providing an exc eptional disc ussion of the history (Darrigol 2005). Robert Batterman Robert Batterm an is Professor of Philosophy at The University of Pittsburgh. He is a Fellow of the Royal Society of Canada. He is the author of _The Devil in the Details: Asym ptotic Reasoning in Explanation, Reduction, and Em ergence_ (Oxford, 2002). His work in philosophy of physics focuses prim arily upon the area of condensed m atter broadly construed. His research interests include the foundations of statistical physics, dynam ical system s and chaos, asym ptotic reasoning, m athem atical idealizations, the philosophy of applied m athem atics, explanation, reduction, and em ergence. Page 7 of 8

For a Philosophy of Hydrodynamics Olivier Darrigol The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter disc usses the need for a philosophy of hydrodynamic s and the lessons that c an be learned from the historic al development of fluid mec hanic s. It explains that hydrodynamic s has been not given attention by philosophers of physic s bec ause of a lac k of detailed historic al studies of hydrodynamic s, and highlights the need for idealiz ations and modeling strategies for this theory to be applic able. The c hapter also c onsiders the struc tures of phenomenologic al theories and the so-c alled modular struc ture of hydrodynamic s. K ey words: h y drody n ami cs, fl u i d mech an i cs, ph i l osoph ers of ph y si cs, h i stori cal stu di es, i deal i zati on s, model i n g strategi es, ph en omen ol ogi cal th eori es, modu l ar stru ctu re Among the major theories of physic s, hydrodynamic s is probably the one that has rec eived the least attention from philosophers of sc ienc e. Until rec ently, three c irc umstanc es easily explained this neglec t. First, there was very little historic al literature on whic h philosophers c ould rely. Sec ond, philosophers tended to foc us on fundamental theories suc h as relativity theory and quantum theory and to neglec t more phenomenologic al theories. Third, they harbored a neo-Hempelian c onc ept of explanation following whic h the foundations of a theory implic itly c ontain all its explanatory apparatus.1 Even Thomas Kuhn, who brought the “normal” phases of sc ienc e to the fore, restric ted c onc eptual innovation to the revolutionary phases.2 Sinc e the fundamental equations of hydrodynamic s have remained essentially the same for about two c enturies, this view reduc es the development of this theory to a matter of technical prowess in solving the equations. In rec ent years these three c irc umstanc es have lost muc h of their weight. We now have fairly detailed histories of hydrodynamic s.3 The superiority of fundamental theories over lower sc ale or phenomenologic al theories has been multiply c hallenged, both within sc ienc e and in the philosophy of sc ienc e.4 And there has been a growing awareness that explanation mostly resides in devic es that are not c ontained in the bare foundations of a theory. For example, Mary Morgan and Margaret Morrison have emphasiz ed the role of models as mediators between theory and experiment; Jeffry Ramsey has argued the c onc eptual signific anc e of approximations and “transformation reduc tions”; Robert Batterman has made explanation depend on strategies for the elimination of irrelevant details in the foundations; Paul Humphreys has plac ed c omputability at the c enter of his assessment of the nature and value of sc ientific knowledge. Eric Winsberg has shown the importanc e of extratheoretic al c onsiderations in judging the validity of numeric al simulations based on the fundamental equations. Already in 1983, Ian Hac king and C. W. F. Everitt, who were more in touc h with the ac tual prac tic e of physic ists than average philosophers, introduc ed “theory artic ulation” or “c alc ulation” as an essential “semantic bridge between theory and observation.”5 Granting that theory artic ulation is as philosophic ally important as the building of foundations, hydrodynamic s bec omes a topic of exc eptional philosophic al interest largely bec ause of the huge time span between the establishment of its foundations and its suc c essful applic ation to some of the most pressing engineering problems. This delay is an indirec t proof of the c reativity needed to expand the explanatory power of theories. It enables us Page 1 of 22

For a Philosophy of Hydrodynamics to observe a ric h sample of the devic es through whic h explanatory expansion may oc c ur. Margaret Morrison, Mic hael Heidelberg, and Moritz Epple have rec ently given philosophic al studies of two of these devic es: Ludwig Prandtl's boundary-layer theory and his wing theory. The present essay is c onduc ted in the same spirit.6 The first sec tion gives a few historic al examples of the means by whic h hydrodynamic s bec ame applic able to a growing number of c onc rete situations. The sec ond provides a tentative c lassific ation of these means. The third c ontains a definition of physic al theories that inc ludes their evolving explanatory apparatus. Spec ial emphasis is given to a “modular struc ture” of theories that makes them more amenable to tests, c omparisons, c ommunic ation, and c onstruc tion.7 1. So me Histo ry In the mid-eighteenth c entury, Jean le Rond d'Alembert and Leonhard Euler formulated the general laws of motion of a nonvisc ous fluid. In Euler's form, c alling v the veloc ity of the fluid, P its pressure, ρ its density, and f an impressed force density, these laws are given by the equation of motion the equation of continuity, and the boundary c ondition that the fluid veloc ity next to the walls of a rigid c ontainer should be parallel to these walls. If the fluid has a free surfac e at whic h it touc hes another fluid, the boundary c onditions (later provided by Lagrange) are the equality of the pressures of the two fluids, and the c ondition that a partic le of the surfac e of one fluid should remain on its surfac e.8 Euler's derivation of the equation of fluid motion assumes the pressure between two c ontiguous fluid parts to be perpendic ular to the separating surfac e, as is the c ase in hydrostatic s. In 1822 Claude Louis Navier implic itly dropped this assumption by c omparing the internal fluid forc es with the molec ular forc es of his general theory of elasticity. The resulting equation of motion is the Navier-Stokes equation whic h involves the visc osity μ. This equation was reinvented several times. There was muc h hesitation on the proper boundary c onditions, although in 1845 George Gabriel Stokes c orrec tly argued for a vanishing relative veloc ity of the fluid next to rigid bodies.9 From a mathematic al point of view, the most evident goal of the theory is to integrate the equations of motion for any given initial c onditions and boundary c onditions. There are at least three reasons not to c onfine fluid mechanics to this goal: 1. In the c ase of a c ompressible fluid, the system of equations is not c omplete bec ause one needs the relation between pressure and density. This relation implies thermodynamic c onsiderations, and therefore forc es us to leave the narrow c ontext of fluid mec hanic s. 2. It is generally impossible to solve the equations by analytic al means bec ause of their nonlinear c harac ter. Moreover, the few restric ted c ases in whic h this is possible may have little or no resemblanc e with ac tual flow bec ause of instabilities. Nowadays, numeric al integration is often possible and is, indeed, suffic ient for some engineering problems. This leads us to the third c aveat. 3. The answer to most physic al questions regarding fluid behavior is not to be found in the solution of spec ific boundary-value problems. Rather, the physic ist is often interested in generic properties of c lasses of solutions. In mathematic al terms, we need to have a handle on the struc ture of the spac e of solutions. What do physicists do when the solution of boundary problems no longer serves their interests? In order to answer this question, we will c onsult some of the historic al evolution of hydrodynamic s. 1.1 Bernoulli's Law Page 2 of 22

For a Philosophy of Hydrodynamics From a prac tic al point of view, the main result that Euler c ould derive from his new hydrodynamic s was the law relating the pressure P, the position r, and the veloc ity v for the steady motions of an inc ompressible fluid that admit a veloc ity potential (g is the ac c eleration of gravity). This ac hievement may seem meager for the following reasons: 10 the law had already been derived by Daniel Bernoulli in the 1730s as an applic ation of the c onservation of live forc e (energy) to steady, parallel-slic e, inc ompressible fluid motion; the law requires a narrow spec ializ ation of the theory; one aspec t of this spec ializ ation, the existenc e of a veloc ity potential, is (or was) physic ally obsc ure (its original purpose was to simplify the equations of motion and to permit their integration); under this spec ializ ation, the law is a straightforward mathematic al c onsequenc e of Euler's equations. From these remarks, one might be tempted to judge that Bernoulli's law adds nothing signific ant to the fundamental equations of hydrodynamic s. Yet the prac tic e of physic ists and engineers suggests the c ontrary: This law is used in many c irc umstanc es, surely more often than Euler's equations themselves. There are several good reasons for this: (1) Bernoulli's law relates easily ac c essible parameters of fluid motion in a simple manner, without any referenc e to the subtleties of the underlying dynamic s; (2) it is related to the general princ iple of energy c onservation, whic h bridges hydrodynamic s with mechanics; (3) it provides the basis for the hydraulic ians’ language of pressure head, veloc ity head, and gravity head; and (4) this language is still used when the law is violated. Although this last point may seem paradoxic al, it illustrates a highly important mode of c onc ept formation in the post-foundational life of a theory: the solutions of the general theory are c harac teriz ed with referenc e to the solutions of amore workable spec ializ ation of this theory. The c onc epts engendered by the spec ializ ation thus enrich the language of the general theory. They are useful as long as the law is valid in parts of the investigated system and as long as the loc i of its violations are suffic iently understood. In typic al hydraulic systems, there are regular pipes and reservoirs in whic h the law applies with a known c orrec tion (visc ous or boundary-layer retardation in pipes) and there are phenomenologic ally or theoretic ally known “losses of head” when some ac c idents, suc h as pipe-to-pipe c onnec tions or sudden enlargements of the sec tion of a pipe, oc c ur. 1.2 Surface Waves Historic ally, the sec ond suc c essful applic ation of Euler's equations was to the problem of water waves. In this c ase, spec ializ ation is also nec essary: the fluid is taken to be inc ompressible and a veloc ity potential is assumed. Moreover, some approximations must be introduc ed to c irc umvent the nonlinearity of the equations. In a memoir of 1781, Joseph Louis Lagrange originally assumed waves of small amplitude and of length muc h larger than the depth of the water. In the mid-1810s, Siméon Denis Poisson and Augustin Cauc hy did without the latter approximation. The resulting differential equation for the deformation of the water surfac e is linear, and it admits sine-wave solutions whose propagation veloc ity depends on the wavelength. At this (first-order) approximation, one may use an autonomous language of sine waves that is no longer reminisc ent of the underlying fluid dynamic s and that is equally applic able to other kinds of linear waves. All one needs to know is how to c ombine (superpose) various sine waves in order to ac c ommodate given initial shapes or perturbations of the water surfac e. We here enc ounter a sec ond c ase of bridging of hydrodynamic s with other theories: the introduc tion of c onc epts that apply to similar modes of motion in different theories (optic s, hydrodynamic s, ac oustic s …).11 This is not to say that all linear wave problems are understood onc e we know the dispersion law (how the veloc ity of a sine wave depends on its wavelength). Historic ally, muc h effort was needed to understand the struc ture of a superposition of sine waves. Employing stric tly mathematic al methods, Poisson and Cauc hy only suc c eeded in desc ribing the wave c reated by a stone thrown into a pond. John Sc ott Russell (in 1844) and William Froude (in 1873) later observed that the front of a group of waves traveled at a smaller veloc ity than individual waves in the group. In 1876, Stokes gave the modern theoretic al explanation in terms of phase and group veloc ity. Ten years later, William Thomson (Lord Kelvin) determined the form of ship waves by a c lever applic ation of these c onc epts. On the physic al side of his deduc tion, he relied on the optic al “princ iple of interferenc e.” On the mathematic al side, Page 3 of 22

For a Philosophy of Hydrodynamics he invented the method of the stationary phase, whic h is now c ommonly used in various domains of physic s. Again, we have a c ase of c onc epts and tools generated in a region of a given theory but ultimately applied to regions of many other theories (by region, I mean a restric tion of the theory to a limited c lass of systems and boundary c onditions). These c onc epts were partly derived by a mathematic al proc ess of spec ializ ation and approximation, partly by observation, partly by analogy with other domains of physic s.12 Similar remarks apply to the c ase of nonlinear waves. George Biddell Airy and Stokes tamed nonlinear periodic waves by suc c essive approximations to the fundamental equations, with applic ations to oc ean waves and river tides. This was a mostly mathematic al proc ess of a c umbersome but fairly automatic nature. In c ontrast, Sc ott Russell observed solitary waves (isolated swells) of invariable shape long before theorists admitted their possibility. When Joseph Boussinesq and Lord Rayleigh at last deduc ed suc h waves from theory, it bec ame c lear that qualitative results (suc h as the deformation of traveling waves) derived by c onsidering separately a small-depth (nondispersive) approximation and a small-amplitude (linear) approximation, no longer obtained when the depth and amplitude were both large. The c ompensation of the dispersive and nonlinear c auses of deformation for waves of a properly selec ted shape is a mec hanism whic h, again, applies to many other domains of physic s.13 1.3 Vortex Motion Early fluid mec hanic s usually assumed the existenc e of a veloc ity potential bec ause it greatly simplified the fundamental equations and also bec ause Lagrange had shown that it resulted from the equations of motion for a large c lass of boundary c onditions (motion started from rest and c aused by moving solids). Another reason, emphasiz ed by British fluid theorists, was the fac t that the veloc ity potential of an inc ompressible fluid obeys the same differential equation (Laplac e's equation) as the gravitational, elec tric , and magnetic potentials. This formal analogy was a c onstant sourc e of inspiration for Stokes, Thomson, and James Clerk Maxwell. It permitted an intuitive demonstration of some basic theorems of the abstrac t “potential theory,” and it provided fluid-mec hanic al analogs of elec trostatic , elec trokinetic , and magnetostatic phenomena. Figure 1.1 A portion of a vortex filament. The product of the vorticity (indicated by the arrows) by the normal sec tion of the filament is a c onstant along the filament. It is also invariable during the motion of the fluid. The general c ase in whic h no veloc ity potential exists was judged intrac table until 1858 when Hermann Helmholtz disc overed a few remarkable theorems that pushed this c ase to the forefront of the theory. As Cauc hy and Stokes had earlier proved, the infinitesimal evolution of a fluid element c an be regarded as the superposition of three kinds of motion: a translation of the c enter of gravity of the element, a dilation of the element along three mutually orthogonal axes, and a rotation. Formally, the rotation per unit time is half the vec tor ω = ∇ ×v whic h has the c omponents ∂ υz/ ∂y − ∂ υy / ∂ z etc . This vec tor, now c alled vorticity, vanishes if and only if there exists a veloc ity potential (in a c onnec ted domain). This kinematic analysis of infinitesimal fluid motion is part of the c onc eptual furniture of modern fluid mec hanic s. Maxwell used it to develop the physic o-mathematic al c onc epts of c url and divergenc e that apply to any field theory. Helmholtz reinvented it to interpret the non-existenc e of the veloc ity potential and the vec tor ω = ∇ ×v geometric ally.14 Helmholtz extended the geometric al interpretation to the “vortic ity equation,” whic h derives from Euler's equations when the fluid is inc ompressible. For this purpose, he defined vortex filaments as thin bundles of lines everywhere tangent to the vortic ity, and the intensity of a filament as the produc t of a normal sec tion of this filament by the value of the vortic ity in the sec tion (see figure 1.1). He then showed that the intensity of a filament was a c onstant along a filament and that the vortic ity equation was equivalent to the statement that the vortex filaments moved together with the fluid without altering their intensity. This theorem implies that the distribution of vortic ity in a perfec t liquid is in a sense invariant: it travels together with the fluid without any alteration.15 Page 4 of 22

