The Routledge Companion to Philosophy of Physics (Routledge Philosophy Companions) [1 ed.] 1138653071, 9781138653078

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THE ROUTLEDGE COMPANION TO PHILOSOPHY OF PHYSICS

The Routledge Companion to Philosophy of Physics is a comprehensive and authoritative guide to the state of the art in the philosophy of physics. It comprises 54 self-contained chapters written by leading philosophers of physics at both senior and junior levels, making it the most thorough and detailed volume of its type on the market – nearly every major perspective in the field is represented. The Companion’s 54 chapters are organized into 12 parts. The first seven parts cover all of the major physical theories investigated by philosophers of physics today, and the last five explore key themes that unite the study of these theories.         I. Newtonian Mechanics       II. Special Relativity    III. General Relativity    IV. Non-Relativistic Quantum Mechanics      V. Quantum Field Theory   VI. Quantum Gravity VII. Statistical Mechanics and Thermodynamics VIII. Explanation   IX. Intertheoretic Relations     X. Symmetries    XI. Metaphysics  XII. Cosmology The difficulty level of the chapters has been carefully pitched so as to offer both accessible summaries for those new to philosophy of physics and standard reference points for active researchers on the front lines. An introductory chapter by the editors maps out the field, and each part also begins with a short summary that places the individual chapters in context. The volume will be indispensable to any serious student or scholar of philosophy of physics. Eleanor Knox is Reader in Philosophy of Physics at King’s College London. She works in philosophy of physics, particularly the philosophy of spacetime physics, and is also interested in issues of reduction and emergence, and how these two come together in quantum gravity. Alastair Wilson is Professor of Philosophy at the University of Birmingham and Senior Adjunct Research Fellow at Monash University. He works on philosophy of physics, philosophy of science, metaphysics, and epistemology, with special interests in the philosophy of quantum theory and the metaphysics of dependence. He is the author of The Nature of Contingency: Quantum Physics as Modal Realism (2020).

ROUTLEDGE PHILOSOPH Y COMPA NIONS

Routledge Philosophy Companions ofer thorough, high quality surveys and assessments of the major topics and periods in philosophy. Covering key problems, themes and thinkers, all entries are specially commissioned for each volume and written by leading scholars in the feld. Clear, accessible and carefully edited and organised, Routledge Philosophy Companions are indispensable for anyone coming to a major topic or period in philosophy, as well as for the more advanced reader. Also available: THE ROUTLEDGE COMPANION TO SHAKESPEARE AND PHILOSOPHY Edited by Craig Bourne, Emily Caddick Bourne THE ROUTLEDGE COMPANION TO THE FRANKFURT SCHOOL Edited by Peter E. Gordon, Espen Hammer, Axel Honneth THE ROUTLEDGE COMPANION TO FEMINIST PHILOSOPHY Edited by Ann Garry, Serene J. Khader, and Alison Stone THE ROUTLEDGE COMPANION TO PHILOSOPHY OF PSYCHOLOGY, SECOND EDITION Edited by Sarah Robins, John Symons, and Paco Calvo THE ROUTLEDGE COMPANION TO MEDIEVAL PHILOSOPHY Edited by Richard Cross and JT Paasch THE ROUTLEDGE COMPANION TO PHILOSOPHY OF PHYSICS Edited by Eleanor Knox and Alastair Wilson

For more information about this series, please visit: https://www.routledge.com/Routledge-Philosophy-Companions/book-series/PHILCOMP

THE ROUTLEDGE COMPANION TO PHILOSOPHY OF PHYSICS

Edited by Eleanor Knox and Alastair Wilson

First published 2022 by Routledge 605 Third Avenue, New York, NY 10158 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2022 Taylor & Francis The right of Eleanor Knox and Alastair Wilson to be identifed as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifcation and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this title has been requested ISBN: 978-1-138-65307-8 (hbk) ISBN: 978-0-367-76961-1 (pbk) ISBN: 978-1-315-62381-8 (ebk) Typeset in Bembo by codeMantra

CONTENTS

List of Figures, Boxes, and Tables Notes on Contributors

xi xiv 1

Introduction Eleanor Knox and Alastair Wilson SECTION A: THEORIES PART I

Newtonian Mechanics

6 8

1 Newtonian Mechanics Ryan Samaroo 2 Formulations of Classical Mechanics Jill North

21

3 Classical Spacetime Structure James Owen Weatherall

33

4 Relationism in Classical Dynamics Julian Barbour

46

PART II

Special Relativity

59

5 Relativity and Space-Time Geometry Tim Maudlin

61

v

Contents

6 The Dynamical Approach to Spacetime Theories Harvey R. Brown and James Read

70

7 Relativity and the A-Theory Antony Eagle

86

8 Relativistic Constraints on Interpretations of Quantum Mechanics Wayne C. Myrvold

99

PART III

General Relativity

123

9 The Equivalence Principle(s) Dennis Lehmkuhl

125

10 The Hole Argument Oliver Pooley

145

11 Relativistic Spacetime Structure Samuel C. Fletcher

160

PART IV

Non-Relativistic Quantum Teory

181

12 Bell’s Theorem, Quantum Probabilities, and Superdeterminism Eddy Keming Chen

184

13 Quantum Decoherence Elise M. Crull

200

14 The Everett Interpretation: Structure Simon W. Saunders

213

15 The Everett Interpretation: Probability Simon W. Saunders

230

16 Collapse Theories Peter J. Lewis

247

17 Bohmian Mechanics Roderich Tumulka

257

vi

Contents PART V

Quantum Field Teory

273

18 The Quantum Theory of Fields David Wallace

275

19 Renormalization Group Methods Porter Williams

296

20 Locality in (Axiomatic) Quantum Field Theory: A Minority Report Laura Ruetsche

311

21 Particles in Quantum Field Theory Doreen Fraser

323

PART VI

Quantum Gravity

337

22 The Development of Quantum Gravity: From Feelings to Phenomena Dean Rickles

339

23 String Theory Richard Dawid

351

24 Quantum Gravity from General Relativity Christian Wüthrich

363

25 Spacetime “Emergence” Nick Huggett

374

26 The Problem of Time Karim P.Y. Thébault

386

PART VII

Statistical Mechanics and Termodynamics

401

27 Equilibrium in Boltzmannian Statistical Mechanics Roman Frigg and Charlotte Werndl

403

28 Equilibrium in Gibbsian Statistical Mechanics Roman Frigg and Charlotte Werndl

414

vii

Contents

29 Quantum Foundations of Statistical Mechanics and Thermodynamics Orly Shenker

425

30 Entropy Asymmetry Arianne Shahvisi

439

SECTION B: THEMES PART VIII

Explanation

452

31 Causal Explanation in Physics Mathias Frisch

454

32 Non-Causal Explanations in Physics Juha Saatsi

466

33 Mechanistic Explanation in Physics Laura Felline

476

34 The Explanatory Value of Selecting the Appropriate Scale(s) Lina Jansson

487

PART IX

Intertheoretic Relations

497

35 Nagelian Reduction in Physics Foad Dizadji-Bahmani

499

36 Phase Transitions Sorin Bangu

512

37 Universality Robert W. Batterman

524

38 Chance and Determinism Nina Emery

536

PART X

Symmetries

549

39 Symmetry and Superfuous Structure: A Metaphysical Overview Shamik Dasgupta

551

viii

Contents

40 Symmetry and Superfuous Structure: Lessons from History and Tempered Enthusiasm Jenann Ismael

563

41 Permutations Adam Caulton

578

42 Gauge Theories Nicholas J. Teh

595

43 Time Reversal Bryan W. Roberts

605

44 Symmetry Breaking Elena Castellani and Radin Dardashti

620

PART XI

Metaphysics

633

45 Laws Marc Lange

635

46 Chance Mauricio Suárez

644

47 Holism Richard Healey

655

48 Dimensions Susan G. Sterrett

666

49 Fundamentality Steven French

679

PART XII

Cosmology

689

50 Why Is There Something, Rather Than Nothing? Sean M. Carroll

691

51 Time in Cosmology Craig Callender and C. D. McCoy

707

ix

Contents

52 The Fine-Tuning of the Universe for Life Luke A. Barnes

719

53 Dark Matter and Dark Energy Melissa Jacquart

731

54 Evidence in Astrophysics Sibylle Anderl

744

Index

753

x

FIGURES, BOXES, AND TABLES

FIGURES 2.1 Two-dimensional tangent bundle 2.2 Plane pendulum 3.1 (a) An arrow u relating two points p and q of space. (b) “Scaling” an arrow u at a point p by various amounts, including fipping its direction. (c) Adding arrows u and v: here u+v is the arrow relating p and r, where r is the point determined by q and v, and q is the arrow determined by p and u 3.2 Galilean spacetime may be thought of as consisting of copies of three-dimensional space stacked on top of one another to form a four-dimensional structure 3.3 The motion of a body is described by its 4-velocity, which is an arrow u with temporal length one 5.1 A four-dimensional representation of Newtonian space-time 5.2 Light cone structure associated with points of space-time 5.3 Trajectories through p with equal interval lengths 5.4 The “twin paradox” in Lorentz coordinates 8.1 The hypersurfaces considered in the proof 8.2 Spacelike separated regions between two spacelike hypersurfaces 8.3 A region common to two spacelike hypersurfaces 16.1 Three quantum states 16.2 Collapse for a single particle 16.3 GRW collapse for two correlated particles 17.1 Several possible trajectories for a Bohmian particle in a double-slit setup, coming from the left 22.1 Penrose’s impossible triangle as a model for the problem of quantum gravity: locally (e.g., focusing on general relativity or quantum mechanics in isolation), we can make sense of picture, but not so globally in which the situation becomes paradoxical 22.2 Quantum theory as containing general relativity 22.3 General relativity as containing quantum theory 22.4 Quantum theory and general relativity as equally fundamental 22.5 Quantum gravity as containing both quantum theory and general relativity ˆ † ( g1 , g 2 , g 3 , g 4 ) 0 25.1 A pictorial representation of a GFT quantum, Φ xi

24 26

34 36 37 62 64 67 67 104 111 112 248 249 250 259

340 342 343 344 344 381

Figures, Boxes, and Tables

29.1 34.1 34.2 36.1 36.2 36.3 37.1 37.2 37.3 37.4 37.5 37.6 40.1 40.2 40.3 40.4 40.5 40.6 40.7 42.1 42.2 43.1 43.2 43.3 43.4 43.5

44.1 46.1 51.1 51.2 51.3 52.1

The interplay between dynamics and macrovariables A bead sliding down a frictionless static wire shaped as a helix A Newtonian decomposition into gravitational and normal forces Phase diagram for water in terms of temperature and pressure A singularity in the free energy Singularities in Helmholtz and Gibbs free energies ((a) and (b)) correspond to discontinuities in entropy and volume, respectively ((c) and d)) Cartoon PVT diagram for water Universality of critical phenomena Droplets inside droplets inside droplets Blocking and averaging to yield a new (coarse-grained) efective system Fixed point and universality class Fixed point, universality class, and λ-transformation Wall tiling with repeated pattern A refected scene in water Circular patterns of cactus leaves Nested patterns of tiles and architectural details in a mosque Newtonian spacetime Galilean spacetime Planes of absolute simultaneity in Galilean spacetime Two patches H1 and H 2 intersecting in a belt H12 around the equator of a spherical spacetime Gauge orbits within a constraint surface in phase space Some properties of the time reversal operator T Time reversal viewed as reversal of temporal orientation: the black lobe represents the “future direction” in each case The Albert-Callender “pancake” approach to time reversal: order reversal of inert spatial slices with no temporal properties Brave or cowardly? This instantaneous property of the soldier appears to depend on the direction of time Time reversal of a harmonic oscillator’s phase space: the pancake account only includes the frst (order-reversing) transformation, leading to momentum and velocity in opposite directions; the standard account avoids this by including the second (instantaneous state) transformation as well Plot of potential in equation (44.11) Norton’s dome A depiction of York time in a spacetime with singularities Aeons of time in a conformal cyclic universe The arrow of time aligning with increasing entropy in “two-sided” universes The Ptolemaic model of the Solar System

431 493 493 513 514 515 525 526 528 532 532 533 565 565 566 566 569 569 569 600 602 606 610 612 614