For a Philosophy of Hydrodynamics In this light, Helmholtz argued that the vortic ity field (as today's physic ists say) better represented arbitrary flows than the veloc ity field: its invariant properties c ompletely determine the rotational c omponent of the flow, while the irrotational c omponent is ruled by the theorems of potential theory. With the help of an elec tromagnetic analogy, Helmholtz then determined the veloc ity fields assoc iated with simple distributions of vortic ity: straight vortex lines, vortex sheets, and vortex rings. He also c alc ulated the interac tions of vortic es and verified his predic tions experimentally. The vortex sheets played an important role in Helmholtz 's later writings. They are mathematic ally equivalent to a finite slide of fluid over fluid, and they should oc c ur, ac c ording to Helmholtz , whenever a fluid is forc ed to pass the edge of an immersed body. As an illustration of this proc ess, Helmholtz gave the formation of smoke jets when he blew the smoke of a c igar through his lips. Through ingenious reasoning, he proved the instability of the disc ontinuity surfac es or vortex sheets: any small bump on them must roll up spirally. This mec hanism, now c alled Helmholtz -Kelvin instability, plays an important role in many hydraulic and meteorologic al phenomena, as Helmholtz himself foresaw.16 Helmholtz not only meant to improve the applic ability of hydrodynamic s but also to equip this theory with a new mode of desc ription for fluid motion in whic h vortic es and disc ontinuity were the leading struc tural features. The enormous suc c ess of this projec t in the later history of hydrodynamic s is somewhat paradoxic al, bec ause Helmholtz 's theorems only hold in the unrealistic c ase of a perfec t liquid. The physic ists’ use of the vortic ity c onc ept in muc h more general situations is c omparable to the hydraulic ians’ use of the c onc ept of hydraulic head in situations in whic h Bernoulli's theorem does not apply. In some c ases of vortex motion, the effec ts of c ompressibility and visc osity c an be shown to be negligible. In all c ases, one c an take Helmholtz 's theorems as a referenc e and c orrec t them through terms derived from the Navier-Stokes equation, as Vilhelm Bjerknes did in the late nineteenth c entury. As for the vortex sheets, we will see in a moment that in the early twentieth c entury Ludwig Prandtl used them to approximately desc ribe important aspec ts of fluid resistanc e at high Reynolds number (low visc osity).17 In the historic al examples disc ussed so far, it bec ame inc reasingly diffic ult to produc e the needed new c onc eptual apparatus. The degree of diffic ulty c an be taken to be proportional to the time elapsed between the invention of Euler's equations and the introduc tion of this apparatus. For example, Bernoulli's law was easiest to derive, as it only requires a simple integration. But pure mathematic s did not suffic e to disc over the laws of wave propagation on a water surfac e. Some intuition of interferenc e proc esses (borrowed from optic s), and also a few experimental observations (groups of waves, solitary waves), were instrumental. The disc overy of the laws of vortex motion was even more diffic ult. A c entury elapsed from the time when d'Alembert and Euler gave the vortic ity equation to the time when Helmholtz interpreted it through his theorem. Experiments or observations did not by themselves suggest this interpretation, though Helmholtz 's efforts were, in fac t, part of a projec t for improving the theoretic al understanding of organ pipes. Helmholtz 's suc c ess primarily depended on his ability to c ombine various heuristic devic es inc luding algebraic manipulation in the style of Lagrange, geometric visualiz ation in the style of Thomson and Maxwell, and a foc us on invariant quantities as exemplified in his own work on energy c onservation. 1.4 Instabilities Exac t solutions of Euler's or Navier's equations under given boundary c onditions may differ widely from the flow observed in a c onc rete realiz ation of these c onditions. For instanc e, the flow of water in a pipe of rapidly inc reasing diameter never has the smooth, laminar c harac ter of exac t steady solutions of the Navier-Stokes equation in this c ase. As Stokes already suspec ted in the 1840s, this disc repanc y has to do with the instability of the exac t steady solutions: any small perturbation of these solutions will induc e wide departures from the original motion. Consequently, the knowledge of exac t solutions of the fundamental equations or (more realistic ally) the knowledge of some features of these solutions under given boundary c onditions is not suffic ient for the predic tion of observed flows. One must also determine whether these solutions or features are stable.18 In princ iple this question c an be mathematic ally dec ided, by examining how a slightly perturbed solution of the equations evolves in time. As we saw, a first suc c ess in this direc tion was Helmholtz 's predic tion of the spiral rolling up of a bump on a disc ontinuity surfac e. Later in the c entury, Lord Rayleigh and Lord Kelvin treated the more diffic ult problem of the stability of plane parallel flow. Their results were only partial (Rayleigh's inflec tion theorem in the nonvisc ous c ase), or wrong (Kelvin's predic tion of stability for the plane Poiseuille flow). Most of these questions Page 5 of 22

For a Philosophy of Hydrodynamics exc eeded the mathematic al c apac ity of nineteenth-c entury theorists, and some of them have remained unresolved to this day. The efforts of Rayleigh and others nonetheless yielded a general method and language of perturbative stability analysis. Rayleigh lineariz ed the equation of evolution of the perturbation, and sought plane-wave solutions. These solutions are “proper modes” whose osc illatory or growing c harac ter depends on the real or imaginary c harac ter of the frequenc y. This proper-mode analysis of stability goes beyond hydrodynamic s: it originated in Lagrange's c elestial mec hanic s and it c an be found in many other parts of physic s.19 As the mathematic al disc ussion of stability was nearly as diffic ult as the finding of exac t solutions of the fundamental equations, the most important results in this domain were reac hed by empiric al means. Plausibly, the observed instability of jets motivated Helmholtz 's derivation of the instability of disc ontinuity surfac es. Certainly, Tyndall's observations of this kind motivated Rayleigh's c alc ulations for parallel flow. Most important, Gotthilf Hagen (1839) and Osborne Reynolds (1883) disc overed that pipe flow, for a given diameter and a given visc osity, suddenly c hanged its c harac ter from laminar to turbulent when the veloc ity passed a c ertain c ritic al value. The sharpness of this transition was a surprise to all theorists. From Reynolds to the present, attempts to mathematic ally determine the c ritic al veloc ity (or Reynolds number) in c ylindric al pipes have failed. This is a question of ac ademic interest only, bec ause unpredic table entranc e effec ts (the way the fluid is introduc ed into the pipe), not the inherent instability in a pipe of infinite length, usually determine the transition.20 In the twentieth c entury, signific ant progress has been made in understanding the transition from laminar to turbulent flow. In the first half of the c entury, Ludwig Prandtl, Walter Tollmien, Werner Heisenberg, and Chia Chiao Lin proved the instability of the plane Poiseuille flow and unveiled the spatial periodic ity of the mec hanism of this instability.2 1 In the sec ond half of the c entury, developments in the theory of dynamic al systems at the intersec tion between pure mathematic s, meteorology, and hydrodynamic s permitted a detailed qualitative understanding of the transition to turbulenc e, with intermediate osc illatory regimes, bifurc ations, and strange attrac tors.2 2 It remains true that most of the prac tic al applic ations of hydrodynamic s only require a rough knowledge of the c onditions under whic h turbulenc e oc c urs. The sourc e of this knowledge is partly theoretic al and partly empiric al. There is no easy way to gather it from the fundamental equations. In most c ases, the best that c an be done is to repeat Reynolds's rough argument that the full vortic ity equation has two terms, a visc ous term that tends to damp any eddying motion, and an inertial term whic h preserves the global amount of vortic ity. The laminar or turbulent c harac ter of the motion depends on the ratio of these two terms, whose order of magnitude is given by the Reynolds number. 1.5 Turbulence The state of motion that follows the turbulent transition is even more diffic ult to analyz e than the transition itself. Casual observation of turbulent flow reveals its c haotic and multi-sc ale c harac ter. The detailed desc ription of any motion of this kind seems to require a huge amount of information, muc h more than is humanly ac c essible (without c omputers at least). As Reynolds pondered, we are here fac ing a situation similar to that of the kinetic theory of gases: the effec tive degrees of freedom are too numerous to be handled by a human c alc ulator. Unfortunately, turbulent motion is more often enc ountered in nature and in manmade hydraulic devic es than laminar motion. Engineers and physic ists have had to invent ways of c oping with this diffic ulty.2 3 One strategy is to design the hydraulic or aeronautic artifac ts so that turbulenc e does not oc c ur. When turbulenc e c annot be avoided, one may adopt a purely empiric al approac h and seek relations between measured quantities of interest. For instanc e, nineteenth-c entury engineers gave empiric al laws for the retardation (loss of head) in hydraulic pipes. A sec ond approac h is to find rules allowing the transfer of the results of measurements done at one sc ale to another sc ale. Stokes, Helmholtz , and Froude pioneered this approac h in the c ontexts of pendulum damping, balloon steering, and ship resistanc e, respec tively. They derived the needed sc aling rules from the sc aling symmetries of the Navier-Stokes equation or of the underlying dynamic al princ iples. This is an example of a hybrid approac h, founded partly on the fundamental equations, and partly on measurements of theoretic ally unpredic table properties.2 4 In a third approac h, one may c ompletely ignore the foundations of fluid mec hanic s and c ook up a model based on a grossly simplified pic ture of the flow. An important example is the laws for open c hannel flow disc overed in the 1830s and 1840s by a few Frenc h Polytec hnique-trained engineers: Jean Baptiste Bélanger, Jean Vic tor Ponc elet, and Gaspard Coriolis. They assumed the flow to oc c ur through parallel slic es that rubbed against the bottom of the c hannel ac c ording to a phenomenologic al fric tion law, and they applied momentum or energy balanc e to eac h 25 Page 6 of 22

For a Philosophy of Hydrodynamics slic e.2 5 In the 1840s Adhémar Barré de Saint-Venant emphasiz ed the “tumultuous” c harac ter of the fluid motion in open c hannel flow and suggested a distinc tion between the large-sc ale average motion of the fluid and the smaller-sc ale tumultuous motion. The main effec t of the latter motion on the former, Saint-Venant argued, was to enhanc e momentum exc hange between suc c essive (large-sc ale) fluid layers. Based on this intuition, he replac ed the visc osity in the Navier-Stokes equation with an effec tive visc osity that depended on various mac rosc opic c irc umstanc es suc h as the distanc e from a wall. In the 1870s, Boussinesq solved the resulting equation for open c hannels of simple sec tion and thus obtained laws that resembled Bélanger's and Coriolis's laws, with different interpretations of the relevant parameters.26 In 1895, Reynolds relied on analogy with the kinetic theory of gases to develop an explic itly statistic al approac h to turbulent flow. In the spirit of Maxwell's kinetic -molec ular derivation of the Navier-Stokes equation, he derived a large-sc ale equation of fluid motion by averaging over the small-sc ale motions governed by the Navier-Stokes equation. Reynolds's equation depends on the “Reynolds stress,” whic h desc ribes the turbulent exc hange between suc c essive mac ro layers of the fluid. Like Saint-Venant's effec tive visc osity, the Reynolds stress c annot be determined without further assumptions c onc erning the turbulent fluc tuation around the large-sc ale motion. There have been many attempts to fill this gap in the twentieth c entury. The most useful ones were Kármán's and Prandtl's derivations of the logarithmic veloc ity profile of a turbulent boundary layer. The assumptions made in (improved) versions of these derivations are simple and natural (uniform stress, matc hing between the turbulent layer and a laminar sublayer next to the wall), and the resulting profile fits experiments extremely well (muc h better than earlier phenomenologic al laws). The logarithmic profile is the basis of every modern engineering c alc ulation of retardation in pipes or open c hannels.2 7 Despite powerful studies by Geoffrey Taylor, Andrey Nikolaevic h Kolmogorov, and many others, the prec ise manner in whic h turbulenc e distributes energy between different sc ales of fluid motion remains a mystery.2 8 There is no doubt, however, that the general idea of desc ribing turbulent flow statistic ally has been fruitful sinc e its first intimations by Saint-Venant, Boussinesq, and Reynolds. In the c ase of turbulent fluid mec hanic s, as in statistic al mec hanic s, a new c onc eptual struc ture emerges at the mac rosc ale of desc ription. Similar questions c an be raised in both c ases c onc erning the nature of the reduc tion or emergenc e. Does the mic rosc ale theory truly imply the mac rosc ale struc ture? Is this struc ture uniquely defined? Can this struc ture be used without further referenc e to the mic rosc ale? Are there singular situations in whic h the reduc tion fails? The answers to these questions tend to be more positive in the c ase of statistic al mec hanic s than in the c ase of the statistic al theory of turbulenc e, bec ause the relevant statistic s are better known in the former than in the latter c ase. 1.6 Boundary Layers From a prac tic al point of view, two outstanding problems of fluid mec hanic s are fluid resistanc e and fluid retardation. Fluid resistanc e is the dec elerating forc e experienc ed by a rigid body moving through a fluid. Fluid retardation is the fall of pressure or loss of head experienc ed by a fluid during its travel along pipes or c hannels. The two problems are related, sinc e they both involve the mutual ac tion of a fluid and an immersed solid. In 1768, d'Alembert c hallenged “the sagac ious geometers” with the paradox that resistanc e vanished for a perfec t liquid in his new hydrodynamic s. There were various strategies to c irc umvent this theoretic al failure. Some engineers determined by purely empiric al means how the resistanc e depended on the veloc ity and shape of the immersed body. Others retreated to Isaac Newton's naïve theory by the impac t of fluid partic les on the front of the body, although some c onsequenc es of this theory (suc h as the irrelevanc e of the shape of the end of the body) had already been refuted. In the mid-nineteenth c entury, Saint-Venant, Ponc elet, and Stokes trac ed resistanc e to visc osity and the produc tion of eddies. With the damping of pendulums in mind, Stokes suc c essfully determined the resistanc e of small spheres and c ylinders by finding solutions to the lineariz ed Navier-Stokes equation. For most prac tic al problems, the larger siz e of the immersed body and the smallness of the visc osities of air and water imply that the nonlinear term of this equation c annot be neglec ted (the Reynolds number is too high). Stokes had nothing to say in suc h c ases beyond the qualitative idea of dissipation by the produc tion of eddies.2 9 Page 7 of 22