615 626 648 712 715 716 720

BOXES 48.1 Defnition of the International System of Units (from Bureau International des Poids et Mesures)

xii

675

Figures, Boxes, and Tables

TABLES 12.1 Possible assignments of properties to the two photons 13.1 Degrees of freedom of system and environment 37.1 Scaling exponents for diferent transitions

xiii

187 205 529

CONTRIBUTORS

Sibylle Anderl is Guest Researcher at the Institut de Planétologie et d’Astrophysique de Grenoble and works as a science editor for the Frankfurter Allgemeine Zeitung. She is the author of Das Universum und ich: Die Philosophie der Astrophysik (Carl Hanser, 2017). Sorin Bangu is Professor in the Department of Philosophy at the University of Bergen, Norway. He has published on the relation between mathematics and physics, and recently edited Naturalizing Logico-Mathematical Knowledge (Routledge, 2018). Julian Barbour supported himself as an independent theoretical physicist for decades. He is a past Visiting Professor in Physics at the University of Oxford and the author of The End of Time (Orion, 1999) and The Janus Point (Bodley Head, 2020). Luke A. Barnes is Lecturer in Physics at Western Sydney University, Australia, researching cosmology, galaxy formation, and the fne-tuning of the Universe for life. He is the co-author (with Prof. Geraint Lewis) of A Fortunate Universe: Life in a Finely-Tuned Cosmos (Cambridge University Press, 2016) and The Cosmic Revolutionary’s Handbook (Cambridge University Press, 2020). Robert W. Batterman is Distinguished Professor of Philosophy at the University of Pittsburgh. He is the author of The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence (Oxford University Press, 2002) and A Middle Way: A Non-Fundamental Approach to Many-Body Physics (Oxford University Press, 2021), and the editor of The Oxford Handbook of Philosophy of Physics (Oxford University Press, 2013). Harvey R. Brown  is Emeritus Professor of Philosophy of Physics at Faculty of Philosophy, Oxford University, and Emeritus Fellow of Wolfson College, Oxford, UK. He is the author of Physical Relativity: Space-Time Structure from a Dynamical Perspective (Oxford University Press, 2007). Craig Callender is Professor of Philosophy and Co-Director of the Institute for Practical Ethics at the University of California San Diego. He is the author of What Makes Time Special? (Oxford University Press, 2017). Sean M. Carroll is Research Professor of Physics at Caltech and External Professor at the Santa Fe Institute. His most recent book is Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime (E.P. Dutton, 2019). xiv

Contributors

Elena Castellani  is Associate Professor in Philosophy of Science at the University of Florence, Italy. She co-edited Symmetries in Physics: Philosophical Refections (Cambridge University Press, 2003), The Birth of String Theory (Cambridge University Press, 2012), and a special issue of Studies in History and Philosophy of Modern Physics on Dualities in Physics (2017). Adam Caulton is Associate Professor and Clarendon University Lecturer at the Faculty of Philosophy, University of Oxford, and Fellow and Tutor in Philosophy at Balliol College, Oxford. Eddy Keming Chen is Assistant Professor of Philosophy at the University of California, San Diego, an associate editor of the journal Foundations of Physics, and a fellow of the John Bell Institute for the Foundations of Physics. Elise M. Crull is Assistant Professor of Philosophy at The City College of New York. Her books include The Einstein Paradox: The Debate on Nonlocality and Incompleteness in 1935 (Cambridge University Press, 2021; co-authored with Guido Bacciagaluppi) and Grete Hermann: Between Physics and Philosophy (Springer, 2017; co-edited with Guido Bacciagaluppi). Radin Dardashti is Junior Professor in Philosophy of Physics at the University of Wuppertal. His research focuses on the various methods used in theory development and assessment in modern physics. He is co-editor of Why Trust a Theory? Epistemology of Fundamental Physics (Cambridge University Press, 2019). Shamik Dasgupta is Associate Professor of Philosophy at the University of California, Berkeley. He works in metaphysics, philosophy of science, and value theory. Richard Dawid is Professor of Philosophy of Science at Stockholm University, Sweden. His research focuses on the philosophy of physics and the general philosophy of science. He is the author of String Theory and the Scientifc Method (Cambridge University Press, 2013) and the co-editor of Why Trust a Theory? Epistemology of Fundamental Physics (Cambridge University Press, 2019). Foad Dizadji-Bahmani is Associate Professor of Philosophy at California State University, Los Angeles. He works on issues in general philosophy of science, with a focus on intertheoretic relations and the realism debate, and on philosophy of physics. Antony Eagle is Senior Lecturer in Philosophy at the University of Adelaide. He works in metaphysics, philosophy of science, and adjacent felds.  Nina Emery  is Associate Professor of Philosophy at Mount Holyoke College and Afliated Graduate Faculty at the University of Massachusetts, Amherst. She writes on topics at the intersection of metaphysics and philosophy of physics. Laura Felline is an independent scholar. She lives in Seneghe (Sardinia) and her work focuses on the role of diferent (causal and non-causal) varieties of explanations in physics. Samuel C. Fletcher is Assistant Professor in the Department of Philosophy at the University of Minnesota, Twin Cities, a Resident Fellow of the Minnesota Center for Philosophy of Science, and an External Member of the Munich Center for Mathematical Philosophy. Doreen Fraser is Associate Professor of Philosophy at the University of Waterloo. She has published on topics including quantum feld theory, Newton, underdetermination, and scientifc realism. xv

Contributors

Steven French  is Professor of Philosophy of Science at the University of Leeds. He has published numerous books and papers in the philosophy of science and philosophy of physics, including mostly recently Applying Mathematics: Immersion, Inference, Interpretation, with Otávio Bueno (Oxford University Press, 2018), and There Are No Such Things As Theories (Oxford University Press, 2020). Roman Frigg is Professor of Philosophy in the Department of Philosophy, Logic and Scientifc Method at the London School of Economics and Political Science. He is the winner of the Friedrich Wilhelm Bessel Research Award of the Alexander von Humboldt Foundation. He co-authored, together with James Nguyen, Modelling Nature. An Opinionated Introduction to Scientifc Representation (2020). Mathias Frisch  is Professor at Leibniz University of Hannover. He is the author of two books: Inconsistency, Asymmetry, and Non-Locality: A Philosophical Investigation of Classical Electrodynamics (2005) and Causal Reasoning in Physics (2014). Richard Healey is Professor of Philosophy at the University of Arizona. He is the author of The Quantum Revolution in Philosophy (Oxford University Press, 2017) as well as Gauging What’s Real (Oxford University Press, 2007). Nick Huggett is LAS Distinguished Professor of Philosophy at the University of Illinois at Chicago. His work focuses on philosophy of physics, especially regarding theories of quantum gravity. His recent work includes Everywhere and Everywhen (2009), Beyond Spacetime (2020), Philosophy Beyond Spacetime (2021), and Out of Nowhere (forthcoming). Jenann Ismael  is Professor of Philosophy at Columbia University. Her books include Time: A Very Short Introduction (Oxford University Press, forthcoming), How Physics Makes Us Free (Oxford University Press, 2016), and Essays on Symmetry (Routledge, 2001). Melissa Jacquart is Assistant Professor at the University of Cincinnati and Associate Director of the UC Center for Public Engagement with Science in Cincinnati, Ohio, USA. Lina Jansson  is Associate Professor in the Department of Philosophy at the University of Nottingham. She works on issues related to explanation, laws of nature, and confrmation. Marc Lange is Philosophy Department Chair and Theda Perdue Distinguished Professor of Philosophy at the University of North Carolina at Chapel Hill. His most recent books are Laws and Lawmakers (Oxford University Press, 2009) and Because Without Cause: Non-Causal Explanations in Science and Mathematics (Oxford University Press, 2017). Dennis Lehmkuhl  is Lichtenberg Professor for History and Philosophy of Physics at the University of Bonn. He is also one of the editors of the Collected Papers of Albert Einstein published by Princeton University Press, and a Visiting Associate of the Einstein Papers Project at the California Institute of Technology. Peter J. Lewis  is Professor of Philosophy at Dartmouth College. He is the author of Quantum Ontology (Oxford University Press, 2016). Tim Maudlin is Professor of Philosophy at New York University and is the Founder and Director of the John Bell Institute for the Foundations of Physics. His recent books include Philosophy xvi

Contributors

of Physics: Space and Time (Princeton University Press, 2012), New Foundations for Physical Geometry (Oxford University Press, 2014), and Philosophy of Physics: Quantum Mechanics (Princeton University Press, 2019). C. D. McCoy is Assistant Professor of Philosophy at Underwood International College, Yonsei University. His research falls within the philosophy of physics and general philosophy of science, and is especially focused on philosophical issues in modern cosmology. Wayne C. Myrvold is Professor in the Department of Philosophy at the University of Western Ontario. He is the author of Beyond Chance and (Oxford University Press, 2021). Jill North is Professor in the Department of Philosophy at Rutgers University. She is the author of Physics, Structure, and Reality, (Oxford University Press, 2021). Oliver Pooley  is Fellow and Tutor in Philosophy at Oriel College, Oxford, and an Associate Professor in Philosophy in the Faculty of Philosophy at the University of Oxford. James Read  is Associate Professor in the Faculty of Philosophy, University of Oxford, and a Tutorial Fellow at Pembroke College, Oxford. Dean Rickles  is Professor of History and Philosophy of Modern Physics at the University of Sydney. His recent books include Covered in Deep Mist: The Development of Quantum Gravity, 1916–1956 (Oxford University Press, 2020) and What Is Philosophy of Science? (Wiley, 2020). Bryan W. Roberts  is a philosopher of physics, Associate Professor of Philosophy, Logic and Scientifc Method, and Director of the Centre for Philosophy of Natural and Social Science at the London School of Economics and Political Science. Laura Ruetsche is Louis E. Loeb Collegiate Professor of Philosophy at the University of Michigan. Her Interpreting Quantum Theories (Oxford University Press, 2011) shared the 2012 Lakatos Award with David Wallace’s The Emergent Multiverse (Oxford University Press, 2012). Juha Saatsi  is Associate Professor at the University of Leeds. He works on various topics in philosophy of science, and he has particular interests in the philosophy of explanation and the scientifc realism debate. Ryan Samaroo  is Associate Faculty at the University of Oxford and is currently serving as an advisor on scientifc methodology to the Government of Canada. He is interested in the foundations of physics and mathematics, with a focus on the structure of theories. Simon W. Saunders is Professor of Philosophy of Physics at the University of Oxford, and a Fellow of Merton College, Oxford. He is the lead editor (with Jonathan Barrett, Adrian Kent, and David Wallace) of Many Worlds? Everett, Realism, and Quantum Theory (Oxford University Press, 2010). Arianne Shahvisi  is Senior Lecturer in Ethics at the Brighton and Sussex Medical School. She has broad philosophical interests, including feminist philosophy, bioethics, philosophy of science, and social epistemology.

xvii

Contributors

Orly Shenker is Eleanor Roosevelt Chair in History and Philosophy of Science and Director of the Sidney M. Edelstein Centre for History and Philosophy of Science, Technology and Medicine at the Hebrew University of Jerusalem. With Meir Hemmo, she is the co-author of The Road to Maxwell’s Demon: Conceptual Foundations of Statistical Mechanics (Cambridge University Press, 2012) and the co-editor of Quantum, Probability, Logic: the work and infuence of Itamar Pitowsky (Springer, 2019). Susan G. Sterrett is the Curtis D. Gridley Distinguished Professor of History and Philosophy of Science at Wichita State University. She has published widely on models, including Wittgenstein Flies a Kite: A Story of Models of Wings and Models of the World (Pi Press, 2005). Mauricio Suárez  is a philosopher of science and probability. He edited Probabilities, Causes and Propensities in Physics (Springer, 2011), and is the author of Philosophy of Probability and Statistical Modelling (Cambridge University Press, 2020). He is currently Full Professor at Complutense University of Madrid, and a research associate at the Centre for Philosophy of Natural and Social Science at the London School of Economics and Political Science. Nicholas J. Teh  is Associate Professor in the Department of Philosophy and the History and Philosophy of Science Program at the University of Notre Dame. His research interests include the philosophy of physics, issues surrounding scientifc representation, modeling and idealization, and the philosophy of painting. Karim P.Y. Thébault is Senior Lecturer in Philosophy of Science in the Department of Philosophy at the University of Bristol. His research is principally within the philosophy of physics, with a particular emphasis on time and symmetry in classical and quantum theories of gravity. Roderich Tumulka  is a faculty member in the mathematics department of the Eberhard Karls University of Tübingen. He works on foundational questions in quantum mechanics, quantum feld theory, and quantum statistical mechanics. David Wallace  is A.W. Mellon Professor of Philosophy of Science at the University of Pittsburgh. He has written on a wide range of topics in philosophy of physics, including statistical mechanics, symmetry, spacetime, quantum mechanics, quantum feld theory, and quantum gravity. James Owen Weatherall  is Professor of Logic and Philosophy of Science at the University of California, Irvine. He is the author of The Misinformation Age: How False Beliefs Spread (with Cailin O’Connor) and Void: The Strange Physics of Nothing, both published by Yale University Press. Charlotte Werndl  is Professor for Logic and Philosophy of Science in the Department of Philosophy at the University of Salzburg, Austria. Her book, The Legacy of Tatjana Afanassjewa: Philosophical Insights from the Work of an Original Physicist and Mathematician (Springer, 2020), was edited together with Jos Ufnk, Giovanni Valente, and Lena Zuchowski. Porter Williams  is Assistant Professor of Philosophy at the University of Southern California. He works primarily in the philosophy of science and the history and philosophy of physics, particularly quantum theories. Christian Wüthrich  is Associate Professor of Philosophy at the University of Geneva. He has two forthcoming monographs, Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity with Nick Huggett and Time and Again: On the Logical, Metaphysical, and Physical Possibility of Time Travel with J.B. Manchak and Chris Smeenk. xviii