For a Philosophy of Hydrodynamics Figure 1.2 Discontinuity surface (ee′) formed when a downward flow encounters the disk A. From Thomson (1894, 220). In the ideal c ase of vanishing visc osity, the proof of d'Alembert's paradox implic itly assumes the c ontinuity of the fluid motion. However, Helmholtz 's study of vortex motion implies that finite slip of fluid over fluid is perfec tly c ompatible with Euler's equations. Around 1870, Kirc hhoff and Rayleigh realiz ed that Helmholtz 's disc ontinuity surfac es yielded a finite resistanc e for an immersed plate. Ac c ording to Helmholtz , a tubular disc ontinuity surfac e is indeed produc ed at the sharp edges of the plate. The water behind the plate and within this surfac e is stagnant, so that its pressure vanishes (when measured in referenc e to its uniform value at infinite distanc es from the plate) (see figure 1.2). Sinc e the pressure at the front of the plate is positive, there is a finite resistanc e, whic h Kirc hhoff and Rayleigh determined by analytic al means. The result roughly agreed with the measured resistanc e.3 0 In the c ase of ships, the resistanc e problem is c omplic ated by the fac t that ships are not supposed to be c ompletely immersed. Consequently, wave formation at the water surfac e is a signific ant c ontribution to the resistanc e. The leading nineteenth c entury experts on this question, William John Mac quorn Rankine and William Froude, distinguished three c auses of resistanc e: wave resistanc e, skin resistanc e, and eddy resistanc e. Skin resistanc e c orresponds to some sort of fric tion of the water when it travels along the hull. Eddy resistanc e c orresponds to the formation of eddies at the stern of the ship; it is usually avoided by proper profiling of the hull. Rankine and Froude traced skin resistance to the formation of an eddying fluid layer next to the hull. They derived this notion from the observation that the flow of water around the ship, when seen from the deck, appears to be smooth everywhere expect for a narrow tumultuous layer next to the hull and for the wake. Rankine assumed the validity of Euler's equations in the smooth part of the flow and solved it to determine the hull shapes that minimiz ed wave formation. Froude gave a fairly detailed desc ription of the mec hanism of retardation in the eddying layer, although he was not able to draw quantitative conclusions. In the end, Froude measured skin friction on plates, total resistanc e on small-sc ale ship models, and then used separate sc aling laws for skin and wave resistanc e in order to determine the resistanc e of a prospec tive ship hull.3 1 Figure 1.3 Formation of a discontinuity surface behind a cylinder. From Prandtl (1905, 579–80). In sum, Rankine and Froude distinguished two different regions of flow amenable to different theoretical or semi- empirical treatments and combined the resulting insights to determine the total resistance. Froude thus obtained the first quantitative successes in the problem of fluid resistance at a high Reynolds number. Although his and Rank- ine's considerations appealed to higher theory in several manners, they also required considerable empirical input. The next and most famous progress in the high Reynolds-number resistance problem occurred in Göttingen, under the leadership of Ludwig Prandtl. Impressed by the qualitative suc c ess of Helmholtz 's surfac es of disc ontinuity, Prandtl assumed that the solution of the Navier-Stokes equation for high-Reynolds flow around a body somewhat resembled a solution of Euler's equation (with strictly vanishing viscosity). In the latter solution, the fluid slides along the surface of the body, whereas for a viscous fluid the relative velocity of the fluid must vanish at the surface of the body. Consequently, for the real flow Prandtl assumed a thin (invisible) layer of intense shear that Page 8 of 22

For a Philosophy of Hydrodynamics imitated the finite slide of the Eulerian solution. He also assumed that in some c ases this layer c ould shoot off the surfac e of the body to mimic a Helmholtz ian surfac e of disc ontinuity (with its c harac teristic instability resulting in an eddying trail). This is the so-c alled separation proc ess. Outside the boundary layer, Prandtl naturally applied Euler's equations. Within the boundary layer, the intense shear allowed him to use an approximation of the Navier-Stokes equation that c ould be integrated to determine the evolution of the veloc ity profile along the body. For suffic iently c urved bodies, Prandtl found that at some point the flow was inverted in the part of the boundary layer c losest to the body. He interpreted this point as the separation point from whic h a (quasi) disc ontinuity surfac e was formed. In the c ase of a flat or little c urved surfac e (for whic h separation does not oc c ur), he determined the resistanc e by integration of the sheer stress along the surfac e of the body. He illustrated the separation proc ess through experiments done with a tank and a paddle-wheel mac hine (figure 1.3).3 2 Comparison with Froude's earlier c onc ept of eddying layer leads to the following remarks. Unlike Froude, Prandtl was able to determine theoretic ally and prec isely the flow within the boundary layer. This determination requires a previous solution of the Eulerian flow problem around the body, bec ause the evolution of the boundary layer depends on the pressure at its c onfines. Conversely, this evolution may induc e separation, whic h nec essarily affec ts the Eulerian part of the flow. Prandtl himself emphasiz ed this interac tion between the Eulerian flow and the boundary layer. Whereas Froude had no interest in separation (whic h ship builders systematic ally avoided), Prandtl had a prec ise c riterion for its oc c urrenc e. Whereas Froude c ould only measure the sheer stress of the boundary layer, Prandtl c ould determine it theoretic ally. So far the c omparison seems to favor Prandtl. In reality, in many c ases inc luding ship resistanc e, the boundary layer has an internal turbulenc e that is not taken into ac c ount in Prandtl's original theory. In 1913, Prandtl's former student Heinric h Blasius found that beyond a c ertain c ritic al Reynolds number, the edgewise resistanc e of a plate obeyed Froude's empiric al law and not Prandtl's theoretic al law. Prandtl explained that the profile of a laminar boundary layer c ould bec ome unstable and thus lead to a turbulent boundary layer à la Froude. He used this notion to explain the biz arre reduc tion of the resistanc e of spheres that Gustave Eiffel had observed at a c ertain c ritic al veloc ity: turbulenc e in a boundary layer, Prandtl explained, delays the separation proc ess and thus sharply dec reases the resistanc e. Paradoxic ally, it is when the boundary layer is turbulent that the global flow mostly resembles the smooth Eulerian flow.33 As the boundary layers around airplane wings are turbulent, Prandtl needed to know the sheer stress along suc h layers in order to determine the drag of the wings. He originally relied on plate resistanc e measurements, as Froude had done in the past. As was already mentioned, it bec ame possible to c alc ulate this stress in the 1830s when Kármán and Prandtl disc overed the logarithmic veloc ity profile of turbulent layers. It is now time to reflec t on the relation that boundary-layer theory has to the foundational theory of Navier-Stokes. Prandtl's idea (if we believe his own plausible ac c ount) has its theoretic al origin in the idea of using solutions to Euler's equations as a guide for solving the Navier-Stokes equation at a high Reynolds number. This is only a heuristic , bec ause Prandtl had no mathematic al proof that the low-visc osity limit of a solution of the Navier-Stokes equation is a solution of Euler's equation. Yet the motion imagined by Prandtl, with its Eulerian, high-sheer, and stagnant regions, c learly is an approximate solution of the Navier-Stokes equation. What is missing is a proof of the uniqueness of this solution (under given boundary c onditions), as well as a general proof of its existenc e for any shape of the immersed body. With this c onc ession, the boundary-layer theory c an legitimately be regarded as an approximation of the Navier-Stokes theory. An interesting feature of the boundary-layer theory is its use of different approximate equations in different regions of the flow. Our disc ussion of Bernoulli's law showed that this law is often used regionally (i.e., in laminar regions of the flow) with head losses loc aliz ed in turbulent regions. Boundary-layer theory similarly introduc es different regions of flow, although it does so in a more interac tive manner. Eac h region is desc ribed through c omputable solutions of appropriate equations of motion, and the prec ise c onditions for the matc hing of the regional solutions are known (c ontinuity of pressure, stress, and veloc ity). These matc hing c onditions imply c ausal relations between features of the two regions: for instanc e, the pressure distribution in the Eulerian region determines the evolution of the veloc ity profile in the boundary layer, and in the c ase of separated flow, the position of the separating surfac e affec ts the Eulerian region.3 4 In qualitative applic ations, Prandtl's theory may be restric ted to the general ideas of a boundary layer, a free fluid, Page 9 of 22

For a Philosophy of Hydrodynamics and their interac tion sometimes leading to separation. In quantitative engineering applic ations, this pic ture must be supplemented with a law for the evolution of the sheer stress along a boundary layer (laminar or turbulent), and with quantitative c riteria for separation and for the transition between laminar and turbulent layer. Granted that this supplementary information is available, the theory c an be used without referenc e to the Navier-Stokes theory. The gain in predic tive effic ienc y is enormous, as verified by the immense suc c ess of Prandtl's theory in engineering applic ations. Yet one should not forget that muc h of the supplementary information c omes from the intimate c onnec tion between the boundary-layer theory and the Navier-Stokes theory. In fac t the legitimac y of the whole pic ture depends upon this intimate c onnec tion. The boundary-layer theory, unlike the early Frenc h models of open c hannel flow, is not an ad hoc model that owes its simplic ity to c ounterfac tual assumptions. It is a legitimate articulation of the Navier-Stokes theory. 2. Explanato ry Pro gress The above examples make c lear that in the c ourse of its history, hydrodynamic s has ac quired a sophistic ated explanatory apparatus without whic h it would remain merely a “paper” theory. The explanatory apparatus is presented in various c hapters in modern textbooks. We will now reflec t on the ways this apparatus was obtained, on its c omponents, and on its func tions. 2.1 The Sources of Explanatory Progress In some c ases, explanation was improved through blind mathematic al methods. For instanc e, a simple integration yielded Bernoulli's law (after proper spec ializ ation), the symmetries of the Navier-Stokes equation yielded sc aling laws, and standard approximation proc edures yielded the theory of waves of small amplitude. Despite the relatively easy and automatic way in whic h these results were obtained, they c onsiderably improved the explanatory power of the theory by direc tly relating quantities of physic al interest. In other c ases, more intra- or intertheoretic al heuristic s was needed. Kinematic analysis of the vortic ity equation led to Helmholtz 's vortex theorems; asymptotic reasoning led to Prandtl's notions of laminar boundary layer and separation; sc aling and matc hing arguments led to the logarithmic veloc ity profile of turbulent boundary layers. These heuristic s required an unusual amount of c reativity; they involved intuitions bound to personal styles of thinking. Suc h intuitions are tentative and may lead to erroneous guesses. For instanc e, the great Kelvin erred in his stability analysis of parallel flow. A rigorous c hec k of the c ompatibility of the c onc lusions with the fundamental equations is always needed.35 In still other c ases, observations or experiments suggested new c onc epts suc h as group veloc ity, solitary waves, the stability or instability of laminar flow, and turbulent boundary layers. The very fac t that pure theory was historic ally unable to lead to these c onc epts (and sometimes even resisted their introduc tion) shows the vanity of regarding them as implic it c onsequenc es of the fundamental equations. They nevertheless belong to fundamental hydrodynamic s inasmuc h as their c ompatibility with the fundamental equations c an be verified a posteriori. Lastly, the impossibility of solving the fundamental equation and the evident c omplexity of observed flows sometimes forc ed engineers and even physic ists to arbitrarily and drastic ally simplify aspec ts of the flow. This happened for instanc e in early models of open c hannel flow. These models c annot be stric tly regarded as parts of fundamental hydrodynamic s, sinc e some of their assumptions c ontradic t both observed and theoretic al properties of the flow. Yet their suc c ess suggests a looser sort of relation with the Navier-Stokes theory. In the c ase of open- c hannel flow, the models c an be reinterpreted as re-parametriz ations of the true equations for the approximate, large-scale motion derived from turbulent solutions of the Navier-Stokes equations. In every c ase, the theoretic al developments oc c urred with spec ific applic ations in mind: some kind of flow frequently observed in nature needed to be explained or the func tioning of some instruments or devic es needed to be understood. Purely mathematic al methods broadly applied to general flow were of little avail. Insight was gained as a result of investigation direc ted at c onc rete and restric ted goals. This is why the heroes of nineteenth-c entury and early twentieth-c entury fluid mec hanic s were either mathematic ally fluent engineers or physic ists who had a foot in the engineering world.

For a Philosophy of Hydrodynamics 2.2 The Components of Explanation A first alley toward better explanation involves the restric tion of the sc ope of a theory. The Navier-Stokes equations, regarded as the general foundation of hydrodynamic s, c an be spec ializ ed in various ways. There are homogeneous spec ializ ations or idealiz ations in whic h the restric ted c hoic e of parameters and kinds of systems (boundary c onditions) leads to more trac table integration problems or suc c essful statistic al approac hes. Typic al examples are irrotational Eulerian flow, low Reynolds-number flow, and fully turbulent flow. There are also heterogeneous spec ializ ations in whic h the restric tions on parameters and systems lead to flows that have different regions, eac h of whic h depends upon a spec ific simplific ation of the Navier-Stokes equations. This is the c ase for the high-Reynolds resistanc e problem and the airplane wing problem ac c ording to Prandtl. As was already mentioned, suc c ess here requires proper matc hing between the different regions. Another explanatory resourc e is the identific ation of invariant struc tures of a flow belonging to a given c lass. The most impressive example of this sort is Helmholtz 's demonstration of the c onservation of vortex filaments. As the mind tends to foc us on invariant aspec ts of our environment, the identific ation of new invariants often shape our desc riptive language. As Helmholtz predic ted, this has, in fac t, happened in fluid mec hanic s: the vortic ity field is now often preferred to the veloc ity field as a desc ription of flow. Third, instead of seeking struc ture in a given solution, we may attend to the struc ture of the spac e of solutions of the fundamental equation when the boundary c onditions vary. For instanc e, we may ask whether laminar solutions are typical, whether small perturbations lead to different sorts of solutions: this is the issue of stability. We may also ask whether some c lasses of solution share c ommon large-sc ale features, as we do in the statistic al theories of turbulenc e. And, we may ask whether some properties or laws are generic in some regime of flow: this is the issue of universality, whic h we briefly touc hed with the logarithmic profile of turbulent boundary layers. Lastly, explanation and understanding may c ome from linking hydrodynamic s to other theories. We have enc ountered a few examples of this kind: potential theory, wave interferenc e, group veloc ity, solitary waves, field kinematic s, and proper-mode analysis of stability. In half of these c ases, c onc epts of hydrodynamic origin were brought to bear on other theories and not vic e versa. The c ross-theoretic al sharing of c onc epts nonetheless remains a token of their explanatory value. 2.3 A Pragmatic Definition of Explanation As was stated above, the goal of fluid mec hanic s c annot be reduc ed to finding integrals of the fundamental equations that satisfy given boundary c onditions. This is usually impossible by analytic al means, and modern numeric al means require a different simulation for eac h c hoic e in the infinite variety of boundary c onditions. As Batterman, Ramsey, and Heidelberger have argued, bare foundations do not answer the questions that truly interest physic ists and engineers. Prac titioners want to be able to c harac teriz e a physic al situation by a humanly ac c essible number of physic al parameters and to possess a pic ture of the situation that enables them to derive relations between these parameters in a reasonable amount of time. In other words, they need a c onc ept of explanation that integrates our human c apac ity at representing and intervening. As Batterman emphasiz es, this requires means for eliminating irrelevant details in our desc ription of systems. This also implies the elaboration of a desc riptive language, the c onc epts of whic h direc tly refer to c ontrollable features of the system.3 6 With this pragmatic definition of explanation, it bec omes c lear that the earlier desc ribed developments of hydrodynamic s served the purpose of inc reasing the explanatory power of the theory. Homogeneous spec ializ ations do so by offering adequate c onc epts and methods for c ertain kinds of flow. Heterogeneous spec ializ ations do so by c ombining the former spec ializ ations to desc ribe flows that oc c ur in problems of great prac tic al import. The identific ation of invariant struc tures for c ertain c lasses of motion improves the ec onomy of the representation. Attention to struc ture in the spac e of solutions enables us to dec ide to what extent smaller details of the motion affec t the features of prac tic al interest, and to what extent their effec t c an be smoothed out by some averaging proc ess. Intertheoretic al links produc e familiar c onc epts that c an indifferently be used in various domains of physic s. In this light, the prac tic e of physic s has more similarity with engineering than is usually assumed. The remark is not unc ommon in rec ent writings in the philosophy of sc ienc e. For instanc e, Ramsey revives J. J. Thomson's old