INTRODUCTION Eleanor Knox and Alastair Wilson

What is philosophy of physics? We can give plausible-looking but uninformative answers of the form “the intersection of physics and philosophy,” but any serious investigation into the character of the feld has to look at its historical context. Depending on what we identify as the same intellectual tradition, we obtain very different conceptions of philosophy of physics and of its beginnings. Attempts to understand the most general features of the physical world probably go back beyond recorded history, but by the frst millennium BCE, the Babylonians and Greeks were engaged in systematic theorizing and hypothesis testing about the natural world and in putting forward global metaphysical explanations for natural phenomena. Should we count the atomism of Democritus as a contribution to philosophy of physics, or to pure metaphysics? We leave that to the reader to decide. Physics itself did not emerge as a recognizable discipline, distinct from the broader category of natural science, until after the Renaissance in Europe. Philosophical refection accompanied the new discipline from the beginning. In the writings of Galileo, Newton, and Leibniz, we can recognize central questions that still animate contemporary philosophers of physics. Works like the ˆ commentary on her translation into French Leibniz-Clarke correspondence (1717) and du Chatelet’s of Newton’s Principia (1756) served as interpretive guides to the new physics and formed the core of a nascent foundational literature. Does this period mark the start of philosophy of physics proper? An interesting question no doubt, but not one we will address here: our focus is on contemporary philosophy of physics. We aim to take a pluralist approach to what counts as philosophy of physics, one which links the nature of the feld to the varied institutions within which it is taught and researched. By 1900, physics and philosophy had been institutionalized in Western universities as distinct disciplines within faculties of science and humanities, respectively, but there were clear examples of ´ Broad, Russell, Noether, and Eddington had cross-overs between these disciplines: Mach, Poincare, very different academic careers but they worked on the same cluster of topics. For the frst decades of the 20th century, physics was respected within philosophy, and vice versa. That relationship has become strained at times in the post-war world; generations of physicists were raised with a distrust of philosophy under the enduring infuence of logical positivist and logical empiricist traditions. In recent decades, though, philosophy of physics has found institutional security as a subfeld in its own right. Today there are numerous undergraduate and postgraduate degree programs which offer courses and qualifcations in philosophy of physics, substantial research clusters at leading universities, well-established recurrent conferences, prizes, specialist journals, and the inevitable concomitant rankings. Our goal with the Companion has been to provide a representative cross section of this wide-ranging research activity. Limiting our focus to the contemporary practice of philosophy of physics does not of course guarantee a unifed conception of what philosophy of physics is and how it should be pursued. Philosophers are fond of prescriptive defnitions and divisions, and physicists are not much better. So it will hardly surprise the reader to hear that, even in a feld as small as the philosophy of physics, methodological disagreements are rife. Given its essential engagement with so mathematical a feld as physics, some gatekeeping is inevitable: without a solid understanding of the physics, things can 1

Eleanor Knox and Alastair Wilson

go badly off the rails. But we are of the view that imposing divisions has largely hindered rather than helped the feld: at times mathematical technical ability is mistaken for philosophical (or indeed physical) insight, while at others towering metaphysical structures are built on fimsy physical foundations. Likewise, philosophy of physics is at its most important and infuential when it looks to the future, but work on contemporary theories suffers if the lessons of history are ignored. This volume is constructed with the assumption that good philosophy of physics can be done in physics departments, philosophy departments, HPS departments, and independently. The very best work in contemporary philosophy of physics combines a deep understanding of the physics with philosophical, and sometimes historical, subtlety: we hope some of that work is showcased here. While a form of pluralism is at the heart of this volume, some carving up of the feld is inevitable. Our chosen primary division, between theories and themes, aims to divide without excluding, and thereby to capture important work even where it falls at the edges of philosophy of physics. Many, but not all, philosophers of physics consider themselves “philosophers of X”, where X is a theory or family of theories: quantum mechanics, general relativity, or the like. The frst half of this volume aims to capture a range of this kind of theory-focused work with respect to the main theories that form the core of modern physics: specifcally, classical mechanics, relativistic mechanics, quantum theories in their various forms, statistical mechanics, and thermodynamics. But this focus on particular theories can obscure the wider philosophical questions that arise from physics; much exciting work, particularly in recent years, lies at the boundary between philosophy of physics and other areas of philosophy. The second half of the volume encompasses this kind of thematic work: philosophy that refects on the discoveries and the practice of physics to reach conclusions about topics of wider philosophical interest such as explanation, causation, or reduction. We hope that this two-fold structure helps to make the book suitable for a wide audience: for the physicist interested in the foundations of their feld, for the philosophy of physics graduate student looking for a thesis topic, but also for the philosopher seeking to understand how physics bears on their areas of interest. We’ve aimed for a range of levels of accessibility within the sections, and at least some chapters in each section should be readable by an advanced philosophy undergraduate without physics training. That said, in areas such as quantum feld theory, authors inevitably presuppose more specifc knowledge of physics. In others, like spacetime physics, important results cannot be expressed without a certain level of technical sophistication. Typically, chapters increase in degree of technicality through each part, especially in the “Theories” section. But generally, we have not tried to impose too much uniformity on the style or structure of the chapters; part of the strength of philosophy of physics, as we have tried to highlight, is diversity of individual approach among its practitioners. We hope that this shows through in the varied character of the contributions. It is traditional in an introduction like this to opine on the future of the feld. Such crystalball-gazing has its drawbacks: 20 years ago, for example, one might not have pegged the study of Newtonian spacetime for a comeback. Nonetheless, recent years have seen substantial developments in our understanding of various Newtonian spacetime structures and the relations between them. But some trends in the discipline are easier to spot. Several spring from the renewed relationship between physicists and philosophers mentioned above: as physics pushes new boundaries, new philosophical problems come to light, and old ones spring back up from the dusty corners to which physics tried to relegate them. A host of new physics research programs, and the grants that accompany them, involve physicists asking for active input from philosophers. Some of these new research programs are featured in this volume. Developments in quantum gravity, or perhaps the lack of them, raise questions about progress in science, as well as revealing underlying disputes along old philosophical lines (it is hard to do quantum gravity without implicitly taking a stand on the measurement problem, for example; in cosmology, no external observer is available to collapse the wavefunction). But quantum gravity also reveals exciting new philosophical questions, for example: what might it mean for the spacetime of our experience not to be fundamental? How should we think about the “dualities” posited by string 2

Introduction

theory? In a theory without a primitive notion of time, do we need a primitive notion of causality to replace it? The philosophy of cosmology is likewise a relatively young feld, and one that looks set to grow in the coming years. Here the pressing questions are often obviously connected to more general philosophical issues: how do we deal with “one-shot” theories for which experimentation is impossible? What role might anthropic reasoning play in such a theory? Can non-empirical values play a role in theory choice here that they might not play elsewhere? It will not escape the reader’s attention that many of the questions addressed in this Companion seem to sit in general philosophy of science and/or in metaphysics rather than in the philosophy of physics, narrowly conceived. That is certainly true, although it’s also true that dealing adequately with these problems in the context above requires a good understanding of both physics and philosophy of physics. But the expertise that physicists sometimes lack and seek elsewhere is often distinctively philosophical. So here, perhaps, is the moral for a graduate student in search of a topic: philosophers of physics can sometimes underestimate the importance of their general philosophical training, but it is also the unique skill they bring to the table. Work at the outward-facing edges of philosophy of physics is as diffcult and signifcant as technical and theory-centered work. Inevitably in a project of this scale, we have not managed to achieve as uniform a coverage as we would have liked. In particular, we regret the lack of chapters on quantum information theory and black holes. We hope to fll some of these gaps in future editions, as well as keeping the treatment of cutting-edge topics up to date. We’ve aimed to take a snapshot of the philosophy of physics community which is reasonably diverse with respect to geography, disciplinary affliation, gender, and other social variables; however, there is no getting away from the fact that philosophy of physics as it is currently practiced is lacking in demographic diversity. We hope and expect that this situation will improve in the coming years. Philosophy of physics, overall, is in as good a shape as it has ever been, and we hope that this Companion will provide a useful resource for existing practitioners and for newcomers alike.

3

Section A

Theories

PART I

Newtonian Mechanics Introduction to Part I In both physics and philosophy of physics, our study usually begins with Newtonian physics. Whether it’s resolving the forces on objects on a slope, or contemplating the justifcation for Newton’s postulation of absolute space, the theory provides a fertile ground for pedagogical exploration. And within physics, of course, knowledge of this theory is crucial; despite being superseded in various domains by relativistic and quantum physics, Newtonian physics remains accurate enough in many domains that it still provides us with many of our best physical models. But why should we study the foundations and philosophy of a superseded theory? What might the aims and outcomes of such an enterprise be? One thing is clear: if we think of the philosophy of Newtonian physics as an exploration of the metaphysics of a non-actual possible world governed by Newtonian laws, the ink spilled on the topic is wasted. After all, Newtonian theories can’t account for the existence of stable matter. So what justifes the enduring popularity of the subject? In part, the answer lies with the theory’s familiarity: after all that exposure in school we take ourselves to have an intuitive understanding of Newtonian mechanics. This means that thinking about the foundations is more straightforward; even though in many ways the structures of Newtonian theories are more complex than those of relativistic ones, the concepts come more easily. Moreover, the connection between Newtonian theory and empirical results is largely unproblematic. This value goes well beyond pedagogy. In order to think about theories that test the very bounds of our physical and mathematical understanding, we need clear heuristics for theory interpretation: how should we think about differences that are, in principle, not empirically detectable? What role do concepts like “force” or “inertia” play in our theories? What kinds of consideration determine what we should say about the spacetime of a theory? The chapters in this section suggest answers to many of these questions, and hence hold morals for further theories discussed in subsequent chapters. In Chapter 1, Ryan Samaroo explores the conceptual structure of Newton’s theory. One might be forgiven for thinking that Newton’s primary achievement lies in giving an empirically accurate model of terrestrial and astronomical motion, but Samaroo argues that the conceptual achievements of the theory go beyond this: Newton provides principles that have a special character and that articulate and apply in a special way concepts of quantity of matter, motion, space, and time. While these principles and the concepts they articulate require adjustment in the light of more recent theories, they represent nonetheless a paragon of the kind of theorizing required for modern physics as we understand it. Samaroo works primarily in the familiar framework of forces and acceleration used by Newton himself, but part of the enduring importance of the theory stems not from Newton’s own mathematical framework, but from the great re-renderings of the theory offered by Lagrange and Hamilton in the late 18th and early 19th centuries. The Lagrangian and Hamiltonian formulations of classical mechanics look, at least on the surface, quite different, both from each other and from Newton’s original theory. The later formulations are particularly important because they provide a framework not

Newtonian Mechanics

just for Newtonian physics, but for the relativistic and quantum theories that were later developed. But although all three formulations are usually thought of as notational variants, the relation between them is far from obvious. Jill North’s chapter explores these relations and asks widely applicable questions about the notions of theory, formulation, equivalence, and fundamentality. The 19th century held further mathematical developments of relevance to Newtonian mechanics, although it took some years for this relevance to be widely recognized. Geometrical techniques more commonly used in relativity can be equally well applied to Newtonian theories, and can help to elucidate the spacetime structures of the theory. Jim Weatherall’s chapter explores these structures; using geometrical ideas more usually encountered in relativity, he considers what we should say about Newtonian spacetime structure. This involves a modern look at the debate between Leibniz and Clarke over the existence of absolute substantival space. As Howard Stein pointed out in 1967, thinking about the structure of Newtonian spacetime allows us to understand that it’s possible for both Leibniz and Clarke to be right: Newton overreached in postulating absolute space but was nonetheless correct that some absolute structure is required by his theory to support the existence of absolute accelerations. While Newton may have been right about the need for absolute structures in his own theory, contemporary strategies do not entirely close the debate between substantivalism (which holds that space or spacetime is an entity in its own right) and relationism (which holds that space is reducible to the relations between bodies). What becomes clear is that for Leibniz (and later Mach) to be correct, a new, relationist theory is needed which can replicate the empirical results of Newtonian mechanics without commitment to spatiotemporal structure. Mach’s legacy had to wait until the early 1980s for a spatially relationist theory to emerge through the work of Julian Barbour and Bruno Bertotti. In Barbour’s chapter for this volume, he explores the possibility of going beyond their original theory, and developing a classical theory which gives a truly relational account of both space and time. This chapter is more technical than the others in this section, so the reader not familiar with the physics and mathematics is advised to start with the earlier entries.

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1 NEWTONIAN MECHANICS Ryan Samaroo

The theory that we now know as “Newtonian mechanics” is Newton’s science of matter in motion, and its philosophical signifcance, in a sentence, is this: Newton gave us more than just an empirically successful theory of mechanics – he gave us an account of what knowledge of the physical world should look like, one that remains with us. But what is this account of physical knowledge? What is it that remains with us? Various answers to these questions have been given and they concern the methodological character of the laws of motion. What is methodologically rational about them? What is their distinctive feature? These are the questions on the table. The structure of this chapter is as follows. I will begin by introducing the laws of motion, the relations among them, and the spatiotemporal framework that is implicit in them. Then I will turn to the question of their methodological character. This has been the locus of philosophical discussion from Newton’s time to the present, and I will survey the views of some of the major contributors. A theme running through this section is that there is something in the spirit of Kant’s analysis of Newtonian physics that is worth preserving, though distilling what that is is an open problem. I will conclude by showing that while Newtonian mechanics motivates a number of philosophical ideas about force, mass, motion, and causality – and through this, ideas about space and time – the laws are themselves the outcome of a philosophical or critical conceptual analysis. Therefore, taking some care to understand how the theory grew out of Newton’s analysis of the conceptual frameworks of his predecessors and contemporaries is valuable for its insights into the nature of that activity. A word about the scope of this chapter is in order. It is worth recalling that Principia contains two theories: the theory of mechanics and the theory of universal gravitation. The former is found in a few pages right at the start of Principia in “Axioms, or The Laws of Motion,” immediately following Newton’s articulation of a few basic notions in “Defnitions.” The latter is a derived theory within the mechanical theory, that is, once that theory has been extended, through a number of assumptions, to encompass planetary systems. My focus will be on the theory of mechanics. What is this theory? It is a theory of causal interaction: it is about motion and the forces producing motion. Newton dealt with matter in resisting and non-resisting media. My focus will be the mechanics of point particles in non-resisting media – the most basic subject matter with which the theory deals. But it is worth mentioning, if only in passing, the formulations of Newtonian mechanics of Euler and Cauchy. (See Truesdell (1977) for details.) And there are still other formulations of Newtonian mechanics, notably those of Lagrange and Hamilton. These formulations are based on the principle of least action, and they incorporate insights into the conservation of momentum and energy. They reveal a deep layer of structure exhibited by physical systems of many kinds and make them amenable to a similar treatment; in this way, these formulations extend Newtonian mechanics and greatly increase its computational power. They also provide a point of contact between classical and quantum theory. I will not discuss any of these formulations – my focus will be on 8

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“old-fashioned” Newtonian mechanics. (See Curiel (2014) and North (2019) for a detailed account of the relation of the Lagrangian and Hamiltonian formulations to classical systems and each other.) It should also be noted that there is a reconstruction of Newtonian gravitation, patterned on Einstein’s theory of gravitation, in which the basic account of motion is inseparable from the gravitational feld: Cartan’s reconstruction. On that reconstruction, gravitation is not a force causing acceleration but a manifestation of the curvature of space-time; as in Einstein’s theory, the trajectories of free particles are geodesics (or “straight lines”) of the (curved) space-time. Cartan’s proposal is in several respects a natural and instructive way of thinking about Newtonian theory. (See Malament (2012) for details.) But I will deal only with Newtonian mechanics, independent of the gravitation theory.

1.1 Background: The Theory and the Spatiotemporal Framework Implicit in It Newton’s theory of causal interaction has three axioms, the laws of motion: Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and opposite in direction. (Newton, 1726 [1999], pp. 416–417) The laws, taken together, defne and interpret the concept of inertial motion. This concept is the backbone of the theory, and, by examining how it is articulated, we will come naturally to all core aspects of the theory. The frst law defnes an ideal force-free trajectory, one from which a particle can be defected by the action of some force, an objective cause. The frst law alone, however, does not provide an account of inertial motion since we do not yet have a defnition of force – it is a precondition for such a defnition. It is the second law that defnes and interprets the concepts of force and mass, tying them to acceleration. The acceleration of mass is the measure of the action of some force. The second law expresses a criterion for distinguishing free particles from particles acted upon by a force. Now, one might be tempted to suggest that the second law alone is enough for giving an account of inertia – one might suggest that the frst law is a limiting case of the second, that is, when there is no force impressed. But the frst law associates or coordinates a free particle with a particular kind of trajectory, a straight line. Hence, the frst and second laws are interdependent. The frst and second laws provide a complete account of inertial motion, provided that one is interested only in ideal point particles. For actual bodies – bodies that are themselves composed of particles – the third law is a necessary condition for inertial motion. The forces among the constituent particles must be equal and opposite, failing which the body by its internal forces will accelerate of its own accord. (See the Scholium to the Laws of Motion, where Newton gives a proof of this; see also Samaroo (2018).) This is what the third law establishes and it is the basis for formulating a principle of conservation of momentum: in an isolated system the total momentum is conserved. Hence, the third law is also a criterion for distinguishing free particles from those acted upon by a force. What should be clear from this brief account is that inertia depends on all three laws for its articulation. It should also be clear that the laws of motion are mutually complementary. Only taken together do they determine Newton’s theory of causal interaction. Now, we can associate with any particle in inertial motion a reference frame. A reference frame is a space, one in which we can describe the motions of bodies in the space among themselves, for example, using a coordinate system. But we can also perform mechanical experiments and calculate 9

Ryan Samaroo

their outcomes using the laws of motion. In any such space, the outcomes will be the same. These “inertial frames” that are picked out by the laws of motion are the basis for empirical investigation in Newtonian mechanics. With this brief account of Newtonian mechanics in hand, let us consider the spatiotemporal framework that is implicit in it. The laws of motion defne inertial motion as that state in which a body unacted upon by forces, or on which the net force is zero, moves in uniform rectilinear motion. In other words, a body in inertial motion moves equal distances in equal times. In this way, the laws of motion defne an ideal clock that marks the “equable fow” of time. Furthermore, it is implicit in the theory that all inertial observers, and all ideal clocks, will measure proportional time intervals and agree on which events are simultaneous. The concept of space that is implicit in Newtonian mechanics is tied to the inertial frame concept. As we have seen, an inertial frame is one in uniform rectilinear motion – one furthermore in which the outcomes of mechanical experiments, calculated using the laws of motion, are the same. And Newton noted that the same outcome would be obtained in any frame in uniform rectilinear motion relative to it – this is the Galilean principle of relativity. By way of this invariance property, we obtain an equivalence class of inertial frames – the class of frames in which the outcomes of mechanical experiments are the same. The equivalence-class structure so determined is the structure of the space-time of Newtonian mechanics. It is important to note that this structure admits of no distinction between rest and uniform motion – both are states of inertial motion. By contrast, Newton held that while inertial frames are empirically indistinguishable, they are not theoretically equivalent. They move with various velocities relative to what he called “absolute space,” even if those velocities cannot be known. It was only in the 19th century, through the work of Neumann (1870), Thomson (1884), Lange (1885), and others, that absolute space was shown to be superfuous. It is sometimes said that the concept of space-time has its origin in Einstein’s special theory of relativity in 1905. But something that should be evident from the foregoing is that already in Newtonian mechanics there is a concept of space-time. The very concept of an inertial trajectory, which is the basis for Newton’s theory of causal interaction, appeals not just to places and times but to places connected at times. And not only does an inertial trajectory connect places at times but it connects them in such a way that certain states of motion are well defned. (See Stein (1967), DiSalle (2006), and Malament (2012) for careful accounts of this; see also Earman (1989) and Weatherall (2016) for other accounts of the spatiotemporal framework that Newtonian mechanics is sometimes held to motivate.) To summarize, what we fnd in the laws of motion is not only an account of force, motion, and causality but also a spatiotemporal framework.

1.2

The Methodological Character of the Laws of Motion

The theory, now introduced in overview, is a paragon of empirical success. What is the rational justifcation for this success? This is the question with which methodological analysis is concerned. A methodological analysis asks two basic questions. What kind of principles are the laws of motion? What is their role in the conceptual framework of physics? For example, are the laws entirely determined by empirical evidence? Or do they refect elements of choice, for example, considerations of simplicity? Or is their methodological character more complex? And, if so, what is their character? One answer is that given by Hume (1777 [1975]), who took Newtonian science to be a revolutionary advance – he took it as the model for his “science of human nature.” Hume regarded the laws of motion as empirical generalizations that are inductively derived from constant and regular experience, that is, from a set of empirical facts. Consider the following remark about the second law: “Geometry helps us apply this law . . . , but the law itself is something we know purely from 10

Newtonian Mechanics

experience, and no amount of abstract reasoning could lead us one step towards the knowledge of it” (Enquiry, IV). It is useful to recall Hume’s claim that all objects of human reason are either “relations of ideas,” of which his examples are the propositions of arithmetic and geometry, or “matter of fact,” namely contingent empirical propositions such as “the Sun will rise tomorrow.” Evidently, Hume regards the laws of motion as matters of fact. Hume’s view of the laws of motion singles out an obvious feature, namely that experience has a role to play in their formulation. But there are a number of criticisms one might raise against his view. One might say that we do not know the laws “purely from experience.” For example, there are no truly force-free bodies: inertial motion is an ideal state and we have no “impressions.” Hence, one could hardly say that the frst law derives from mere induction. One might also take issue with Hume’s remark that geometry merely “helps us” apply the laws. The formulation of the laws presupposes a number of mathematical concepts, notably concepts belonging to Euclidean geometry and the calculus. In this way, one might argue that Hume’s division of the objects of reason into relations of ideas and matters of fact is inapt for the analysis of the laws of motion. Hume’s failure to give a satisfactory characterization of the laws is naturally contrasted with Kant’s. Kant’s account is not without its own diffculties, but it captures a feature of the laws that has remained part of subsequent discussions. Much like Hume, Kant saw in Newton’s theory not only a revolutionary scientifc discovery but a revolutionary philosophical advance. He saw a basis for criticizing the reigning Leibnizian tradition in which concepts of force and motion, space and time, substance and causality are applied to the “intelligible” world of monads. In the First Critique (1787 [1998]), Kant asked, how has science achieved universal assent, while philosophy is the subject of endless dispute? What distinguishes scientifc reasoning from philosophical reasoning, so that the former leads to principles that are necessary and universal, whereas the latter remains arbitrary and particular? How can philosophy start on “the secure path of a science”? Kant argued that nothing less than a Copernican Revolution in philosophy is needed. No longer should philosophy be done after the fashion of Leibniz and Wolff, or following an earlier empiricism: philosophy’s task is to reveal the structure of our faculty of understanding – the structure that the very possibility of knowledge implicitly presupposes. Kant’s theory of the constitution of experience provides an account of the concepts of this faculty and, of particular interest to us, the principles both “constitutive” and “regulative” by which they are applied to possible experience. The principles are rules that the understanding imposes on the appearances, in order to submit them rules. Without such rules, experience would be impossible. We would have nothing but a chaos of sensory appearances. The principles are about the world, and therefore “synthetic,” but known through transcendental deduction, and therefore “a priori.” Kant regarded the account of the constitution of experience as the true subject matter of metaphysics. In the Metaphysical Foundations (1786 [2011]), Kant held the laws of motion to be just such principles. They have a constitutive function: they determine the concepts of the objects of enquiry; they make it possible for objects of knowledge to be objects of knowledge. The laws of motion are constitutive not only of a particular conception of force, mass, inertia, and causal interaction but of a spatiotemporal framework relative to which true motion can be understood. Breaking with the Leibnizian tradition, Kant argued that our metaphysical concepts of force and motion, causal interaction, and space and time have no content at all except through their “sensible” counterparts, that is, through their articulation in the laws of motion. Kant’s account is a landmark in the theory of knowledge, but it is problematic in at least one respect: Kant took Newton’s laws to be the only ones that constitute the concepts of force, mass, and motion, and, furthermore, space, time, and causality. But though physics did not end with Newton, the idea that the laws of motion and certain other physical principles have a constitutive function was developed in the work of Kant’s ´ successors, notably in the work of Poincare.