For a Philosophy of Hydrodynamics c harac teriz ation of theories as tools for solving physic s or engineering problems; Epple c ompares the formation of Prandtl's wing theory to an engineering proc ess c ombining multiple theoretic al and experimental resourc es. In these sc holars’ view, the engineer only differs from the physic ist by (usually) not partic ipating in the invention of the theories and by his more systematic appeal to extra-theoretic al c omponents. Physic ists and engineers not only share the goal of effic ient intervention, they also share some of the means.3 7 Ramsey and Heidelberger insist that the artic ulation of theories implies the formation of new, adequate c onc epts. One c ould even argue that the bare Navier-Stokes theory has no physic al c onc epts. It harbors only mathematic al c onc epts suc h as the veloc ity field that c orrespond to an ideal desc ription of the flow, ignoring molec ular struc ture and presuming indefinite resolution. A c onc ept, in the etymologic al sense of the word (concipio in Latin, or begreifen in German), is a mental means to grasp some c onc rete objec t or situation. Hydraulic head, vortic es, wave groups, solitary waves, the laminar-turbulent transition, boundary layers, separation, etc . are c onc epts in this prac tic al sense. The detailed veloc ity field or the various terms of the Navier-Stokes equation are not. What Thomas Kuhn onc e belittled as the “mopping up” of theories in the normal phases of sc ienc e truly is c onc ept formation.38 3. Theo ries And Mo dules 3.1 Defining Physical Theories Onc e we rec ogniz e the c ognitive impotenc e of the bare foundations of a theory, we need a general definition of “theory” that is not limited to the fundamental equations and a few naïve rules of applic ation. The definition must allow for evolving c omponents, sinc e the c ognitive effic ienc y of any good theory always inc reases in time. It must inc lude explanatory devic es and it must allow the intertheoretic al c onnec tivity found in mature theories. The following is a sketc h of suc h an enric hed definition.3 9 A physic al theory is a mathematic al c onstruc t inc luding: (a) a symbolic universe in whic h systems, states, transformations, and evolutions are defined by means of various magnitudes based on Cartesian powers of R (or C ) and on derived func tional spac es. (b) theoretical laws that restric t the behavior of systems in the symbolic universe. (c) interpretive schemes that relate the symbolic universe to idealiz ed experiments. (d) methods of approximation and c onsiderations of stability that enable us to derive and judge the c onsequenc es that the theoretic al laws have on the interpretive sc hemes. The symbolic universe and the theoretic al laws are permanently given. They c orrespond to the “family of models” of the semantic view of physic al theories. In the c ase of hydrodynamic s, the symbolic universe c onsists in the veloc ity, pressure, and density fields for eac h fluid of the system, in the boundaries of rigid bodies that may or may not move, and in forc e densities suc h as gravity. The theoretic al laws are the Navier-Stokes equations, boundary c onditions, and (for c ompressible fluids) a relation between density and pressure that may involve modular c oupling with thermodynamic s (we will return to this point). In the semantic view of theories, the empiric al c ontent of a theory is defined by an isomorphism between parts of the symbolic universe and empiric al data; although the means by whic h this isomorphism is determined are usually left in the dark. The notion of an interpretive sc heme is intended to fill part of this gap. By definition an interpretive scheme consists in a given system of the symbolic universe together with a list of characteristic quantities that satisfy the three following properties.(1) They are selected among or derived from the (symbolic) quantities that define the state of this system. (2) At least for some of them, ideal measuring procedures are known. (3) The laws of the symbolic universe imply relations of a functional or a statistical nature among them. More spec ific ally, interpretive sc hemes are blueprints of c onc eivable experiments whose outc omes depend only on relations between a finite set of mutually related quantities, a suffic ient number of whic h are measurable. In some c ases, the intended experiments may be designed to determine some theoretic al parameters from the measured quantities. In other c ases, the theoretic al parameters are given, and theoretic al relations between the measured quantities are verified. In all c ases, the interpretive sc hemes do not c ontain rigid linguistic c onnec tions between theoretic al terms and physic al quantities; their c onc rete implementation is analogic al, historic al, and subjec t to revisions.40

For a Philosophy of Hydrodynamics The introduc tion of interpretive sc hemes implies a selec tion of systems and quantities from the infinite variety of elements in the symbolic universe of the theory. This selec tion c an evolve dramatic ally with the number and nature of the imagined applic ations of the theory. The two main c lasses of interpretive sc hemes of early hydrodynamic s were the pierc ed vessel, in whic h the efflux of water is related to the height of the water surfac e; and the resistanc e sc heme in whic h a solid body immersed in a stream of water experienc es a forc e related to the veloc ity of the stream. Another interesting sc heme, Bernoulli's pipe of variable sec tion, implied pressure measurement through vertic al c olumns of water. A sample of later sc hemes inc ludes the determination of the veloc ity of surfac e waves as a func tion of depth and wavelength, the visualiz ed motion of vortic es as a func tion of their relative c onfiguration, the visualiz ed lines of flow around an immersed body as a func tion of the asymptotic veloc ity, the drag and lift of a wing as a func tion of asymptotic veloc ity and angle of attac k. Some sc hemes were reac tions to well-identified prac tic al problems and others to some new theoretic al development. In the latter c ategory, we may cite the determination of the separation point for the flow around an immersed sphere, the measurement of instability thresholds, Prandtl's aspiration of the boundary layer to prevent separation, and the post-theoretic al visualiz ation of laminar boundary layers. For an interpretive sc heme to serve its purpose as an experimental blueprint, a few c onditions must be met: one must know how to realiz e c onc retely the system pic ked in the symbolic universe; one must know how to implement the ideal measuring proc edures; one must be able to c ompute the relations between measured quantities and theoretical parameters; and one must know something about the stability of these relations. Point (d) of my general definition of theories is meant to meet these two last requirements. In this regard, the reader may c onsult the growing literature regarding the philosophy of approximation, numeric al analysis, and stability. The following disc ussion is restric ted to aspec ts of the working of interpretive sc hemes that have to do with the modular structure of theories.41 3.2 Modules By definition, a module is a c omponent of a theory whic h is itself a theory, with a different domain of applic ation. Our ability to apply a theory c ruc ially depends on integrated modules. First, there are defining modules that serve to define some of the quantities in the symbolic universe. In the c ase of hydrodynamic s, the list of these modules inc ludes a Euc lidian geometric al module that defines the spatial relations of the systems; a mec hanic al module that defines external forc e densities, external pressures, and the motion of immersed bodies; a thermodynamic module that defines relations between fluid density, pressure, and temperature (sometimes also heat transfer). These modular definitions enable us to transfer already known measuring proc edures into the interpretive sc hemes of hydrodynamic s. In the c ase of c ompressible fluids, they are essential to the c ompleteness of the theory: no predic tion c an be made without knowing how the density varies ac c ording to the thermal properties of the system. Sec ond, there are idealizing modules obtained by simplifying the symbolic universe and retaining similar interpretive sc hemes (of c ourse, the func tional relations between sc hematic quantities are different). In the c ase of hydrodynamic s, the most important modules of this kind are the theory of inc ompressible fluids, the theory of invisc id fluids, and the theory of inc ompressible invisc id fluids. Inc ompressibility enables us to ignore the c oupling of hydrodynamic s with thermodynamic s. Invisc idity eliminates one term in the Navier-Stokes equations and yields Euler's simpler equations. The usefulness of these idealiz ations c omes from the relative smallness of the c ompressibility of water and from the smallness of the visc osities of air and water. Third, there are specializing modules that are exac t substitutes of the theory for subc lasses of sc hemes that meet c ertain c onditions. For instanc e, Lagrange's theory of irrotational inc ompressible fluid motion c an replac e Euler's theory for sc hemes in whic h the fluid motion is started from rest by the motion of walls or immersed bodies; Helmholtz 's theory of vortex motion c an replac e the inc ompressible spec ializ ation of Euler's theory for sc hemes based on the vortex struc ture. Idealiz ing and spec ializ ing modules are not by themselves suffic ient to design effec tive interpretive sc hemes. We also need approximating modules that c an be seen as limits of the theory for a given subc lass of systems when a parameter of this c lass or a parameter of the symbolic universe (or a c ombination of both kinds of parameters) takes extreme but still finite values (the limit may involve statistic al c onsiderations). Hydrodynamic examples of modules of this kind c onc ern the small-depth and small-amplitude limits of surfac e wave sc hemes, the high

For a Philosophy of Hydrodynamics Reynolds-number limit of fluid resistance or fluid retardation schemes (boundary-layer theory), and the low Reynolds-number limit of these schemes (creeping flow). In most cases, it is only at the level of approximating modules that the func tional relations between sc hematic quantities c an be effec tively c omputed.4 2 There is a last kind of modules, the reducing modules, that has more to do with the foundations of the theory than with its applic ations. These are theories diverted from their original domain of applic ation in order to build the whole symbolic universe of another theory.4 3 This is what happens, for instanc e, when the mec hanic s of a system of interac ting mass points is used in Clerk Maxwell's manner as a molec ular-kinetic -theoretic al foundation for the Navier-Stokes equation.4 4 There is a differenc e between saying that T is a reduc ing module of T′ and saying that T′ is an approximating module of T: in the latter c ase, the sc hemes of T′ are a subc lass of those of T, whereas in the former c ase the sc hemes of T have nothing to do with the sc hemes of T′ (they lose their empiric al realiz ability in the reduc ing proc ess). In the c ase of reduc ing modules, the theory T is nec essarily known before the reduc tion is done and the theory T′ may even be invented through the reduction, as was the case with Maxwell's theory of elec trodynamic s. With approximating modules, the theory T may or may not prec ede the theory T′. Whereas the Navier-Stokes theory prec eded its boundary-layer module, Euler's hydrodynamic s postdated its narrow-vase module à la Bernoulli. Maxwell's elec trodynamic s postdated its quasi-stationary module and wave optic s postdated its rays-optic s module. Modules, qua theories, c an have submodules. For instanc e, the inc ompressible idealiz ing module of the Navier- Stokes theory has an invisc id spec ializ ing module. More interestingly, the boundary-layer theory, as an approximating module, relies on defining modules that are idealiz ing, spec ializ ing, or approximating modules of the Navier-Stokes theory. These defining modules respec tively c orrespond to invisc id fluid motion (in the “free fluid”), disc ontinuity surfac es (in the c ase of separation), and the boundary-layer equation. This means that a module of a theory c an also be a submodule of another module of the same theory (see figure 1.4). It also means that the same theory c an be a module of different theories. More evident examples of multiply inserted modules are Euc lidian geometry and Newtonian mec hanic s, whic h are defining modules of all the main theories of c lassic al physic s. Figure 1.4 Some of the modular structure of modern hydrodynamics. The solid arrows correspond to spec ializing or approximating modules, the dotted arrows to defining or idealizing modules. The modular structure varies as the theory develops. The defining modules are there, by necessity, from beginning to end. Reduc ing modules may oc c ur at any stage of the life of the theory: at its birth, in its middle age, or even at its death. An instanc e of the last c ase oc c urred when the elec tromagnetic theory of light replac ed elastic -solid theories of light. Spec ializ ing and approximating modules are gradually introduc ed, for the sake of mathematic al simplific ation and effic ient applic ation. The status of a module may vary. For instanc e, a defining module may become a reducing module or vice versa. In the course of the history of electrodynamics, mechanics was succes- sively a defining module (Coulomb, Ampère, Neumann, Weber), a reducing module (Thomson, Maxwell), and again a defining module (Hertz). This variability of the status of modules is the reason why I have introduced a fairly wide spec trum of modular interrelations. As I have argued elsewhere4,5 modules play an essential role in the applic ation, c onstruc tion, c omparison, and communication of theories. In the case of hydrodynamics, the role of modules in permitting efficient applications of the theory is most evident. They yield conceptual structures that are better adapted to concrete problem situ- ations than the bare Navier-Stokes equation. They instruct us about the choice of accessible, causally interrelated aspects of fluid motion and they tell us how to measure them. Through a nesting hierarchy of modules, we can c apitaliz e on our c onc rete knowledge of the sc hemes of the most basic modules to imagine and c ontrol the

For a Philosophy of Hydrodynamics c omplex experimental environment through whic h the predic tions of higher-level theories are tested. The c onstruc tive role of modules is evident in the c ase of defining and reduc ing modules. Idealiz ing, spec ializ ing, and approximating modules also help theory c onstruc tion when they are known before the projec tive theory. They may play an instrumental role in theoretic al unific ation or in the rejec tion of a tentative unific ation. And they may provide a “c orrespondenc e princ iple” for guiding the design of the symbolic universe of a new theory, as was the c ase when Bohr and Heisenberg appealed to c lassic al elec trodynamic s in the c onstruc tion of quantum theory. The c omparison of two theories requires shared interpretive sc hemes whose c onc rete realiz ation is not tied to either of these theories. This is possible if all the sc hematic quantities c an be defined by means of shared modules. For example, the predic tions of various nineteenth-c entury theories of elec trodynamic s c ould be c ompared thanks to the sharing of elec trostatic , elec trokinetic , and magnetostatic modules. Shared modules are also essential for the c ommunic ation between different subc ultures of physic s and other c ommunities of sc ientists and engineers who use physics in their work. These shared modules enable someone to use results of a theory whose foundation he ignores or even rejec ts. They permit the sharing of apparatus whose func tioning depends only on lower-level modules. Lastly, modular struc ture is essential to the teac hing of theories. A typic al textbook is organiz ed by c hapters that c orrespond to modules of the theory. Thus, the student c an c onnec t the new theory to other theories with whic h he is already familiar, he c an get a grasp on how to apply the theory in c onc rete situations, and he c an learn tec hniques that transc end the domain of this theory. 3.3 Models and Modules In rec ent philosophy of sc ienc e, there has been a strong emphasis on models as mental c onstruc ts that differ both from full-fledged theory and from narrow empiric al induc tion. Mary Morgan and Margaret Morrison regard models as mediating instruments between theory and phenomena. In their view, models are partially autonomous from theory: some of their c omponents have extratheoretic al origins. The models help to shape theories as muc h as they rely on theory. They are more direc tly relevant to the empiric al world than theories, at the pric e of a more limited sc ope. For all these reasons, Morgan and Morrison insist that models are not theories. Yet (physic s) models fit my definition of theories, sinc e they nec essarily have a symbolic universe, internal laws, and interpretive sc hemes. In my view, they differ from other theories only by having a smaller sc ope or less struc tural unity. This differenc e is largely a matter of degree and c onvention. The partial autonomy of models from more fundamental theories results from the modular c harac ter of their interc onnec tion with these theories. Typic ally, fundamental theories are defining or reduc ing modules of models; or else models are approximating modules of a more fundamental theory.4 6 The relation between models and theories is just a partic ular c ase of the modular relation between two theories. It therefore implies the same sort of mutual fitness without fusion. There is no need to sharply disc riminate models from theories onc e the modular struc ture of theories is taken into ac c ount. It is suffic ient to rec ogniz e that some theories are more fundamental than others. We may now revisit Prandtl's boundary-layer theory, whic h has rec eived more attention from philosophers of sc ienc e than any other aspec t of hydrodynamic s. The reason for this interest, no doubt, is the glaring c ognitive superiority of Prandtl's theory c ompared to any earlier approac h to the high Reynolds-number resistanc e problem. Margaret Morrison c alls Prandtl's theory a model and insists on its extratheoretic al origins in c onformity with her general views on models. In her opinion, Prandtl's c onc ept of boundary layer originated in an induc tive inferenc e from the flow patterns that Prandtl observed with his water mill and tank. Mic hael Heidelberger denies this rec onstruc tion and favors an ac c ount in terms of theoretic al heuristic s. As he c orrec tly remarks, laminar boundary layers c ould not be seen in Prandtl's tank, and Prandtl himself c ited asymptotic reasoning as the true sourc e of this c onc ept. However, the sc enario imagined by Morrison is frequently enc ountered in the history of hydrodynamic s. For instanc e, Rankine and Froude's c onc ept of eddying boundary layer did result from c asual observation of the flow around a ship hull.47 Despite his disagreement with Morrison over the origins of Prandtl's theory, Heidelberger c ontinues to c all it a model. Presumably, he means to indic ate that Prandtl's theoretic al heuristic s implied more c reative guessing than would be needed in a mere deduc tion from the Navier-Stokes theory would engender, and that it c reated a new effic ient, and fairly autonomous, c onc eptual struc ture. Prandtl himself did not c all his theory a model. The reasons are not diffic ult to guess. The word was then used in Göttingen as a way to c harac teriz e semi-c onc rete theories