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Ryan Samaroo

The idea that the laws of motion are defnitions has a special place in the analysis of their methodological character. What is meant by “defnition” is important here. For example, we might take a Russellian notion of defnition as a starting point. For Russell (1897), the constituents of a sentence expressing a proposition must have independently grasped meanings. On such an account, the terms appearing in the laws – “force,” “mass,” and “inertial motion” – would already have their meanings, independently of the theory determined by the laws. But there is another way of thinking about the laws of motion: we might regard them as implicit defnitions. We fnd this idea in the work of Poincare´ and Duhem. Poincare´ (1902 [1952]) argues against Russell’s view that we can know the meanings of primitive terms directly, for example, by ´ the primitive terms are implicitly defned by the axioms in intuition or acquaintance. For Poincare, which they fgure. The laws of motion, on his account, are defnitions disguised as claims. But there is a further aspect to his view: Poincare´ pointed out that a geometrical framework must be presupposed for the construction of a mechanical theory and he claimed that we can choose any one of the geometries of constant curvature, namely Euclidean, Bolyai-Lobachevskian or Riemannian. For him, there is no fact of the matter about which of them is the actual space of experience, but, since the laws of mechanics will be simplest on a Euclidean background, he held that Euclidean geometry would always be preferred. He stressed that geometry, on its own, tells us nothing about the behavior of physical objects, only geometry together with physical laws. He held that a geometrical framework and parts of the laws can be chosen arbitrarily – all that is required is that the remaining part of the laws be chosen such that the resulting theory is empirically adequate. For these reasons, he claimed that the laws of motion are conventions. The presupposition of a Euclidean background is evident in the frst law of motion, where “straight” is understood in the Euclidean sense. The frst law defnes the trajectory of a force-free body as a straight line – it establishes a correspondence between a physical object and a geomet´ it is ric notion, as part of a particular way of constructing a mechanical theory. But, for Poincare, important that assuming a Euclidean background does not preclude the possibility that the completed theory or another theory that is in some sense more fundamental may lead us to revise our presuppositions about geometry. What about the second law of motion? Taken on its own, we cannot speak of the truth or falsity of the relation expressed in the law because there is no experiment that settles the question. To speak of the truth or falsity of the law would be to assume that there is something prior to Newtonian mechanics that provides an independent defnition of force – at least a defnition that forms part of an empirically adequate theory of mechanics. For example, we cannot say that the relation between force and acceleration expressed in the law is imprecise, since any imprecision that we might notice while measuring some particular force only suggests to us that we need to look for the forces contributed by some yet-unnoticed bodies. But to say that there is no experiment that settles the question of the truth or falsity of the relation is not to suggest that the force law is not empirically constrained. The law can only be evaluated as part of the entire system of mechanics that it helps defne. What about the third law of motion? Poincare´ notes that there are no perfectly isolated systems, only nearly isolated systems. When we observe such systems, we see that the constituent parts interact with one another such that they satisfy the third law and the center of gravity of the system moves (nearly) uniformly in a straight line. Poincare´ asks, could a more accurate experiment invalidate this? “What, in fact, would a more accurate experiment teach us? It would teach us that the law is only ´ 1902 [1952], p. 105). The third law defnes approximately true, and we know that already” (Poincare, action and reaction to be equal and opposite – it expresses a criterion by which we can determine whether momentum is conserved in an isolated system. ´ s view that the laws of motion are conventions commits him Now, one might think that Poincare’ to the view that they are arbitrary. But he is clear that the laws are not arbitrary:

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Are the laws of acceleration and of the composition of forces only arbitrary conventions? Conventions, yes; arbitrary, no – they would be so if we lost sight of the experiments which led the founders of the science to adopt them, and which, imperfect as they were, ´ 1902 [1952], p. 110) were suffcient to justify their adoption. (Poincare, ´ s recognition that while the laws of motion refect elements What we fnd in this passage is Poincare’ of choice, they are empirically constrained. The laws may of course be revealed to be bad defnitions or to have only limited applicability, but they function nonetheless as implicit defnitions of the basic concepts of mechanics. Before pressing on, it is worth noting a (perhaps obvious) feature of laws of motion: they fail a condition of observational non-creativity, according to which, roughly speaking, a defnition should have no observational consequences. This is a condition that defnitions are required to satisfy if they are to be regarded as analytic. The laws of motion are defnitions, but they are not analytic – they are empirically constrained. The proposal that the laws of motion are correctly understood as defnitions, and furthermore implicit defnitions, takes us much of the way to later proposals. But the notion of implicit defnition has its origin in 19th-century work in the foundations of geometry, where it is discussed without reference to physical theory. It is evident that the laws of motion have a feature that the axioms of geometry do not: not only do they implicitly defne the concepts of mechanics but they interpret them. They coordinate theoretical concepts with empirically measurable correlates. That certain principles have a defning and coordinating function was recognized by Reichenbach (1928 [1958]), even if he did not discuss the laws of motion explicitly. Reichenbach regarded relativity theory as well established, but not well understood. He sought to improve our understanding of it by revealing the physical presuppositions that underlie the application of relativistic geometry and chronometry. Specifcally, he argued that their application depends on principles that he called “coordinative defnitions.” These defnitions establish how the claims of a mathematical theory are transformed from mathematical truths into claims that can be revised on the basis of experience. To take a simple example, Euclidean geometry becomes a theory of applied or physical geometry by means of the principle of free mobility: practically rigid bodies undergo free motions without change of shape or dimension. This principle is a presupposition of our ability to perform the compass-andstraightedge constructions of Euclidean geometry, and in this way it controls the application of the theory. Now, for Reichenbach, the main interest in identifying coordinative defnitions resides in their capacity to isolate which among the assumptions that control the application of geometry and chronometry are conventions and which are factual claims. And it is central to his view that certain principles that control the application of geometry and chronometry are based on stipulations. For this reason, he held coordinative defnitions to be arbitrary. We see this, for example, in his account of special relativity, in which he claims that the Einstein synchronization criterion rests on a stipulation about the to and fro velocities of light; hence, the synchronization of distant clocks is a matter of convention. The interpretive function of coordinative defnitions led Reichenbach to regard them as constitutive principles. Coordinative defnitions serve to apply an uninterpreted conceptual framework – the pure concepts of the understanding – to the world of experience. But while Kant held Newton’s laws to be the unique set of principles that constitute the conceptual framework of physics, Reichenbach regarded them not as absolute but relative. He recognized that experience might lead us to mutually inconsistent coordinations that are relativized to particular contexts of enquiry, but have nonetheless a constitutive function. Reichenbach’s notion of coordination is incorporated into the recent work of Michael Friedman (e.g., 2001, 2010). Friedman’s account of the laws of motion is found in his analysis of Newton’s 13

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and Einstein’s gravitation theories. According to Friedman, a satisfactory methodological analysis of these theories requires us to distinguish between three levels of enquiry. The frst level is comprised of principles that are epistemologically distinguished by the fact that they defne a space of intellectual and empirical possibilities, and so determine a framework of investigation. They articulate theoretical concepts and their physical interpretations. The second level is comprised of empirical hypotheses that are formulable within the framework. The third level is comprised of distinctly philosophical principles that motivate discussions of the framework-defning principles and the transition from one theory to another. Friedman calls the frst-level principles “constitutive principles.” He includes in this category both mathematical principles or presuppositions and coordinating principles. The mathematical principles defne a space of mathematical possibilities; they allow certain kinds of physical theories to be constructed. Among other examples, we fnd the calculus, linear algebra, and Riemann’s theory of manifolds. The coordinating principles, which Friedman understands in Reichenbachian terms, interpret theoretical concepts. They express mathematically formulated criteria for the application of concepts such as force, mass, motion, electric feld, magnetic feld, and others. On Friedman’s analysis, Newtonian gravitation has as its constitutive component Euclidean geometry, the calculus, and the laws of motion. This component defnes the space of intellectual and empirical possibilities that allows us to conceive of gravitation as a force, and that makes it possible to formulate the law of universal gravitation – an empirical hypothesis. For this reason, Friedman regards the principles comprising this component as relativized but nonetheless constitutive principles. They are not a priori, as Kant held them to be, but they are prior to the development of hypotheses about particular systems. Friedman’s approach to the analysis of physical theories is intended as a corrective to Quine’s (1951) account of scientifc knowledge. Quine took aim at the logical empiricists’ account of theories with its distinction between the analytic and synthetic components of a theoretical framework. Quine represented scientifc knowledge as a web of belief in which no satisfactory analytic-synthetic distinction can be drawn. Strands of the web are not subject to confrmation or disconfrmation as individuals – the web is confrmed or disconfrmed as a whole. And he claimed that in the case of a derivation where the conclusion conficts with experience, there is nothing to prevent us from holding on to the conclusion by revising the logical and mathematical principles that were assumed in the derivation. It is Friedman’s principal goal to show that there are distinctions between the components of our frameworks of physical knowledge, and that these components are stratifed. Friedman argues, furthermore, that certain components of our frameworks can hardly be said to be revisable, as Quine maintained. It makes little sense to speak of revising the constitutive component of a theory in the case of a conclusion that conficts with experience since constitutive principles determine the framework of empirical investigation – the framework without which an empirical hypothesis could be neither formulated nor tested. Friedman’s proposal is signifcant for its restoration of the logical empiricists’ idea that frameworks of physical knowledge are stratifed. But I have argued (Samaroo, 2015) that Friedman’s account of a constitutive principle is too broad: only coordinating principles should be regarded as constitutive. Friedman’s inclusion of both mathematical and coordinating principles in the category of constitutive principles is intended to address Quine’s contention that the mathematics involved in formulating a theory is just another element in the web of belief. Friedman argues that this view of the role of mathematics in physics fails to account for the way in which mathematics makes certain kinds of empirical theories intellectual possibilities; it fails to account for the way in which mathematics supplies some of the concepts required for formulating a theory and for deriving predictions. While I agree with Friedman about this, there are good reasons for taking constitutive principles to be only those principles that constitute or interpret theoretical concepts by expressing criteria for their application. 14

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First, one might argue that including mathematical principles in a theory’s constitutive component opens the notion of a constitutive principle to trivialization. One might argue that what is constitutive is relative to some particular formulation of a theory, and since what is constitutive in one is not constitutive in another, the very idea of a constitutive principle is undermined. For example, Newtonian mechanics admits of various formulations, some of which rest on radically different mathematical frameworks from others. Take, for example, the mathematical frameworks peculiar to analytic mechanics, which are very different from the one in which Newton worked. But, however the theory is formulated, Newtonian mechanics is the theory whose basic structure at least is constituted by the laws of motion. Second, one might argue that including mathematical principles in a theory’s constitutive component lends support to a main feature of Quine’s account of theories. A Quinean might argue that if, e.g., the calculus and Euclidean geometry are constitutive components of Newtonian mechanics, then they are confrmed or infrmed along with the rest of the theory. Friedman argues against Quine that constitutive principles can hardly be said to be tested along with the empirical hypotheses whose formulation they permit – they are principles without which empirical hypotheses would make neither mathematical nor empirical sense, and without which no test would be possible. But the principles that establish Friedman’s argument against Quine are not the mathematical principles, which, on their own, are subject to neither empirical confrmation nor disconfrmation, but the coordinating principles that interpret theoretical concepts and control and application of the mathematics. Hence, distinguishing the mathematical principles from the coordinating principles strengthens the case against Quine. Most importantly, however, the inclusion of both mathematical principles and coordinating principles in a theory’s constitutive component blurs the distinction between the theory’s factual and non-factual components. By taking only coordinating principles to be constitutive, we can distinguish clearly between those components of our theories that are empirically constrained and those that are not; we can distinguish between those principles that defne and articulate our epistemic relation with the world and those that are part of the formal background or language. The proposed limitation to the account of a constitutive principle is in no way intended to diminish the role of mathematical principles in the articulation and application of physical theories, but to clarify the fact that mathematical principles and coordinating principles have different criteria of truth. This proposal benefts the account of the stratifcation of our theoretical knowledge and allows a still stronger criticism of Quine’s account to be given; it aims in this way to vindicate something close to the analytic-synthetic distinction that Quine rejected. The foregoing is only a brief overview; see Samaroo (2015) for a sustained critical analysis. In light of these criticisms of Friedman’s program, what remains of the notion of a constitutive principle? And what of the methodological character of the laws of motion? I have argued that the notion of a constitutive principle – a principle that constitutes or interprets a theoretical concept by expressing a criterion of its application – has something to offer the account of the laws of motion. On this account, the laws of motion express criteria for the application of the concepts of force, mass, and inertial motion – and those that depend on these. It is worth noting that this account of the laws of motion – that is, of the laws as empirical criteria – is essentially Einstein’s. In a short but suggestive article, Einstein (1919 [2002]) sketched a distinction between theories that provide a general framework for physics (“principle theories” or “framework theories”) and specifc theories constructed within such a framework (“constructive theories”). Although Einstein’s focus was relativity theory, Newtonian mechanics and Einstein’s special and general theories are all framework theories. That is, they provide frameworks of constraints in which physical quantities can be constructed and whose evolution can be determined. As Einstein put it, these theories are based on “empirically discovered . . . general characteristics of natural processes” and they express “mathematically formulated criteria” that physical processes satisfy (Einstein, 1919 15

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[2002], p. 213). These criteria enable us to articulate theoretical concepts such as force, mass, inertia, acceleration, rotation, and simultaneity; furthermore, they motivate spatiotemporal frameworks. These theories must be presupposed for the construction of theories of special systems, for example, the theory of a point particle or that of a perfect fuid; in the Newtonian context, the theory of the gravitational feld. So, while the frameworks articulated by Newtonian mechanics, special relativity, and general relativity are not a priori in any Kantian sense, they are prior in this particular sense. Nonetheless, the presupposition of these frameworks does not preclude the possibility that some new theory will motivate their replacement. The laws of motion, in sum, are founded on experience and are in that sense synthetic, but they are not mere empirical generalizations, derived by induction. Nor are they synthetic a priori propositions, though they function as “constitutive a priori principles” in a particular sense of that term. They are certainly defnitions, but they are not analytic in the sense of being true by mere stipulation or convention, because they are responsible to a body of observation and experiment, and to the pre-analytic concepts of which Newton gave an analysis. They are “analytic of ” the concepts they determine. They implicitly defne the basic concepts of mechanics that appear in them, but they do more than that: they interpret those concepts. They function as “coordinative defnitions,” but they are not arbitrary in the Reichenbachian sense of that term. Still, they have the constitutive function that Reichenbach sought to capture: they constitute or interpret theoretical concepts by expressing criteria for their application; furthermore, they control the application of a number of mathematical theories. More simply, perhaps, they are empirical criteria for the application of the basic concepts of mechanics. Newtonian mechanics, then, is a “framework theory” in Einstein’s sense. It is a framework of investigation that is “prior” to the theories of special systems that we might pursue and evaluate, but it is evidently not a priori in the usual sense of the term.

1.3

The Laws of Motion as the Outcome of a Conceptual Analysis

I will close with a refection on the nature of Newton’s “activity” in constructing his theory of mechanics. I wish to make the following point: while the laws of motion motivated philosophical ideas about motion, force, and causality, they come from Newton’s analysis of what is presupposed in the dynamical reasoning of his predecessors and contemporaries. More specifcally, the laws are formulations of the principles that Newton thinks are explicitly, as in the case of the frst law, or implicitly, as in the case of the second and third laws, presupposed in the reasoning of these fgures – when they are reasoning properly, that is, solving problems successfully. In this way, the laws of motion are the outcome of a philosophical or critical conceptual analysis of the conceptual framework of the mechanical philosophers. I will give a brief account of a few of the main parts of that framework. This terrain has been covered by others and more carefully. I will introduce only as much as is needed to make my point. The Early Modern current known as “the mechanical philosophy” represents the Universe as a mechanism, one subject to mechanical laws governing all matter and implying determinism. The central tenet of the mechanical philosophy is a principle of action or causality: all physical action is mechanical action, that is, action through pressure or impact. This is the basis for the reductionist view that all natural phenomena can be explained by mechanical processes. Some, though not all, of the mechanical philosophers held that all matter is composed of minute corpuscles and aggregates of them. Their confgurations determine bodies’ primary and secondary qualities. This, in overview, is the mechanical philosophy. The mechanical philosophy was intended as a corrective to the hierarchical and teleological aspects of Aristotelian and scholastic-Aristotelian science that persisted. Its proponents took it to be a development of Galileo’s program of mechanical explanation. For this reason, one might begin with Galileo – after all, it was his arguments for the heliocentric hypothesis, and his proposals along the 16

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way, that undermined much of Aristotelian physics. From Galileo, we might move from fgures such as Mersenne and Gassendi, through Hobbes and Boyle, to Descartes, Huyghens, and Leibniz. But I will focus on Descartes, whose Principles of Philosophy (1644 [1983]) was the standard work in 17th-century natural philosophy and whose physical principles Newton sought to refute. Let us consider a few aspects of the mechanical philosophers’ views on inertial motion, force, and the conservation of momentum. I will organize my discussion around their embryonic versions of these concepts – the concepts that Newton would subject to a critical analysis. The frst set of ideas is bound up with inertial motion. Two ideas are commonly held: that all motion is relative; a certain conception of inertial motion. Consider frst the relativity of motion, an idea with which Descartes, Huyghens, and Leibniz are generally associated, though they understood “relativity of motion” differently. Descartes’ criterion of “true motion” is in many respects singular and worth distinguishing from the others: a unique standard of motion that is also relative – relative, that is, to immediately contiguous bodies, which, however, provide a univocal reference. This is unlike the Leibnizian view, or “standard” relativism, according to which any two descriptions that agree on the relative distances, and so on changes of instantaneous “situation,” are equivalent. This might aptly be called a “general principle of relativity.” Now, on the surface, Huyghens shared with Leibniz the view that all motion is relative. But where Leibniz defended a “general relativity,” Huyghens recognized that determining accelerations and rotations implicitly depends on a privileged state of uniform rectilinear motion relative to which they can be referred. This was also recognized by Newton – see, e.g., his criticism of Cartesian motion in De grav (c1660 [2004]). Both Newton and Huyghens saw clearly that such a state of motion is necessary for a satisfactory expression of the principles of mechanics. It is the recognition of this privileged state of motion – a state of “true motion” over and above the merely relative motions – that led Huyghens to formulate the frst law of motion, which Newton embraced. For Newton, then, the frst law expresses what was explicit in Huyghens’ work and implicitly presupposed in other 17th-century accounts. (See Stein (1977) for references to the original sources and for translations of previously unpublished fragments of Huyghens.) Consider also the idea of inertial motion. This is commonly held to have come from Descartes, though it was again Huygens who frst stated it properly. For all that, it is worth considering the view of Descartes, who helped lay the foundations of mechanics by taking motion and rest to be primitive states of bodies that do not require further explanation. We fnd this in his frst two laws of nature: The frst law of nature: that each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move . . . (Principles, II, §37) The second law of nature: that all movement is, of itself, along straight lines; and consequently, bodies which are moving in a circle always tend to move away from the centre of the circle which they are describing. (Principles, II, §39) Many of the salient features of Newtonian inertia are there: a body at rest remains at rest; a body in motion perseveres in its state of motion; bodies move in straight lines. But the concept of motion that Descartes articulates in the frst and second laws differs from Newtonian inertia in that motion and rest are different states. Descartes also fails to make clear the connection between motion and force: there is, for example, no recognition that in the cases of both rest and motion the net force on a body is zero. Nor is there any requirement that a body’s state of motion be uniform with respect to time, that is, unaccelerated. The requirement of uniformity is essential: without it, there is no notion that a body moving inertially moves equal distances in equal times, and hence no notion of an ideal clock that marks time. On Descartes’ account, there is no concept of inertial motion in any Newtonian sense, and hence no basis for articulating viable concepts of force and mass. Newton’s formulation of the frst law of motion refects his understanding of the very features that the Cartesian account lacks; his formulation of the law is an explication of what a satisfactory account of force demands. 17

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The second set of ideas I will consider is concerned with the concept of force or action. Here, too, Cartesian physics was a starting point for Newton’s analysis. Descartes sought to free mechanics from the hierarchical and teleological aspects of scholastic science; he set out to rid physics of qualitative properties and to reduce everything to “certain dispositions of size, fgure, and motion.” For all that, Descartes discusses force in a way that is reminiscent of the scholastic-Aristotelian framework of impetus and resistance: he appeals to the power or tendency needed to maintain bodies in their state of rest or to keep them in rectilinear motion. There is a question whether a body’s tendency to move or to remain at rest is an essential, God-given characteristic or whether it derives from extension and from the interactions of bodies among themselves. Newton’s critical analysis of Cartesian force follows naturally in several respects from his articulation of the concept of inertial motion. Newtonian inertia is a necessary presupposition for saying what an objective cause or force is: it is that which defects a body from its uniform rectilinear trajectory. Through this account, we also gain the concept of inertial mass, which is understood as a body’s resistance to changes to its state of motion. All three concepts – inertial motion, force, and inertial mass – are articulated at once. Only with Newton’s laws, therefore, do we fnd an explication of force that is a full realization of the mechanical philosophers’ ideal: a physical quantity known only through its effects. As is well known, however, Newton’s account of force encompasses the action of felds of force on distant matter – something the mechanical philosophy cannot comprehend. The third set of ideas is bound up with the conservation of a certain “quantity of motion” in collisions and interactions. Here, again, Cartesian physics is in the background. This is the subject of Descartes’ third law: The third law: that a body, upon coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body. (Principles, II, §40) Descartes’ account of the quantity of motion that is transferred from one body to another in a collision is clarifed in §43. He takes that quantity to be the product of the size and speed of the body, where “size” appears to be understood as volume or bulk; there is no mention of the vector quantity velocity, only the scalar quantity “speed.” Descartes’ account falls short of Newton’s in several respects – see, e.g., the seven rules of impact (§46–52) where this is manifest – but the principle at issue is an embryonic version of the principle of conservation of momentum. Descartes’ account is signifcant for being one of the frst attempts at formulating the principle. But, for Descartes, the conservation of momentum has as much a theological as an empirical foundation: when God created the Universe, He gave to all bodies a certain quantity of motion, a quantity that He preserves at every successive moment, even when it is transferred (§62). For a strictly empirical account of the conservation of momentum, we need to look elsewhere. The germ is already there in the work of Galileo and his contemporaries. But the frst systematic accounts are found in the work of Wallis (1668), Wren (1668), and Huyghens (1669) on the laws of impact: Wallis dealt with inelastic collisions; Wren and Huyghens dealt with elastic ones. Their work is mentioned by Newton in the Scholium to the Laws and the third law of motion is based on it. Given two bodies A and B, the third law of motion defnes their interaction as FA on B = −FB on A , and, in an isolated system, the corresponding expression for the conservation of momentum is dpdtA = − dpdtB . In this way, Newton not only incorporates the contributions of his contemporaries but explicates them in his theory of mechanics. What should be clear from the foregoing is that the laws of motion, far from being radical, are implicitly and explicitly presupposed in the work of the mechanical philosophers. This was Newton’s reason for taking them to be axioms. Newton’s activity, then, is an eminently philosophical one: it is a critical conceptual analysis of confused concepts; it is also an analysis of successful practice that 18

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aims to discover the principles on which that practice depends. This culminates in the explication of the basic concepts of mechanics and the articulation of criteria for their application. These criteria are the basis for an empirically adequate theory of mechanics. We fnd in Newton’s construction of his theory an exemplar of an approach to conceptual analysis that has been at the heart of the foundations of physics, at least since Galileo, and at the heart of the analytic tradition, at least since Frege. Conceptual analysis, so understood, is the practice of identifying central features of a concept by revealing the assumptions on which use of the concept depends. (This way of expressing the basic idea of conceptual analysis is due to Demopoulos (2000, p. 220).) This practice proceeds by examining the use, misuse, and limitations of pre-existing concepts – in the case of interest to us, inertia, force, and mass – and revealing the assumptions on which their pre-analytic use depends. Therefore, while Newton’s theory of mechanics motivates philosophical ideas about matter and motion, space-time and causality, and philosophical debates about them, its philosophical signifcance goes deeper: the theory is a reminder of what conceptual analysis in the foundations of physics might aspire to. The laws of motion are the result of that analysis.