For a Philosophy of Hydrodynamics that saved the phenomena without pretending to reac h the true c auses. In c ontrast, Prandtl's boundary-layer theory was meant to represent the true flow around bodies at a high Reynolds number; it did not imply any c ounterfac tual hypothesis; and it was demonstrably c ompatible with the Navier-Stokes equation. In my terminology, Prandtl's theory was an approximating module of the Navier-Stokes theory. In c onformity with the physic ists’ parlanc e, I would rather reserve the word “model” for theories that imply c onsc ious simplific ations of the system under c onsideration, for instanc e, the early nineteenth-c entury “models” of open c hannel flow.4 8 These terminologic al subtleties matter inasmuc h as an overly generous use of the word “model” implies a neglec t of the modular struc ture of theories, whic h I regard as pervasive and essential. Morrison's and Heidelberger's insights into the func tion of what they prefer to c all models are nevertheless important. They both emphasiz e the impotenc e of bare fundamental theories and the need to supplement them with c onc eptual struc tures that somehow mediate between theory and experiment. And they both understand that unific ation, in the c ontext of a fundamental theory, remains a desideratum. In a witty allusion to Nanc y Cartwright's c ritic ism of fundamental theories, Heidelberger c laims that the Navier-Stokes theory “does not even lie about the world.” At the same time, he understands that the boundary-layer theory, whic h so muc h improves the explanatory power of hydrodynamics, is an approximation of the Navier-Stokes theory. In my view, the moral is that the Navier-Stokes theory, or any other of the great theories of physic s, should not be c onsidered independently of its ever-inc reasing modular struc ture. Although the result of this evolution c an never fulfill the dream of a transparent and automatic applic ation of the fundamental equations to every c onc eivable situation, it has the organic unity and effic ienc y that we need in order to understand and c ontrol some of the physic al world.4 9 References Airy, George Biddell (1845). Tides and waves. In Encyclopedia Metropolitana, 5: 291–396. Batterman, Robert (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. Oxford: Oxford University Press. Bernoulli, Daniel (1738). Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Strasbourg: J. R. Dulsecker. Birkhoff, Garrett (1950). Hydrodynamics: A study in logic, fact, and similitude. Princ eton: Princ eton University Press. Bjerknes, Vilhelm (1898). Über einen hydrodynamisc hen Fundamentalsatz und seine Anwendung besonders auf die Mechanik der Atmosphäre und des Weltmeeres. Kongliga Svenska, Vetenskaps-Akademiens, Handlingar, 31: 3–38. Boussinesq, Joseph (1871). Théorie de l'intumesc enc e liquide appelée onde solitaire ou de translation, se propageant dans un c anal rec tangulaire. Ac adémie des Sc ienc es, Comptes rendus hebdomadaires des séances, 72: 755–59. ———. (1877). Essai sur la théorie des eaux c ourantes. Ac adémie des Sc ienc es de l'Institut de Franc e, Mémoires présentés par divers savants, 23: 1–680. Cartwright, Nancy (1983). How the laws of physics lie. Oxford: Oxford University Press. ———. (1999). The dappled world: A study of the boundaries of science. Cambridge: Cambridge University Press. Cat, Jordi (1998). The physic ists' debates on unific ation in physic s at the end of the 20th c entury. Historical Studies in the Physical and Biological Sciences 28: 253–99. ———. (2005). Modeling c rac ks and c rac king models: Struc ture, mec hanisms, boundary c onditions, c onstraints, in inconsistencies and the proper domain of natural laws. Synthese 146: 447–87. Cauc hy, Augustin (1827). Théorie de la propagation des ondes à la surfac e d'un fluide pesant d'une profondeur indéfinie. Ac adémie des Sc ienc es de l'Institut de Franc e, Mémoires présentés par divers savants, 1: 1– 123.

For a Philosophy of Hydrodynamics D'Alembert, Jean le Rond (1768). Paradoxe proposé aux géomètres sur la résistanc e des fluides. In Opuscules mathématiques, vol. 5, 34th memoir, 132–38. Paris: David. Darrigol, Olivier (2005).Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl. Oxford: Oxford University Press. ———. (2007). On the nec essary truth of the laws of c lassic al mec hanic s. Studies in the History and Philosophy of Modern Physics 38: 757–800. ———. (2008). The modular struc ture of physic al theories. Synthese 162: 195– 223. Draz in, Philip, and William Reid (1981). Hydrodynamic stability. Cambridge: Cambridge University Press. Eckert, Michael (2005). The dawn of fluid dynamics: A discipline between science and technology. Berlin: Wiley. ———. (2008). Turbulenz : ein problemhistorisc her Abriss. NTM 16: 39–71. Epple, Moritz (2002). Präz ision versus Exaktheit: Konfligierende Ideale der angewandten mathematisc hen Forschung. Das Beispiel der Tragflügeltheorie. Berichte zur Wissenschaftsgeschichte 25: 171–93. Euler, Leonhard (1755) [printed in 1757]. Princ ipes généraux du mouvement des fluides. Ac adémie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires, 11: 274–315. Farge, Marie, and Etienne Guyon (1999). A philosophic al and historic al journey through mixing and fully-developed turbulenc e. In Mixing: Chaos and turbulence, ed. Hugues Chaté et al., 11– 36. New York: Kluwer Ac ademic /Plenum Publishers. Franc esc helli, Sara (2007). Construc tion de signific ation physique pour la transition vers la turbulenc e. In Chaos et systèmes dynamiques: Eléments pour une épistémologie, ed. S. Franc esc helli, M. Paty, and T. Roque, 213– 37. Paris: Hermann. Frisch, Uriel (1995). Turbulence: The legacy of A. N. Kolmogorov. Cambridge: Cambridge University Press. Froude, William [1868] 1955. Observations and suggestions on the subjec t of determining by experiment the resistanc e of ships, Memorandum sent to E. J. Reed, Chief Construc tor of the Navy, dated Dec ember 1868. In The papers of William Froude, ed. A. D. Duc kworth, 120– 27. London: Institution of Naval Arc hitec ts. ———. (1874). Reports to the Lords Commissioners of the Admiralty on experiments for the determination of the fric tional resistanc e of water on a surfac e, under various c onditions, performed at Chelston Cross, under the authority of their Lordships. British Assoc iation for the Advanc ement of Sc ienc e, Reports, 249– 55. ———. (1877). The fundamental princ iples of the resistanc e of ships. Royal Institution, Proceedings, 8: 188– 213. Hacking, Ian (1983). Representing and intervening: Introductory topics in the philosophy of natural science. Cambridge: Cambridge University Press. Hagen, Gotthilf (1839). Über die Bewegung des Wassers in engen c ylindrisc hen Röhren. Annalen der Physik 46: 423–42. Heidelberger, Mic hael (2006). Applying models in fluid dynamic s. International Studies in the Philosophy of Science 20: 49–67. Helmholtz , Hermann (1858). Über Integrale der hydrodynamisc hen Gleic hungen, welc he den Wirbelbewegungen entsprechen. Journal für die reine und angewandte Mathematik 55: 25–55. ———. (1868). Über diskontinuirlic he Flüssigkeitsbewegungen. Akademie der Wissensc haften z u Berlin, mathematisc h-physikalisc he Klasse, Sitzungsberichte, 215– 28. ———. (1873). Über ein Theorem, geometrisc h ähnlic he Bewegungen flüssiger Körper betreffend, nebst Anwendung auf das Problem, Luftballons z u lenken. Königlic he Akademie der Wissensc haften z u Berlin, Monatsberichte, 501–14.

For a Philosophy of Hydrodynamics ———. (1888). Über atmospherisc he Bewegungen I. Akademie der Wissensc haften z u Berlin, mathematisc h- physikalische Klasse, Sitzungsberichte, 652. Humphreys, Paul (2004). Extending ourselves: Computational science, empiricism, and scientific method. Oxford: Oxford University Press. Kármán, Theodore von (1930). Mec hanisc he Ähnlic hkeit und Turbulenz . Gesellsc haft der Wissensc haften z u Göttingen, mathematisc h-physikalisc he Klasse, Nachrichten, 58– 76. Kirchhoff, Gustav (1869). Zur Theorie freier Flüssigkeitsstrahlen. Journal für die reine und angewandte Mathematik 70: 289–98. Kragh, Helge (2002). The vortex atom: A Vic torian theory of everything. Centaurus 44: 32– 114. Kuhn, Thomas (1961). The func tion of measurement in modern physic al sc ienc e. Isis 52: 161– 93. ———. (1962). The structure of scientific revolutions. Chic ago: University of Chic ago Press. Lagrange, Joseph Louis (1781). Mémoire sur la théorie du mouvement des fluides. Ac adémie Royale des Sc ienc es et des Belles-Lettres de Berlin, Nouveaux mémoires. Also in Oeuvres (1869), 4: 695–750. Morrison, Margaret (1999). Models as autonomous agents. In Models as mediators: Perspectives on natural and social science, ed. Mary Morgan and Margaret Morrison, 38–65. Cambridge: Cambridge University Press. ———. (2000). Unifying scientific theories: Physical concepts and mathematical structures. Cambridge: Cambridge University Press. Morrison, Margaret, and Mary Morgan (1999). Models as mediating instruments. In Models as mediators: Perspectives on natural and social science, ed. Mary Morgan and Margaret Morrison, 10–37. Cambridge: Cambridge University Press. Navier, Claude Louis (1822). Sur les lois du mouvement des fluides, en ayant égard à l'adhésion des moléc ules [read on 18 Marc h 1822]. Annales de chimie et de physique 19 (1821) [in fac t 1822]: 244– 60. Poisson, Siméon Denis (1816). Mémoire sur la théorie des ondes. Ac adémie Royale des Sc ienc es, Mémoires, 1: 71– 186 (read on 2 Oc tober and 18 Dec ember 1815, published in 1818). Ponc elet, Jean Vic tor (1839). Introduction à la mécanique industrielle, physique ou expérimentale. 2d ed. Paris. Prandtl, Ludwig (1905). Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In III. internationaler Mathematiker- Kongress in Heidelberg vom 8. bis 13. August 1904, ed. A. Kraz er, Verhandlungen (Leipz ig), 484–91. Also in Gesammelte Abhandlungen 2: 575–84. ———. (1914). Der Luftwiderstand von Kugeln. Gesellsc haft der Wissensc haften z u Göttingen, mathematisc h- physikalische Klasse, Nachrichten. Also in Gesammelte Abhandlungen 2: 597–608. ———. (1931). On the role of turbulenc e in tec hnic al hydrodynamic s. World Engineering Congress in Kyoto, Proceedings. Also in Gesammelte Abhandlungen 2: 798–811. Putnam, Hilary (1974). The “c orroboration” of theories. In The philosophy of Karl Popper, ed. P. A. Sc hilpp, vol. 2, La Salle. Also in H. Putnam, Philosophical papers, vol. 1: Mathematics, matter and method (Cambridge University Press, 1975), 250–69. Ramsey, Jeffry (1992). Towards an expanded epistemology for approximations. In PSA 1992: Proc eedings of the 1992 Biennial Meeting of the Philosophy of Sc ienc e Assoc iation, ed. K. Okruhlik, A. Fine, and M. Forbes, 1: 154– 64. East Lansing, MI: Philosophy of Sc ienc e Assoc iation. ———. (1993). When reduc tion leads to c onstruc tion: Design c onsiderations in sc ientific methodology. International Studies in the Philosophy of Science 7: 239–51. ———. (1995). Construc tion by reduc tion. Philosophy of Science. 62: 1– 20.

For a Philosophy of Hydrodynamics Rankine, William John Mac quorn (1858). Resistanc e of ships, Letter to the editors of 26 August 1858. Philosophical Magazine 16: 238–39. ———. (1865). On plane water-lines in two dimensions. Royal Soc iety of London, Philosophical Transactions, 154: 369–91. ———. (1870). On stream-line surfac es. Royal Institution of Naval Arc hitec ts, Transactions, 11: 175– 81. Rayleigh, Lord (William Strutt) (1876a). On waves. Philosophical Magazine. Also in Scientific Papers 1: 251–71. ———. (1876b). On the resistanc e of fluids. Philosophical Magazine 11: 430– 41. ———. (1880). On the stability, or instability, of c ertain fluid motions. London Mathematic al Soc iety, Proceedings, 11: 57–70. Reynolds, Osborne (1883). An experimental investigation of the c irc umstanc es whic h determine whether the motion of water shall be direc t or sinuous, and of the law of resistanc e in parallel c hannels. Royal Soc iety of London, Philosophical Transactions, 174: 935–82. ———. (1895). On the dynamic al theory of inc ompressible visc ous fluids and the determination of the c riterion. Royal Society of London, Philosophical Transactions, 186: 123–64. Ritz , Walther (1903). Zur Theorie der Serienspektren. Annalen der Physik. Also in Oeuvres (Paris, 1911), 1–77. Russell, John Sc ott (1839). Experimental researc hes into the laws of c ertain hydrodynamic al phenomena that ac c ompany the motion of floating bodies, and have not previously been reduc ed into c onformity with the known laws of the resistanc e of fluids. Royal Soc iety of Edinburgh, Transactions, 14: 47– 109. Saint-Venant, Adhémar Barré de (1843). Note à joindre au mémoire sur la dynamique des fluides, présenté le 14 avril 1834. Ac adémie des Sc ienc es, Comptes-rendus hebdomadaires des séances, 17: 1240– 43. Stokes, George Gabriel (1843). On some c ases of fluid motion. Cambridge Philosophic al Soc iety, Transactions, 8: 105–37. ———. (1847). On the theory of osc illatory waves. Cambridge Philosophic al Soc iety, Transactions. Also in Mathematical and Physical Papers, 1: 197–225. ———. (1849) [read in 1845]. On the theory of the internal fric tion of fluids in motion, and of the equilibrium and motion of elastic solids. Cambridge Philosophic al Soc iety, Transactions, 8: 287– 319. ———. (1850). On the effec t of the internal fric tion of fluids on the motion of pendulums. Cambridge Philosophic al Society, Transactions. Also in Mathematical and Physical Papers, 3: 1–141. ———. [1876]. Smith priz e examination papers for 2 Feb. 1876. In Mathematical and Physical Papers, 5: 362. Thomson, William (1887a). Rec tilinear motion of visc ous fluid between two parallel planes. Philosophical Magazine 24: 188–96. ———. (1887b). On ship waves [lec ture delivered at the “Conversaz ione” in the Sc ienc e and Art Museum, Edinburgh, on 3 Aug. 1887]. Institution of Mec hanic al Engineers, Minutes of Proceedings, 409– 34. ———. (1894). On the doc trine of disc ontinuity of fluid motion, in c onnec tion with the resistanc e against a solid moving through a fluid. Nature 50: 524–25, 549, 573–75, 597–98. Truesdell, Clifford (1954). Rational fluid mec hanic s, 1657– 1765. In Euler, Opera Omnia, ser. 2, 12: ix– c xxv. Lausanne: Orell Füssli. ———. (1968). The c reation and unfolding of the c onc ept of stress. In Essays in the history of mechanics, 184– 238. Berlin: Springer. Tyndall, John (1867). On the ac tion of sonorous vibrations on gaseous and liquid jets. Philosophical Magazine 33:

For a Philosophy of Hydrodynamics 375–91. Winsberg, Eric (1999). Sanc tioning models: The epistemology of simulation. Science in Context 12: 275– 92. Wright, Thomas (1983). Ship hydrodynamic s 1770– 1880. Ph.D. dissertation (Sc ienc e Museum, South Kensington, London). Yamalidou, Maria (1998).Molec ular ideas in hydrodynamic s. Annals of Science 55: 369– 400. Notes: (1) For a c ritic ism of the Hempelian view, c f. Heidelberger 2006, 49– 50. (2) Kuhn 1961. (3) Darrigol 2005, hereinafter abbreviated as WF; Ec kert 2005. (4) Cf., e.g., Cartwright 1983, 1999; Cat 1998. (5) Morrison and Morgan 1999; Ramsey 1993, 1995; Batterman 2002; Humphreys 2004; Winsberg 1999; Hac king 1983, 215. Kuhn earlier applied the word “artic ulation” to the paradigms of normal sc ienc e. In 1974, Hilary Putnam noted “in passing” a pervasive but neglec ted sc hema for sc ientific problems, “sc hema III,” in whic h the fundamental laws of the theory and some auxiliary statements are known but the fac tual c onsequenc es are unknown (Putnam 1974, 261–62). (6) Morrison 1999; Heidelberger 2006; Epple 2002. (7) The foundations of hydrodynamic s, though historic ally stable, are not devoid of philosophic al interest. As Clifford Truesdell pointed out long ago, some of its basic c onc epts, suc h as the c onc ept of internal stress, are indeed problematic (Truesdell 1968; Darrigol 2007). The relation of these foundations to general mec hanic s and to statistic al mec hanic s (for instanc e, the kinetic theory of gases) is another philosophic ally interesting topic (Yamalidou 1998). For the sake of homogeneity, I c onfine this essay to post-foundational developments. (8) Euler 1755. Cf. Truesdell 1954; WF, c hap. 1. (9) Navier 1822; Stokes [1845] 1849. Cf. WF, c hap. 3. (10) Euler 1755; Bernoulli 1738. (11) Lagrange 1781; Poisson 1816; Cauc hy [1815] 1827. Cf. WF, 35– 47. (12) Stokes 1876; Thomson 1887b. Cf. WF, 85–100. (13) Airy 1845; Stokes 1847; Russell 1839; Boussinesq 1871; Rayleigh 1876a. Cf. WF, 69– 84. Similar c omments c ould be made about the c ompression waves studied by Euler and Lagrange. (14) Helmholtz 1858. Cf. WF, 149. (15) Cf. WF, 148–58. (16) Helmholtz 1868, 1888. Cf. WF, 159–71. (17) Bjerknes 1898; Prandtl 1905. (18) Stokes 1843. Cf. WF, 184–87. (19) Rayleigh 1880; Thomson 1887a. Cf. WF, 208–18; Draz in and Reid 1981. (20) Tyndall 1867; Hagen 1839; Reynolds 1883. Cf. WF, 243–63. (21) Cf. Eckert 2008.

For a Philosophy of Hydrodynamics (22) Cf. Franc esc helli 2007. (23) Reynolds 1895. Cf. WF, 259–60. (24) Stokes 1850; Helmholtz 1873; Froude [1868] 1957; 1874. Cf. WF, 256–58, 278–79. (25) Cf. WF, 221–28. (26) Saint-Venant 1843; Boussinesq 1877. Cf. WF, 229–38. (27) Reynolds 1895; Kármán 1830; Prandtl 1831 Cf. Ec kert 2005, c hap. 5; WF, 259– 62, 297– 301. (28) Cf. Farge and Guyon 1999; Frisc h 1995. (29) D'Alembert 1768; Saint-Venant 1843; Ponc elet 1839; Stokes 1850. Cf. WF, 135– 39, 265– 67, 270– 73. (30) Kirchhoff 1869; Rayleigh 1876b. (31) Rankine 1858, 1865, 1870; Froude 1874, 1877. Cf. Wright 1983; WF, 273–82. (32) Prandtl 1905. Cf. Ec kert 2005, c hap. 2; Heidelberger 2006; WF, 283– 89. (33) Prandtl 1914. Cf. Ec kert 2005; WF, 293– 94. (34) Heidelberger 2006 rightly insists on this c ausal struc ture of the boundary-layer theory. (35) On misleading intuitions in fluid mec hanic s, c f. Birkhoff 1950. (36) Batterman 2002; Ramsey 1992, 1993, 1995; Heidelberger 2006. (37) Ramsey 1995, 16; Epple 2002. (38) Kuhn 1962, 24. Hilary Putnam similarly c ritic iz ed another of Kuhn's c harac teriz ations of normal sc ienc e: “The term ‘puz z le solving’ is unfortunately trivializ ing; searc hing for explanations of phenomena and for ways to harness nature is too important a part of human life to be demeaned” (Putnam 1974, 261). (39) For a disc ussion of this definition and a c omparison with the definition of Sneedian struc turalists, c f. Darrigol 2008, 198–203. (40) This is a c onsiderable weakening of the logic al-empiric ist stric tures on the meanings of theoretic al terms. (41) A more detailed disc ussion is given in Darrigol 2008. Interpretive sc hemes supplemented with the requirement of c omputability are similar to Humphreys's “c omputational templates.” Ac c ording to Humphreys 2004, it is at the level of c omputational templates that questions about theoretic al representation, empiric al fitness, realism, and so on must be disc ussed; knowledge “in princ iple” must be subordinated to knowledge “in prac tic e,” whic h involves the available tec hnologies of measurement and c alc ulation. (42) Approximating modules c orrespond to what Jeffry Ramsey c alls transformation reduc tion (Ramsey 1993, 1995). (43) In this c ase, being a module of another theory does not imply a sort of inc lusion; but it remains true that a module of a theory serves this theory. (44) More exac tly, the low-density gas spec ializ ation of the Navier-Stokes theory is an approximating module of the kinetic theory of gases, of whic h the mec hanic s of a set of interac ting molec ules is a reduc ing module. (45) Darrigol 2008. (46) In c onformity with the physic ists’ usage, Morrison and Morgan also c all “models” what I c all a “reduc ing module.” For instanc e, Maxwell's mec hanic al model of 1862 for the elec tromagnetic field is a model in this sense. This kind of model widely differs from ad hoc models for limited c lasses of phenomena.

For a Philosophy of Hydrodynamics (47) Morrison 1999, 53–60; Heidelberger 2006, 60–62. (48) For instanc e, Walther Ritz (1903, 3) c alled his vibrating-square theory of series spec tra a “model” (his quotation marks). (49) Cartwright 1983; Heidelberger 2006, 64. O livier Darrig ol Olivier Darrigol is a CNRS research director in the SPHERE/Rehseis research team in Paris. He investigates the history of physics, m ostly nineteenth and twentieth century, with a strong interest in related philosophical questions. He is the author of several books including From c-num bers to q-num bers: The classical analogy in the history of quantum theory (Berkeley: University of California Press, 1992), Electrodynam ics from Am père to Einstein (Oxford: Oxford University Press, 2000), Worlds of flow: A history of hydrodynam ics from the Bernoullis to Prandtl (Oxford: Oxford University Press, 2005), and A history of optics from G reek antiquity to the nineteenth century (Oxford: Oxford University Press, 2012).

What is “Classical Mechanics” Anyway? Mark Wilson The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter analyz es the c onc ept of c lassic al mec hanic s. It suggests that the interpretation of c lassic al mec hanic s will most likely provide an appropriate theoretic al setting in the c ontext of planetary motions, billiards, and simplified ideal gases in boxes. The c hapter also c ontends that c lassic al mec hanic s is best thought of as c onstituted by various foundational methodologies whic h do not fit partic ularly well with one another and highlights the fac t that a good deal of philosophic al c onfusion has arisen from failing to rec ogniz e the c omplic ated sc ale- dependent struc tures of c lassic al physic s. K ey words: cl assi cal mech an i cs, i n terpretati on , pl an etary moti on s, bi l l i ards, i deal gases, fou n dati on al meth odol ogi es, scal e-depen den t stru ctu res, cl assi cal ph y si cs 1. Preliminary Co nsideratio ns One of the prominent sourc es of unhelpful folklore within philosophy is the historic al c ontroversy whose proper intric ac ies have been underapprec iated. Misunderstood problems beget mistaken “morals” that c an lead philosophic al thinking astray for long epoc hs thereafter. This has oc c urred, to an extent that few philosophers rec ogniz e, with respec t to the so-c alled “foundations of c lassic al mec hanic s.” As matters are c ommonly represented within modern c ollege primers, “c lassic al physic s” appears to be a transparent subjec t matter firmly founded upon Newton's venerable laws of motion. But this plac id appearanc e is dec eptive. Any purc haser of an old home is familiar with parlor walls that seem sound exc ept for a few imperfec tions that “only require a little spac kle and paint.” When those innoc ent dimples are opened up, the anc ient gerry-rigged struc ture c omes tumbling down and our hapless fix-it man finds himself c onfronted with months of dusty rec onstruc tion. So it is with our subjec t, whose basic c onc epts c an seem so “c lear and distinc t” on first ac quaintanc e that unwary thinkers have mistaken them for a priori verities. But the true lesson of “c lassic al mec hanic s” for philosophy should be exac tly the opposite: the c onc eptual matters that initially strike us as simple and pelluc id often unwind into hidden c omplexities when probed more adequately.1 Figure 2.1 Matters have been rendered more c onfusing by the fac t that a c onc eptually simple surrogate for c lassic al doc trine is readily available, even though its formally artic ulated doc trines skirt most of the tric ky c onc eptual problems enc ountered within c lassic al tradition. The tenets of this simple theory c omprise the themes that we shall Page 1 of 45

What is “Classical Mechanics” Anyway? investigate under the heading of “point-mass mec hanic s.” Within this approac h the term point mass designates an isolated, z ero-dimensional point that c arries c onc entrated mass, c harge, and so on. In c ontrast, there are two other sorts of “fundamental objec ts” with whic h a “c lassic al mec hanic s” c an be potentially c onc erned: rigid bodies, understood as extended solids whose points never alter their relative distanc es to one another and flexible bodies suc h as fluids or solids that are c ompletely malleable at every siz e sc ale (figure 2.1). Commonly, the latter are also c alled continua, a prac tic e we shall adopt here. Of c ourse, any of these entities c an be joined together in larger c ombinations, as when individual rods are assembled into a mechanism or one flexible body is embedded within another as a composite (e.g., a jelly doughnut).2 Mathematic ians c ommonly label our c ontinua as fields due to their distributed c harac ter. We will generally avoid this terminology and will not disc uss c lassic al elec trodynamic s at all. In the sequel, I shall employ the phrase material point to designate a z ero-dimension region within a c ontinuously distributed body (either in its interior or along some bounding surfac e). In c ontrast to our point masses, material points are c onnec ted with one another quite densely and (usually) do not c arry finite values of mass or impressed forc e (they, instead, only display mass and charge densities that sum to genuine masses and densities over regions of an adequate measure). The phrase analytic mechanics will serve as a generic title for the sundry formalisms that deal with c onnec ted systems of rigid bodies. As just noted, the “c onc eptually simple surrogate” for c lassic al doc trine that most c ommonly dominates philosophic al disc ussions of “Newtonian mec hanic s” c omprises a set of presc riptions that make c oherent sense only with respec t to isolated point masses that never c ome into c ontac t with one another. We shall disc uss the spec ific features of these doc trines in sec tion 3. From a point-mass perc h, any appeal to rigid bodies or c ontinua merely represents a c onvenient means of disc ussing large swarms of point masses held together through c ohesive bonding at short sc ale lengths. The dec eptive simplic ity available to the point-mass approac h trac es largely to the fac t that, within its frame, matter c an exist only in the form of isolated singularities, thereby sidestepping the substantial mathematic al c onc erns that arise when extended objec ts come in contact with one another (on rare oc c asions, point masses c an c ollide with one another, but these c ontac ts only oc c ur at fleeting moments that c an usually be handled through appeal to c onservation princ iples). As a result, point masses ac t upon one another only through action-at-a-distance forces,3 but higher dimensional objec ts require direc t contact forces as well. As we will learn, getting ac tion-at-a- distanc e forc es and c ontac t forc es to work in tandem is a nontrivial affair, but it bec omes a c onc eptual obligation that vanishes from view if we are allowed to restric t our fundamental ontology to point masses alone. However, there is a wide range of subtle reasons why it c an easily look as if a spec ific c lassic al author embrac es the point-mass viewpoint. As we will observe in sec tion 3, Newton's c elebrated laws of motion are diffic ult to parse c oherently unless terms like “body” are interpreted in a punc tiform manner. A host of signific ant mathematical complexities attac h to the notion of “material point” as it appears within c ontinuum physic s (i.e., as a point-siz ed region within a c ontinuous body), and these are sometimes bypassed by c onfusing embedded c ontinuum points with the simple isolated singularities of the point-mass treatment. We shall survey several of these shifts in the pages to follow. From a formal point of view, it is important to distinguish between the ordinary differential equations (ODEs) pertinent to point masses and analytic mec hanic s and the tric kier partial differential equations (PDEs) required in c ontinuum modeling.4 Figure 2.2 Page 2 of 45