References Curiel, E. (2014). Classical mechanics is Lagrangian; it is not Hamiltonian. British Journal for the Philosophy of Science, 65: 269–321. Demopoulos, W. (2000). On the origin and status of our conception of number. Notre Dame Journal of Formal Logic, 41: 210–226. Descartes, R. (1644 [1983]). The Principles of Philosophy, translated by V.R. Miller and R.P. Miller. Dordrecht: Reidel. DiSalle, R. (2006). Understanding Space-time. Cambridge: Cambridge University Press. Earman, J. (1989). World Enough and Spacetime: Absolute and Relational Theories of Motion. Cambridge, MA: MIT Press. Einstein, A. (1919 [2002]). What is the theory of relativity? In M. Janssen et al. (eds.), The Collected Papers of Albert Einstein, Vol. 7. Princeton, NJ: Princeton University Press, pp. 206–215. Friedman, M. (2010). Synthetic history reconsidered. In M. Domski and M. Dickson (eds.), Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. Chicago: Open Court, pp. 571–813. Friedman, M. (2001). Dynamics of Reason: The 1999 Kant Lectures of Stanford University. Stanford: CSLI Publications. Hume, D. (1777 [1975]). Enquiries Concerning Human Understanding and Concerning the Principles of Morals, reprinted from the 1777 edition with Introduction and Analytical Index by L.A. Selby-Bigge. Oxford: Clarendon Press. Huyghens, C. (1669). A summary account of the laws of motion. Philosophical Transactions of the Royal Society, 4: 925–928. Kant, I. (1787 [1998]). Critique of Pure Reason, translated and edited by P. Guyer and A. Wood. Cambridge: Cambridge University Press. Kant, I. (1786 [2011]). Metaphysical Foundations of Natural Science, edited by M. Friedman. Cambridge: Cambridge University Press. Lange, L. (1885). Ueber das Beharrungsgesetz. Berichte der Koniglichen Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-physische Classe, 37: 333–351. Malament, D. (2012). Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. Chicago: University of Chicago Press. Neumann, C. (1870). Ueber die Principen der Galilei-Newton’schen Theorie. Leipzig: B. G. Teubner. Newton, I. (c1660 [2004]). De gravitatione et æquipondio fuidorum. In A. Janiak (ed.), Philosophical Writings. Cambridge: Cambridge University Press, pp. 12–40. Newton, I. (1726 [1999]). The Principia: Mathematical Principles of Natural Philosophy, translated by I.B. Cohen and A. Whitman. Berkeley: University of California Press. North, J. (2019). Formulations of classical mechanics. In E. Knox and A. Wilson (eds.), The Routledge Companion to the Philosophy of Physics. London: Routledge. ´ H. (1902 [1952]). Science and Hypothesis, translated by W. Greenstreet. New York: Dover. Poincare, Quine, W.V. (1951). Two dogmas of empiricism. The Philosophical Review, 60: 20–43. Reichenbach, H. (1928 [1958]). The Philosophy of Space and Time, translated by M. Reichenbach and J. Freund. New York: Dover.

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Russell, B. (1897). An Essay on the Foundations of Geometry. Cambridge, Cambridge University Press. Samaroo, R. (2015). Friedman’s thesis. Studies in History and Philosophy of Modern Physics, 52: 129–138. Samaroo, R. (2018). There is no conspiracy of inertia. British Journal for the Philosophy of Science, 69: 957–982. Stein, H. (1967). Newtonian space-time. Texas Quarterly, 10: 174–200. Stein, H. (1977). Some philosophical prehistory of general relativity. In J. Earman, C. Glymour, and J. Stachel (eds.), Foundations of Space-Time Theories. Minnesota Studies in the Philosophy of Science, Vol. VIII. Minnesota: University of Minnesota Press, pp. 3–49. Thomson, J. (1884). On the law of inertia; the principle of chronometry; and the principle of absolute clinural rest, and of absolute motion. Proceedings of the Royal Society of Edinburgh, 12: 568–578. Truesdell, C. (1977). A First Course in Rational Continuum Mechanics. New York: Academic Press. Wallis, J. (1668). A summary account given by Dr. John Wallis, of the general laws of motion, by way of Letter written to him to the Publisher, and communicated to the R. Society, Novemb. 26. 1668. Philosophical Transactions of the Royal Society, 3: 864–866. Weatherall, J. (2016). Maxwell-Huygens, Newton-Cartan, and Saunders-Knox Spacetimes. Philosophy of Science, 83: 82–92. Wren, C. (1668). Theory concerning the same subject; imparted to the R. Society Decemb. 17. Last, though entertain’d by the Author divers years ago, and verifed by many Experiments, made by Himself and that other excellent Mathematician M. Rook before the said Society, as is attested by many Worthy Members of that Illustrious Body. Philosophical Transactions of the Royal Society, 3: 866–867.

Further Reading from the Editors For an accessible introduction to these topics with extracts from classic readings, see N. Huggett, Space from Zeno to Einstein (MIT Press, 1999). Chapters 7 and 8 discuss Newton, but earlier chapters also give historical context. For a collection with a range of papers on Newton’s philosophy, see Cohen, I.B. and George E. Smith (ed.), The Cambridge Companion to Newton (Cambridge: Cambridge University Press, 2002). Chapters 1, 2, 4, and 8 are particularly relevant. Two papers on Newton with an historical focus are Rynasiewicz, “By their Properties, Causes and Effects: Newton’s Scholium on Time, Space, Place and Motion – I. The Text” (Studies in History and Philosophy of Science 26 1995) and his “By Their Properties, Causes and Effects: Newton’s Scholium on Time, Space, Place and Motion—II. The Context” in the same volume.

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2 FORMULATIONS OF CLASSICAL MECHANICS Jill North

2.1 Introduction Classical mechanics is the physical theory with which we are most familiar, the one we frst encounter in school. Philosophers tend to regard classical mechanics as metaphysically unproblematic. At frst glance, it does appear straightforward: the theory is fundamentally about particles, with intrinsic features like mass,1 that move around in three-dimensional space in response to various forces, which arise via interactions between the particles. It seems as though if any physical theory is metaphysically perspicuous, classical mechanics is. But the theory is not as clear-cut as it initially seems. Our familiarity misleads us. The reason is not just that classical mechanics ultimately runs into the kind of trouble that presaged quantum mechanics. Even taking it to be the true fundamental theory of a world,2 classical mechanics does not offer as candid a picture of things as we tend to think. One reason for this is that there are different formulations, which are generally claimed to be equivalent by physics books, but which are at least not obviously equivalent – neither in terms of the mathematical structure they use, nor in terms of the physical world they describe. What I want to do in this chapter is to outline the three leading formulations of classical mechanics, and to raise some questions about them, the chief one being: are these genuinely equivalent formulations, as usually thought? If so, in what sense are they equivalent? If not, in what way(s) do they differ? Another way to put the focal question of this chapter is by means of a title of Mark Wilson’s (2013): “What Is ‘Classical Mechanics’, Anyway?” Indeed, since the terms “classical mechanics” and “Newtonian mechanics” “are used virtually synonymously” (Spivak, 2010, p. 7), one aim of this chapter is to suggest that it is not right to do so. There are different versions of classical mechanics, which might even amount to distinct theories. A related aim is to show that there are interesting philosophical questions that arise in the context of classical mechanics. Classical mechanics merits the attention of philosophers, who often disregard it as either too perspicuous or too outdated to warrant much discussion.3 Although this chapter is limited to classical mechanics, a host of general questions in the philosophy of physics and science are touched upon, such as the following. What is the right notion of theoretical equivalence: when are two scientifc theories mere notational variants? How do we interpret a scientifc theory: how do we fgure out the nature of the world according to a theory? When faced with different theories or formulations, how do we choose which one to adopt? Indeed, must we choose?

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2.2

Three Formulations

I will outline the three main formulations of classical mechanics – Newtonian, Lagrangian, and Hamiltonian mechanics – in relatively standard ways, before turning to some questions about them.4 My focus will be on the dynamical laws and the quantities that appear in them. This is where much of the action lies in comparing and contrasting the different formulations.

2.2.1

Newtonian Mechanics

Newtonian mechanics might be the only formulation one comes across, the others typically not introduced until more advanced college courses. In the Newtonian mechanics of point particles – point-sized physical objects with intrinsic features like mass5 – two sets of coordinates specify a system’s fundamental state at a time: the positions and velocities (or momenta) of all the particles. Assuming the particles are free to move around in three-dimensional physical space, these coordinates will each have three components, one along each spatial dimension. For a system consisting of n particles, the total state is specified by means of 6n coordinates: three coordinates for the position and three coordinates for the velocity of each particle in the system. It turns out to be extremely useful to represent all the possible states of a system in a mathematical space called the statespace, each point of which represents a different possible fundamental state of the system. Since we need 6n coordinates to specify the state of a system, the statespace will have 6n dimensions. Different curves through the statespace represent different possible histories of the system, different sequences of fundamental states over time. (The curves are parameterized by time.6 ) These histories will be given by a theory’s dynamical laws, in this case, Newton’s second law:7 6Fi = mi ai = mi x¨i .

(2.1)

Here, 6Fi indicates the sum of the forces – which are vector quantities, written in bold – on a given particle labeled by i (i ranges from 1 to n, for n particles in the system); mi is the particle’s mass; ai , or x¨i , is the particle’s acceleration, the second derivative of its position with respect to time, which is also a vector quantity. (A dot over a quantity indicates a derivative with respect to time of that quantity.) In other words, 6j6=i Fij = mi ai , where 6j6=i Fij is the sum of the forces on the given particle due to all the other particles (both in the system and external to it). The above is a vector equation. There is one such equation for each particle in each component direction – three equations per particle in three-dimensional space. These equations can be grouped together into one master equation, which says how the point representing the state of the entire system moves through the statespace over time. Given the initial state of a system and the total forces acting on it, integrating (twice) yields a unique solution or history: the laws are deterministic.8 A solution picks out a trajectory in the statespace, which represents the paths of all the particles through ordinary physical space.9 Equation 2.1 is the fundamental dynamical equation of the theory. Newton’s second law, mathematically represented by this equation, predicts the motion of every particle, in any situation. What forces there are will depend on the types of particles involved, and to calculate the forces, we will need additional rules, like the law of gravitation. But this one dynamical law predicts any system’s behavior, once given those forces. Two other laws of Newtonian mechanics as standardly presented are important to the theory as a whole, but will play a less central role here. Newton’s first law says that an object continues with

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uniform velocity unless acted on by a net external force. This law helps define what it is for an object to not accelerate, or to travel inertially (with the second law saying what happens when an object is subject to a net force that yields an acceleration). Newton’s third law specifies the nature of forces. It is often stated in “action-reaction” form: for every action there is an equal and opposite reaction; when one object exerts a force on a second object, the second simultaneously exerts a force equal in magnitude and opposite in direction on the first. This law tells us that forces come in pairs, as the result of interactions between two objects. It “describes the forces to some extent” (Feynman et al., 2010, sec. 9.1), with the particular force laws further indicating that forces do not depend on anything other than the types of particles involved and their spatial separations, and that they are central forces, directed along the line between the particles. (Conservative forces, derivable from a potential.)10

2.2.2

Lagrangian Mechanics

In Lagrangian mechanics, two sets of what are called generalized coordinates characterize systems’ fundamental states at a time: the generalized positions, qi , and their first time derivatives, the generalized velocities, q˙i (i from 1 to n, for n particles in the system). As in Newtonian mechanics, we need 6n coordinates to completely specify the state of a system of n particles: three generalized position coordinates and three generalized velocity coordinates per particle. But unlike in Newtonian mechanics, these do not have to be ordinary position and velocity coordinates. (They are called generalized positions and velocities by analogy to ordinary positions and velocities.) Generalized coordinates can be any set of independent parameters that completely specify a system’s state.11 Generalized positions can have units of energy, or length squared, or an angle, or can even be dimensionless. We can use any kind of coordinates that are suited to a system, the choice typically guided by the number of degrees of freedom of the system12 and the topology of the spatial region in which the particles are free to move around. For a pendulum, for example, we might use the angle θ the suspending string makes with respect to the vertical as the generalized position, with θ˙ being the generalized velocity (as we will see in Section 2.3). The Lagrangian statespace is a 6n-dimensional space with the structure of a tangent bundle. This space comprises a 3n-dimensional space in which we represent the generalized positions (called the configuration space), plus the 3n-dimensional tangent space at each point (to represent the generalized velocities, which are tangent to the generalized positions). Each point in the statespace picks out a generalized position and generalized velocity for each particle in the system. Standard labels are Q for the configuration space (the “base space” of the tangent bundle), Tq Q for the tangent spaces (the “fibers,” one for each q in Q), and TQ for the entire statespace, sometimes referred to as the velocity phase space. Notice the configuration space is what represents the physical space the particles move around in. Given the freedom in generalized coordinates, this representation needn’t occur in an obvious way, yet the structure of physical space will still be coded up in the structure of Q. The dynamical laws, called the Euler-Lagrange, or simply Lagrange, equations, say how the point representing a system’s state moves through the statespace over time, given a scalar function called the Lagrangian, L. At each point in the statespace, this function assigns a number, typically equal to the system’s kinetic energy, T, minus its potential energy, V .13 Although this gives the Lagrangian as defined on TQ, we can think of this function as coding up information about particles’ ordinary spatial features, those that are relevant to their energies, so that it is ultimately about goings-on in three-dimensional space. The motion of an n-particle system in three-dimensional space is then given by 3n second-order equations, one equation for each particle in each direction – one for each degree of freedom (three per particle in three-dimensional space): d dt



∂L ∂q˙i

 −

23

∂L = 0. ∂qi

(2.2)