What is “Classical Mechanics” Anyway? The fac t that the real world proves quantum mechanical within its small-sc ale behaviors oc c asions c onfusion as well. Although partic les like elec trons appear to be “point-like” in their sc attering behaviors, they also “fill” larger effec tive volumes c ourtesy of the unc ertainty relations. In many c ases, one obtains the requisite Sc hrödinger equation for a system of partic les (whic h is a PDE desc ribing a field spread out within a high dimensional spac e) by “quantiz ing” a parallel set of ODEs for a c lassic al point-mass system.5 But this mathematic al linkage does not entail that nature behaves muc h like any c lassic al point-mass system at a small siz e sc ale (figure 2.2). Quite the c ontrary, c onstruc ting a c lassic al system that c an approximate the “effec tive volumes” of quantum c louds ac c urately at the siz e sc ale of so-c alled “molec ular modeling” often requires c lassic al blobs of extended siz e and flexibility. Most sc ientists working in the final epoc h when c lassic al mec hanic s c ould plausibly c laim to govern the world in its entirety, namely the late nineteenth century, rejected the point-mass viewpoint as empiric ally inadequate for the bloblike c harac teristic s of real-life atoms and molec ules. Nonetheless, there are convenient mathematical associations between the ODEs for classical point-mass models and the Sc hrödinger equation, so many c ontemporary physic ists and philosophers of physic s are familiar with the point-mass formalism alone. However, sc holars hoping to extrac t methodologic al morals from the struggles over “matter,” “atoms,” and “force” that occurred toward the end of the nineteenth century will be misled if they study point-masses only, for it misses the conceptual complexities at the heart of the historical disputes. Viewed retro- spectively, the degree to which the technical arcana of classical mechanics have impacted the development of sc ientific ally attuned philosophy over the past several c enturies is quite striking, even if this influenc e is not always rec ogniz ed by modern readers. In this review, we shall sketc h some of the c hief ways in whic h the subtleties of c lassic al mec hanic s have impac ted philosophy. There are two major arenas in whic h these effec ts have arisen. First, many of our greatest historic al thinkers (Newton, Leibniz , Kant, Duhem, and others) direc tly struggled with the problems of c lassic al matter, and their developed philosophies often prove intimately entangled with the specific foundational pathways they chose to follow.6 Suc h portions of our philosophic al heritage are often misunderstood nowadays simply bec ause the true c ontours of the physic al problems our forebears fac ed have been forgotten. Sec ond, as a result of these struggles, the great philosopher-sc ientists formulated a wide range of philosophic al attitudes inc luding anti-realism and instrumentalism as a response to the tec hnic al oddities they c onfronted. The twentieth-c entury logic al empiric ists who c ame later—after the c hief foc us of ac ademic physic s had shifted to quantum theory and relativity—were influenc ed by those older philosophic al c onc lusions without adequate apprec iation of the c onc rete issues that prompted them. Unfortunately, many philosophers have continued to hew to these old presumptions as if they represented firm verities, illustrating Darwin's celebrated aperccu: “False facts are highly injurious to the progress of sc ienc e, for they often endure long; but false views, if supported by some evidenc e, do little harm, for everyone takes a salutary pleasure in proving their falseness.”7 A large folklore of “false fac ts” c onc erning c lassic al mec hanic s c ontinues to bend c ontemporary philosophy along unprofitable c ontours even today. It is not the c hief intent of this essay to pursue these satellite philosophic al c onc erns with any vigilanc e, but to instead concentrate upon the key tensions that render classical doctrine hard to capture in the first place. Nonethe- less, I hope that our prolegomena on larger themes suggests that signific ant points of general philosophic al edific ation still lodge within the c rac ks of mec hanic s’ hoary edific e. 2. Axio matic Presentatio n It will serve as a c onvenient benc hmark for our investigations to rec all that David Hilbert plac ed the rigoriz ation of mec hanic s on his c elebrated 1899 list of problems that mathematic ians should address in the c entury to c ome (it is his sixth problem). He wrote, “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physic al sc ienc es in whic h mathematic s plays an important part; in the first rank are the theory of probabilities and mechanics.”8 Indeed, Hilbert's own work in geometry and else- where comprised a chief inspiration for the logical empiricist program. Following this lead, we will serially examine the prospects for meeting Hilbert's challenge based upon the three foundational choices identified in section 1: point masses, rigid bodies, and continua. Page 3 of 45

What is “Classical Mechanics” Anyway? Sinc e this essay will c onc lude that Hilbert's objec tives c annot be c ompletely satisfied with respec t to c lassic al mec hanic s in the manner antic ipated, let me first distanc e this evaluation from a popular viewpoint with whic h it might be otherwise c onfused. Many rec ent philosophers have responded to the axiomatic expec tations of the logic al empiric ist sc hool by c onc luding that sc ienc e c annot be usefully studied in a formal manner at all. “Real life physic s represents an ongoing prac tic e,” they c laim, “and any attempt to c apture its free-spirited antic s within the rigid net of mathematic al formaliz ation represents an intrinsic distortion.” But this is not what I shall c laim, for I rejec t suc h a point of view entirely. Writing idly of “prac tic es” in the loose manner of suc h authors offers little prospec t for either apprec iating or c orrec tly identifying the c onc rete c onc eptual diffic ulties to be doc umented in this essay. Indeed, it was prec isely through c areful formal studies in Hilbert's manner that twentieth-c entury prac titioners eventually reac hed a muc h sharper understanding of the fundamental requirements of c ontinuum mec hanic s than was available in 1899. Indeed, Hilbert's own lec tures in 1905 and the pioneering efforts of his student, Georg Hamel, c omprised early landmarks along this long and tortuous development.9 The only anti-Hilbertian moral we will extrac t from our examination is that a desc riptive regime c an often address large-sc ale objec ts more suc c essfully if its underpinnings are struc tured in an overall “theory fac ade” manner somewhat at odds with standard axiomatic expec tations. In every other way, I c ompletely endorse the motivating intent of Hilbert's sixth problem. We c annot apprec iate the old puz z les of c lassic al matter in their historic al dimensions unless we keep the mathematic al diffic ulties of c ontinua firmly in mind. Sc ientists planning bridges or studying the music al qualities of violins in early eras did not have the luxury of waiting until the twentieth c entury to gather the tools they properly require. They simply had to c obble by with the mathematic s they had on hand, even at the pric e of rather dodgy justific ations. For example, due to the lac k of c learly artic ulated PDE equations, Leibniz and his sc hool c ould not deal direc tly with the three-dimensional c omplexities of a shaking beam straight on; they were forc ed to dissec t the problem as illustrated into a c onnec ted sequenc e of one-dimensional tasks loc ally governed by ODEs (figure 2.3). Newton followed a similar proc edure in investigating how rotation affec ts the earth's shape: he began his treatment with a one-dimensional “c anal” through the planet's interior.10 Even today, most textbook problems adopt similar reductive stratagems: witness the standard treatment of the vibrating string. Studying physic s within these reduc ed, lower-dimensional settings c an be very misleading from a “foundational” point of view (enc ouraging one to, e.g., think of stress as simply a kind of forc e). However, it is unlikely that c lassic al physic s c ould have staggered its way to an adequate treatment of c ontinua without relying upon a broad array of results for systems that, from a foundational point of view, c annot represent their proper c onc eptual ingredients. Page 4 of 45

What is “Classical Mechanics” Anyway? Figure 2.3 Finally, to apprec iate the historic al debates over c lassic al physic s in a proper c ontext, we must disentangle the term “foundations” from certain absolutist demands that contemporary philosophers are inclined to make. If we mark out c lear axiomatic “foundations” for point masses, say, have we thereby selec ted an absolute bottom layer of entities from whic h any other objec t or system c onsidered within a c lassic al frame should be c onstruc ted? Many c ontemporary philosophers almost instinc tively answer “yes,” but the more prevalent historic al assumption would have rejec ted “ultimate foundations” for c lassic al mec hanic s in that vein. Indeed, c alls for axiomatiz ation per se need not inherently favor any unique choice of “ideology and ontology” in an absolutist manner, for one may instead believe that different selections of base entities and primitive terms may prove better suited for different agendas. Indeed, nineteenth-century mathematicians influenced by Julius Plücker maintained that traditional Euc lidean geometry lac ks any privileged basic ontology—there is no spec ial reason to regard points as the subjec t's primitive objec ts rather than lines or c irc les. Indeed, a c hief objec tive of traditional “foundational” work within geometry was interested in learning how the subjec t appears when it is dissec ted into alternative c hoic es of elementary forms (points, lines, circles, etc.), under the assumption that each dissection into “primitives” offers fresh insights into the structural relationships that interlace the subject. Hilbert may have approached his sixth- problem axiomatiz ation projec t with similarly tolerant expec tations. O Most of the great sc ientists of Hilbert's time tac itly rec ogniz ed that desc riptive suc c ess in reliable modeling invariably relies upon some tacit choice of scale length. Matter generally reveals a hierarchy of qualities de- pending on how c losely one inspec ts its struc tural details (it is traditional to designate this depth of foc us by a “characteristic scale length” ΔL). For example, on an observational scale ΔL , well made steel obeys simple isotropic rules for stretc h and c ompression under normal loads (figure 2.4). Page 5 of 45

What is “Classical Mechanics” Anyway? Figure 2.4 But c loser inspec tion reveals that this mac rosc opic uniformityG and toughness represents the resultant of a c arefully engineered randomness at the level of the crystalline grain ΔL making upG the material (such a scale length is sometimes dubbed the “mesoscopic level”). Considered at this lowered ΔL length, each component granule will stretc h and c ompress in a more c omplic ated manner than the bulk steel, but their randomiz ed orientations supply the larger body with its simple behGavior at the Omac rosc opic level (so-c alled “homogeniz ation theory” c onc erns itself witLh the details of how this ΔL scale to ΔL scale process operates). Lowering our focus to the molecular lat- tice ΔL composing the grain, we find that its capacity to transmit dislocations supplies the truOe underpinnings of the admirable toughness witnessed in the bulk steel at the much longer characteristic length ΔL . If we attempt to c apture these various sc ale-dependent behaviors individually utiliz ing c lassic al modeling tec hniques alone (as we c an, to a remarkable degree of suc c ess), we will generally find ourselves selec ting different ontologic al base units ac c ordinOg to the implic it sc ale length we have selec ted. In suc h a mode, c ivil engineers usually model a steel beam upon a ΔL scale as a single flexible body of considerable homogeneity, whereas technicians interGested in steel manufacture typically concern themselves with the thermodynamics of structural formation at the ΔL level. As suc h, the latter often adopt an ontology ofL rigid crystalline forms bound together into a c omplex material matrix. Initial efforts in modeling materials at the ΔL scale often employ point-mass atoms bound together in an irregular grid. But a more refined approach to these same lattice “atoms” will instead assign them flexible shapes—at the c ost of c onsiderable c omputational c omplexity. And so the modeling shifts proc eed, eac h alteration in c harac teristic sc ale length c ommonly favoring a different “ontology” in its modeling material. Here is a useful way to think about the relationships between sc ale siz es. In presuming that the point masses within a rigid part retain their comparative distances, we are actually pursuing a rough-hewn stratagem for profitable variable reduction, in the sense that we are attempting to evade consideration of the huge class of descriptive pa- rameters needed to fully fix the position and veloc ity of every point mass within its surrounding rigid-body c loud. By treating the c loud as a united whole, we c an trac k its dominant behaviors with a simple c hoic e of six desc riptive parameters (three to loc ate its c enter of mass; three to mark its angles of rotation around that c enter). But in trac king these values, we are only attending to the dominant behavior of the c loud bec ause any normal c ollec tion of point masses will need to jiggle in very c omplex ways as they move forward. So our six rigid-body c oordinates c ount as an effec tive set of reduced variables for our c omplic ated point-mass swarm. Modern mathematic ians like to pic ture suc h reduc tions as c onsisting of the trajec tories etc hed upon a smallish “reduc ed manifold” sitting inside some muc h larger dynamic spac e. Our point-mass swarm (whic h is symboliz ed within a Page 6 of 45

What is “Classical Mechanics” Anyway? standard high dimensional “phase spac e” as the movements of a single dot) will wander throughout the larger spac e in an exc eedingly c omplic ated way, but it may fly fairly c lose (for c ertain portions of its journey at least) to a smaller “reduced variable” manifold, as illustrated (figure 2.5). If so, we c an gauge its c omplex movements with reasonable ac c urac y by simply trac king its shadow upon the surfac e of the reduc ed manifold. Suc h reduc ed-variable tec hniques have been long employed within c elestial mec hanic s and it remains the hope of modern modelers in, for example, hydrodynamic s that some allied set of reduc ed quantities might be found to simplify the refrac tive c omplexities within those topic s. Figure 2.5 Spec ulative philosophers suc h as Leibniz opined that this alteration of ontologic al units would c ontinue forever as one desc ends to smaller sc ales. More c autious observers have merely observed that experiment had not established any c lear c hoic e of lowest sc ale unit for c lassic al mec hanic s. In this regard, it should be rec alled that the evidenc e for fundamental partic les only bec ame overwhelming at the very end of the c lassic al period, in the guise of Rutherford's experiments on radioac tive sc attering and the like. Onc e quantum mec hanic s enters our desc riptive arena, its perc epts inc reasingly dominate at smaller sc ale lengths and we eventually fall beyond the resourc es of c lassic al modeling tools altogether. Unfortunately, the various c rossover points at whic h c lassic al treatments lose their ac c urac y do not favor any uniform c hoic e of fundamental c lassic al entity. Sometimes point-mass treatments supply the most c onvenient form of lowest-sc ale c lassic al modeling, but more often c ontinua or rigid bodies provide better modeling ac c urac y. So while quantum mec hanic s may selec t c ertain entities as physic ally “bottom level,” it does not follow that c lassic al mec hanic s will do the same when c onsidered upon its own merits. Ac c ordingly, Hilbert's sixth-problem formaliz ation projec t should not be saddled with the burden of satisfying a c ontemporary philosopher's expec tations with respec t to bottom-level ontology. What we will want to investigate c arefully, as part of our “foundationalist” enterprise, is the degree to whic h princ iples applic able on a higher sc ale level ΔL* relate to those applic able at the lower length ΔL. I c all suc h transfers of doc trine ac ross siz e sc ales lifts, and I employ “lift” in the elevator sense: one c an go both up and down in a hoist. Hilbert's own artic ulation stresses the importanc e of understanding these lifts more c entrally than the simpler task of formaliz ing our three starting perspec tives. He wrote: Figure 2.6 Boltz mann's work on the princ iples of mec hanic s suggests the problem of developing mathematic ally the limiting proc esses, there merely indic ated, that lead from the atomistic view to the laws of motion of continua. Conversely, one might try to derive the laws of motion of rigid bodies by a limiting from a system of axioms depending upon the idea of c ontinuously varying c onditions of matter filling all spac e Page 7 of 45

What is “Classical Mechanics” Anyway? c ontinuously, these c onditions being defined by parameters. For the question of equivalenc e of different systems of axioms is always of great theoretic al interest.11 Here Hilbert c alls our attention to the various relationships between sc ale length that have been intensely studied in rec ent times under the general headings of “homogeniz ation” and “degeneration.”12 He observes that the vague invoc ation of “limits” rarely provides an adequately prec ise diagnosis of the relationships involved, an observation that modern investigations heartily undersc ore. Observe that Hilbert's final sentenc e suggests that he did not anticipate that any of his suggested starting points would prove fundamental in the bottom-layer sense just c anvassed. Ac c ording to the applic ational task at hand, different modes of ontologic al dissec tion (e.g., flexible c ontinua or Boltz mannian swarms of rigid bodies) may possess their desc riptive utilities in the same manner in whic h alternative dec ompositions of geometry into “primitive elements” prove fruitful. Even so, Hilbert insists that we must guard against erroneously lifting physic al doc trines from one dec ompositional program to another without adequate precaution (figure 2.6). In standard textbook prac tic e, these lifts usually appear as dubious “derivations” of, for example, rules of c ontinua c onsidered at a ΔL* sc ale level on the basis of rigid body swarms at a ΔL sc ale. As we will later see in detail, suc h improper doc trinal transfers are c ommon in prac tic e and sometimes serve as the sourc e of substantial c onc eptual c onfusion.13 Figure 2.7 Consider a simple example of the problems that c an arise in suc h shifts from ΔL to ΔL* . The term force has a notorious tendenc y to alter its exac t signific anc e as c harac teristic sc ale lengths are adjusted. At a mac rosc opic level, the “rolling fric tion” that slows a ball upon a rigid trac k is a simple Newton-style forc e opposing the onward motion. But at a lower sc ale length, the seemingly “rigid” trac ks are not so firm after all: they elongate under the weight of the sphere to a nontrivial degree. So part of the work required to move our ball against fric tion c onsists in the fac t that it must travel further than is apparent. But when we c onsider the “forc es” on our ball at a mac rolevel, we instinc tively treat the trac k length as fixed and alloc ate the effec ts of its ac tual elongation to a portion of the “forc e of rolling fric tion” budget (figure 2.7). A similar phenomenon oc c urs with the “visc osity” of a fluid. When suc h adjustments in referenc e oc c ur, one c annot legitimately lift a doc trine about “forc es” applic able on sc ale level ΔL to sc ale level ΔL* , for “forc e” does not mean quite the same thing in the two applic ations. Of c ourse, if these innoc ent drifts were the only kinds of problematic lift to whic h mec hanic al prac tic e was liable, serious c onc eptual debates would not have arisen in the subjec t. But these humble illustrations supply a preliminary sense of the problems we must watc h for. The properties we asc ribe to a system with respec t to an upper-sc ale length ΔL* (“rolling on a rigid trac k”) usually represent averages (or some allied form of homogeniz ation or degeneration) over the more elaborate behaviors we will witness at a finer sc ale of resolution ΔL (“stretc hing the molec ular lattic e”). Obtaining a workable sc heme of physical description tailored to ΔL* usually requires that a fair amount of fine detail gets frozen over in our modelings. In other words, we generally hope to c apture only the dominant behaviors of our real-life system within Page 8 of 45

What is “Classical Mechanics” Anyway? in our ΔL* treatment and antic ipate that we will sometimes need to open up the suppressed degrees of freedom whenever the c omplexities of the lower sc ale begin to intrude upon the patterns normally predominant at the coarser scale ΔL*. Generic ally, the use of a smaller set of quantities to c apture system behaviors dominant upon a higher sc ale length ΔL* is c alled a reduced variable treatment. There are a large number of ways in whic h these reduc ed-variable models c an arise. For example, a reasonable polic y of homogenization might adjust its desc riptive terms from those suited to a ΔLG assembly of iron grains to a smoothed-over steel bar desc ribed as c ontinuous at the ΔLO level.14 But a quite different exemplar of reduc ed-variable “freez ing” c an be witnessed in Newton's c elebrated treatment of the planets. At the sc ale lengths appropriate to c elestial mec hanic s, one c an ignore the c omplexities attendant upon the earth's shape and siz e by modeling it as a simple point mass. Rather than smearing out the properties of the planets over wider regions (as oc c urs in homogeniz ation), we instead c onc entrate their extended traits upon much smaller supports. Suc h polic ies of c ompressing c omplex expanses into singularities (or other lower-dimensional struc tures suc h as one-dimensional strings) are sometimes c alled degenerations (a term I regard as preferable to the misleading phrase idealization). Plainly, when very detailed astrophysic al c alc ulations are wanted, one must open up those internal c omplexities and treat the earth as a c ontinuum subjec t apt to distort under rotational effec ts. However, there are many forms of reduc ed-variable lift that involve a mixture of the two polic ies or other sorts of tac tic altogether. Some of the anti-atomism advoc ated by late nineteenth-c entury sc ientists suc h as Duhem and Mac h trac es not to some obtuse dismissal of lower sc ale struc ture per se, but to the widely shared assumption that, in any applic ation, modelers must invariably engage in suc h “freez ing to a sc ale level” proc edures. Their primary disagreement with other mec hanists of their era c onc erns the format that should be regarded as the optimal embodiment of “c lassic al princ iple” within suc h a sc ale-sensitive setting. Spec ific ally, Duhem and Mac h maintained that “basic physic s,” as an organiz ational enterprise, should develop tools that will prove maximally useful at any chosen scale length. This requirement almost automatic ally favors a “thermomec hanic al” approac h of the sort desc ribed in the disc ussion of flexible bodies in sec tion 5. Their opponents, suc h as Ludwig Boltz mann, generally favored the simplest base ontology that c ould plausibly support the more c omplex forms of mec hanic s in a ΔL to ΔL* manner (they often employed point masses or c onnec ted rigid bodies as their base level ingredients). In these respec ts, we might observe that Duhem and Mac h's stric tures better suit the methodologic al perc epts of empiric ists suc h as David Hume, who opined that any postulation of lower-sc ale struc ture must be based upon “laws” direc tly verifiable at the laboratory level. Prima fac ie, we might reasonably expec t that it should prove possible to formaliz e any of our three basic ontologies independently of one another, plac ing them on their own bottoms, as it were. Thus Hilbert probably antic ipated that we should be able to frame distinc t axiomatic enc apsulations for point masses, rigid bodies and flexible bodies and then proc eed to investigate how ably suc h formalisms relate to one another under ΔL to ΔL* lifts. However, a somewhat surprising obstac le impedes suc h projec ts, whose various ramific ations will c omprise the bulk of this essay. They c ollec tively trac e to the simple c onsideration that if we attempt to frame general princ iples applic able to a higher ΔL* sc ale length based upon behaviors operative on a lower sc ale length ΔL, we will find that our ΔL* level princ iples generally display gaps, holes, or gross inaccuracies in spec ial c irc umstanc es. The general explanation for suc h upper-sc ale gaps is quite straightforward: a useful selec tion of “reduc ed variables” at the ΔL* level will foc us upon behaviors that dominate at that siz e sc ale. But, invariably, there will be spec ial ΔL-level arrangements where the effec ts suppressed in our ΔL* treatment obtain equal or greater importanc e than the usual dominant behaviors. I shall sometimes c all suc h shifts “esc ape hatc hes,” for they provide ladders that allow us to evade the inferential instruc tions of a formalism that no longer serves its empiric al purposes. But suc h prac tic es c reate a formal diffic ulty for axiomatiz ation projec ts in Hilbert's vein bec ause the domain of interest frequently bec omes re-ontologiz ed under the sc ale shift. But axiomatic presentations rarely inc lude provisos for ontology shifts. Instead, we antic ipate that their formal tenets will supply behavioral princ iples applic able to its ontology in all c irc umstanc es, even if, in real-life prac tic e, we would normally esc ape suc h desc riptive straitjac kets in favor of some revised treatment operating at a lower length sc ale ΔL. In short, c onventional axiomatiz ed theories are expec ted to supply princ iples that c an govern even the bad spots Page 9 of 45

What is “Classical Mechanics” Anyway? within their ranges of empiric al c overage. Suc h formal expec tations lead many philosophers to further suppose that “c lassic al mec hanic s” must c ompletely spec ify the behaviors tolerated within its own parochial range of possible worlds, in spite of the fac t that we would never apply suc h modelings to real-world dominions of a strongly quantum mec hanic al or relativistic c harac ter. But suc h dogmas presume that some fairly c omplete axiomatiz ation of overall “c lassic al mec hanic s” is available, a thesis we shall c ritic ally examine in this essay. Figure 2.8 Let us now ask ourselves a c ommonsensic al question. Considered from a prac tic al point of view, is it really wise or meritorious to fill out a formalism in a manner that c arries with it no disc ernible empiric al merit? Mightn't it be better to deliberately leave our stoc ks of physic al princ iple somewhat inc omplete, allowing its very holes to signal when we should look for suitable ΔL* to ΔL esc ape hatc hes? Indeed, explic it indic ations in the mathematic s of when modeling problems begin should be greatly c ultivated, for we surely want to avoid the fate of the c omputers who c heerfully c ompute worthless data simply bec ause no one has told them to stop.15 Training in mec hanic s generally inculcates considerable skill in knowing when one should adventitiously shift from one modeling framework to another. So it is sometimes unwise to push a formalism's axiomatiz ed c overage beyond the limits of its real-life modeling effectiveness. This point of view suggests that we might look upon the inherited c ompendium of desc riptive lore we c all “c lassic al mec hanic s” as a series of desc riptive patc hes (c orresponding to our three basic c hoic es of fundamental objec ts) linked together at their desc riptive bad spots by various ΔL* to ΔL esc ape hatc hes. However, whenever manifolds are c onstruc ted through sewing together loc al patc hes in this way, twisted topologies c an potentially emerge in the final result (Klein bottles and Möbius strips provide c lassic illustrations of the phenomenon). In these respec ts, nature shows little favoritism as to whic h of our three basic ontologies of c lassic al objec ts should be viewed as “fundamental” from an applic ational point of view. If we attempt to understand “c lassic al physic s” as a conceptual system closed unto itself, we thereby obtain a struc ture like one of those impossible Esc her etc hings: loc al plates c onnec ted by stairc ases that never stabiliz e upon a lowest landing (figure 2.8). But suc h topographic al oddities do not indic ate that “c lassic al physic s” has not served its desc riptive purposes perfec tly well. As long as the salient esc ape routes are c learly marked, our Esc herish edific e serves a base frame upon whic h a wide range of interc onnec ted forms of reduc ed-variable modeling tec hniques c an be c onveniently loc ated (I sometimes c all struc tures of this sort theory facades). By operating with a proper regard for the requisite level shifts, we c an thereby assemble the most fruitful terminology yet devised for dealing with the c omplex physic al world about us at nonmic rosc opic sc ale lengths: the shared language of “c lassic al physic s.” The twisted topology within its c onnec tion manifold merely reflec ts the “exit from bad patc hes” c onsiderations that allow the sc heme to c over extremely wide swatc hes of applic ation with great effic ienc y. The historic al triumph of “c lassic al mec hanic s” as a desc riptive enterprise would have never oc c urred had the subjec t not lightly skipped over the many problematic transitions of the sort we shall survey. Historic ally, the pric e of a vigorous c onc eptual enlargement is often a lingering residue of c onfusion that c an oc c asionally blossom into full paradox when suitably nurtured. And suc h has been the c areer of c lassic al mec hanic s: full of predic tive glories but c omingled with mystifying transitions that have led some of our greatest philosophic al minds down the garden Page 10 of 45

What is “Classical Mechanics” Anyway? path to strange assessments of our desc riptive position within nature. In the sequel, we c onsider three basic desc riptive patc hes handed down to us in our c lassic al legac y and examine the typic al c onfusions that arise when one shifts from one framework to another without notic ing. 3. Po int- Mass Mechanics Let us first c onsider the point-mass formalism of c lassic al mec hanic s, suggested by Newton's familiar formulation of the fundamental laws of motion. To begin, it is worth noting that substantive foundational issues immediately arise if we sc rutiniz e these laws with a c ritic al eye. In their original form, these princ iples are hard to interpret with any exac titude due to the ambiguous manner in whic h Newton employs his terms. Here they are in Motte's translation: Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, exc ept insofar as it is c ompelled to c hange its state by forc e impressed. Law II: The alteration of motion is ever proportional to the motive forc e impressed; and is made in the direc tion of the right line in whic h that forc e is impressed. Law III: To every ac tion there is always opposed an equal reac tion: or the mutual ac tions of two bodies upon eac h other are always equal, and direc ted to c ontrary parts.16 Look carefully at Law I. If a “body” represents an isolated point mass, then the phrase “moves uniformly straight forward” is not ambiguous. But what is the parallel intent if a rotating rigid object c an be selec ted as a body? Or a pac ket within a c ompressible fluid? One c annot demand that every point within a freely moving boomerang must translate rec tilinearly—at best, they c an rotate around some representative center within the full projec tile (suc h as its c enter of mass). Allied interpretational problems affec t Newton's remaining laws as does the question of prec isely where the “impressed forc es” are supposed to ac t. Indeed, the three laws c an be readily interpreted only if “body” is read as “isolated point mass” throughout. However, this was neither Newton's intent nor that of the many subsequent writers who have c ited the three laws with approval, as illustrated by Peter Tait and Lord Kelvin in their celebrated Treatise on Natural Philosophy: We c annot do better, at all events in c ommenc ing, than follow Newton somewhat c losely. Indeed, the introduc tion to the Principia c ontains in a most luc id form the general foundations of Dynamic s. The definitiones and Axiomata sive Leges Motus, there laid down, require only a few amplific ations and additional illustrations, suggested by subsequent developments, to suit them to the present state of sc ienc e, and to make a muc h better introduc tion to dynamic s than we find in even some of the best modern treatises.17 But suc h c laims are misleading. Why have Newton's laws been allowed to stand so long in suc h a c onfusing form? Newton himself proved somewhat wobbly with respec t to prec ise c ontent of his own first law, in that he offers as an illustration the fac t that a rotating hoop will c ontinue in its angular movements if not ac ted upon by “outside forc es”: “A spinning hoop, whic h has parts that by their c ohesion c ontinually draw one another bac k from rec tilinear motions, does not c ease to rotate, exc ept insofar as it is retarded by the air.”18 Plainly a tac it appeal to some generalized inertia princ iple is implic ated: the ac tivities of the wholly “internal” forc es within a rigid body should not affec t its overall rotation. Newton, of c ourse, knew that this same c laim will not hold for a flexible objec t suc h as the earth or a falling c at. The “rigidity” of the hoop somehow underpins a lift that c onverts an inertial princ iple relevant to isolated point masses into a requirement upon c omposite objec ts operating at the sc ale siz e of a hoop. But shouldn't Newton have properly attended to the c onstitutive modeling assumptions that render the internal c onstitution of a rigid ring different from a c at or a flexible earth? Yes—as we have already observed, suc h forms of ΔL to ΔL* sc ale lift pose the same kinds of justific atory problems as arise when we shift from a ΔL-level swarm of interac ting molec ules to their “averaged” statistic al mec hanic s at level ΔL* . Modern sc holarship generally c redits the standard modern reading of Newton's sec ond law to Euler, who introduc es the expec tation that “F = ma” supplies the c entral framework upon whic h suitable sets of ODE modeling equations for point-mass modeling c an be assembled. This rec ipe of Euler's unfolds as follows: Choose a target system S to model in a point-mass mode. Count the number of masses one needs in S. For eac h i ∈ S write down the following framework for c onstruc ting a well-posed set of (vec torial) ordinary differential equations: Page 11 of 45