Jill North

Given L, these equations uniquely determine the motion for an initial state characterized by the generalized position and generalized velocity of each particle in the system. A solution, found by integrating, gives a function or trajectory on Q, which represents the motions of all the particles through physical space. (Solutions are curves through TQ, which are projected onto Q.) To get a feel for the Lagrangian statespace, picture the statespace for a particle moving on a one-dimensional circle: Figure 2.1. (Keep in mind that this is “just about the only easily visualized nontrivial TQ” (Jose´ and Saletan, 1998, p. 94); with more degrees of freedom, things quickly become diffcult to picture.) This is a two-dimensional space, each point being picked out by two coordinates (q, q˙ ). The circle represents the different possible values of the generalized position coordinate, the lines represent the different possible values of the generalized velocity. Curves through this space represent different possible histories of the system, different sequences of generalized positions and velocities over time. The fgure could represent the statespace of a point-mass pendulum, for instance, with the circle representing the values of θ and the lines the values of θ˙ . Briefy note three interesting, interrelated differences between the Lagrangian and Newtonian formulations.14 First, in Lagrangian mechanics, a scalar energy function is what determines a system’s motion, whereas in Newtonian mechanics, the motion is given by the forces, which are vector quantities. Second, Lagrangian mechanics takes a more “holistic” approach to describing systems’ motions, in terms of the energy of the system as a whole. By contrast, the Newtonian formulation “is intrinsically a particle-by-particle description” (Sussman and Wisdom, 2014, p. 3), given in terms of the forces on each individual particle due to every other particle. Third, Lagrangian mechanics is a more coordinate-independent formulation of the dynamics, in that we can substitute any kind of coordinates for q and q˙ in equation 2.2. The central equation of Newtonian mechanics, however, contains an implicit preference for Cartesian coordinates, those in which it has the form of equation 2.1. We can of course use other kinds of coordinates, but the form of the equation will differ (contrast equation 2.1 with the form in polar coordinates, e.g.: Taylor (2005, eq. 1.48)). This is not the case in Lagrangian mechanics: “Lagrange’s equations, unlike Newton’s, take the same form in any coordinate system” (Taylor, 2005, p. 237). (The form of an equation is the form as a function of its variables, a standard notion in physics.15 )

Figure 2.1 Two-dimensional tangent bundle

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2.2.3

Hamiltonian Mechanics

Hamiltonian mechanics shares a special kinship with Lagrangian mechanics, more so than with Newtonian mechanics. Here, too, a scalar energy function determines the motion, and the central equations are formulated in terms of generalized coordinates. There are also some notable differences. Hamiltonian mechanics uses a different energy function and a different kind of generalized coordinate, with the result that the dynamical equations and statespace also differ. The Hamiltonian coordinates are called canonical coordinates. These are the generalized positions, qi , and the generalized momenta, pi . (Again, i ranges from 1 to n for n particles, three of each coordinate per particle in three-dimensional space.) The Hamiltonian statespace is the cotangent bundle of configuration space, T ∗ Q: the configuration space, Q, together with the cotangent space, T ∗ (dual to the tangent space), at each point in Q (to represent the generalized momenta, which are covectors, or one-forms). This is a 6n-dimensional space, each point of which picks out a generalized position and generalized momentum for each particle in the system. It is often called the momentum phase space, or simply the phase space.16 The scalar function that describes a system’s motion is called the Hamiltonian, H, which is (typically17 ) equal to the total energy of the system – the sum of the potential and kinetic energies, instead of the difference between them, as in Lagrangian mechanics. The dynamical laws are a set of 2n first-order equations, two equations for each particle in each direction; two equations for each degree of freedom: ∂H ∂H q˙i = , p˙i = − . (2.3) ∂pi ∂qi These equations, called the Hamiltonian or canonical equations, uniquely determine a system’s motion given an initial state specified by the canonical positions and momenta of all the particles in the system. The Hamiltonian and Lagrangian formulations are both more coordinate-independent than the Newtonian formulation. Each of them is given in terms of generalized coordinates, with the result that the dynamical equations retain their form regardless of which coordinates we use. The reason is that the Lagrangian and Hamiltonian functions, which determine the motion, are scalar functions. In Newtonian mechanics, by contrast, vector quantities – forces – determine the motion. Although vectors are coordinate-independent objects, their components change with the coordinate system. (Vectors can be defined by means of how their components transform under coordinate changes.) And as Feynman puts it, “The general statement of Newton’s Second Law for each particle. . . is true specifically for the components of force and momentum [or acceleration] in any given direction,” since “any vector equation involves the statement that each of the components is equal” (Feynman et al., 2010, sec. 10.3, 11.6; original italics). Scalars are even more coordinate-independent than that, being completely unaffected by coordinate changes, not even “altering component-wise.” (The form of a scalar function such as L or H may change with the coordinate system, but not the scalar value, nor the form of the equation in which L or H appear.)

2.3

Example: Plane Pendulum

Briefly work through a simple example to get a feel for the different flavor of each formulation. Consider a vertical plane pendulum, which moves through two spatial dimensions, as shown in Figure 2.2. (Assume the usual idealizations: frictionless, massless, rigid suspending string; point-mass bob; negligible air resistance; uniform gravitational field.) Use each formulation to find the equation of motion for the pendulum, the equation that describes the position of the bob as a function of time. We will see that each formulation yields the same equation of motion, but by means of different routes. 25

Jill North

θ

l

mg sin θ mg

Figure 2.2 Plane pendulum.

To use Newton’s law, equation 2.1, frst choose a rectangular coordinate system. Let y be in the radial direction, with x in the direction tangential to the path of the bob. Resolve the forces on the bob into their components in this coordinate system. There are two forces on the bob: the tension directed along the string, and the downward-directed gravitational force. The component of the gravitational force in the direction of the acceleration along the path – the tangential force – is mg sin θ , where θ is the angle the string makes with respect to the vertical, as shown in the fgure. There are two component equations of Newton’s law, one for each direction of our coordinate system: Fx = max and Fy = may . Plugging in the relevant force components yields Fx = −mg sin θ = max (the negative sign because the gravitational force points downward) and Fy = T − mg cos θ = may , with T being the tension in the string. Note that ay = 0; as a result, we effectively ignore this second equation when solving for the equation of motion. (T has no component in the direction of nonzero acceleration: it is merely a “constraint force.”) The arclength, which measures the distance traveled by the bob along the curved path, is given by s = lθ . The second derivative of this quantity, ¨s = lθ¨ , is the acceleration along the path. Plug into the x-component equation of Newton’s law, and we obtain the following equation of motion for the pendulum: −g sin θ = lθ¨ . (2.4) We obtain the same equation of motion, in a different way, using Lagrangian mechanics. We could use rectangular coordinates as we did above; but things are simpler if we instead use generalized coordinate θ, with θ˙ being the generalized velocity. We can plug these coordinates directly into equation 2.2 to fnd the solution. We can effectively treat θ and θ˙ as ordinary position and velocity coordinates, respectively, and, perhaps surprisingly, this yields the right answer. First calculate the Lagrangian, L = T − V . The kinetic energy T = 12 mv2 = 12 m(lθ˙ )2 . (The arclength is s = lθ , the velocity its frst time derivative.) The potential energy V = −mgl cos θ , setting the zero at the height of the pivot point where θ = π2 . (Gravitational potential energy = mgy, with y being the vertical distance from a chosen zero.) Thus, L = 12 m(lθ˙ )2 + mgl cos θ . Calculate the following derivatives (in effect treating θ and θ˙ as independent variables, even though one is really ∂L ∂L ∂L 2˙ defned as the time derivative of the other): ∂L ∂q = ∂θ = −mgl sin θ and ∂q˙ = ∂θ˙ = ml θ , so that   d ∂L = ml2 θ¨ . Finally, plug into equation 2.2: ml2 θ¨ − (−mgl sin θ ) = 0, i.e., lθ¨ + g sin θ = 0, dt ∂θ˙ which, rearranged, is equation 2.4. In Hamiltonian mechanics, we frst fnd the Hamiltonian, H = T + V . Given L above, we can see that H = 21 m(lθ˙ )2 − mgl cos θ , but we need to rewrite this in terms of canonical coordinates. To fnd the generalized momentum, pθ , which is “conjugate” to the position variable θ , use this 18 Using the equation: p = ∂L ∂q˙ , often taken to be the defnition of the generalized momentum. pθ ˙ equation pθ = ∂L , we fnd that pθ = ml2 θ˙ , so that θ˙ = ml 2 , which we can use to eliminate θ ∂θ˙  2 2 p pθ from the expression for H. Thus, H = 12 m l ml − mgl cos θ = 2mlθ 2 − mgl cos θ . Now we can 2 26

Formulations of Classical Mechanics

find the equation of motion for the pendulum using the Hamiltonian equation p˙ = − ∂H ∂q ; that is, 2 θ˙ to obtain p˙ = ml2 θ, ¨ p˙θ = − ∂H = −mgl sin θ. Differentiate p = ml and plug into the equation θ θ ∂θ 2 ¨ ¨ for p˙θ to obtain ml θ = −mgl sin θ; i.e., lθ = −g sin θ, which again yields equation 2.4.

2.4

Equivalent Formulations?

We find the same equation of motion for the pendulum regardless of which formulation we use. This turns out to be true in general. It is often simpler to use Lagrangian or Hamiltonian mechanics rather than Newtonian mechanics, since we do not have to calculate the various component forces on each particle. Nonetheless, it is generally agreed that each formulation suffices for describing the motion of any classical mechanical system.19 The difference seems to be merely a matter of calculational convenience. Indeed, physics books typically state, and go on to prove, an equivalence among the three formulations, by showing that their dynamical equations are all inter-derivable.20 A typical route is to begin with Newton’s laws, derive the Lagrangian and Hamiltonian equations from them, and then show that the derivation can go the other way. Thus, Jose´ and Saletan, at the beginning of their chapter on Lagrangian mechanics, following the one on Newtonian mechanics, write, “In this chapter we show how the equations of motion can be rewritten....We should emphasize that the physical content of Lagrange’s equations is the same as that of Newton’s” (1998, p. 48). They then show that Hamilton’s equations, in turn, can be derived from Lagrange’s, and vice versa, concluding that these all “contain the same information” (1998, p. 207). Another book concludes, “From the point of view of the physicist this division [into the three formulations] is rather artificial.. . . The segregation is based entirely on the mathematical methods used” (Talman, 2000, p. 163). It certainly seems like these are simply “alternative statements of the laws” (Marion and Thornton, 1995, p. 213), with “nothing new. . . added to the physics involved” (Goldstein et al., 2004, p. 334) as we pass from one formulation to another. That is the standard view: the three formulations are completely equivalent, mere notational variants; they say all the same things, just in different ways. I want to urge caution in adopting the standard view. The alleged equivalence is not as straightforward as the above statements would have us believe. The reason is that there are some differences among the formulations, and it is not obvious that they are as superficial as usually thought. Draw a rough distinction between two kinds of differences: mathematical and metaphysical. I won’t go into these in detail, but will point to places where there is a case to be made that the differences go deeper than ordinarily claimed.

2.4.1

Mathematical Differences

It is important to keep in mind that two things can be similar or equivalent in some ways while differing in other ways. Two objects can share a shape yet have different colors or patterns. Two spaces can share a distance structure yet differ in whether they have a privileged location. In mathematics more generally, two mathematical objects are considered equivalent when there is the relevant structurepreserving mapping between them, in which case they are said to be equivalent with respect to that structure. Two such objects can still differ with respect to other kinds of structure. All of which is to say that, even if the three formulations of classical mechanics are equivalent in all the ways that physics books suggest, the formulations could still be inequivalent in other ways. The question is whether they are equivalent, full stop. The answer depends on whether what differences there are matter in any way. There is one patent mathematical difference among them: the formulations use different symbols, in equations that do not “look” the same. The standard view is that this difference does not matter. Consider the change from Cartesian to polar coordinates to describe a Euclidean plane, or from one 27

Jill North

set of Cartesian coordinates to another that is rotated or translated with respect to the frst. Some things will be different when we switch to the other coordinate system – the points will get different numerical labels, for example – but we know that nothing has really changed. The plane remains the same; we have simply used a different, equally legitimate way of describing it. The standard view is that the differences among the three formulations of classical mechanics are just like the differences among the coordinate-based descriptions of the plane: just a change in the coordinates or variables being used to describe the very same physics. However, there are some reasons to question this idea. Take Newtonian mechanics, on the one hand, and Lagrangian and Hamiltonian mechanics, on the other. The latter are comparatively coordinate-independent formulations of classical mechanics. This suggests that they more directly get at the nature of classical mechanical reality, apart from our descriptions of it – just as the metric tensor on the Euclidean plane, rather than any coordinate-dependent distance formula, more directly captures the intrinsic structure of thepplane. (The familiar form of the distance formula stemming from the Pythagorean theorem, d = 1x2 + 1y2 , for instance, assumes Cartesian coordinates and won’t work in other types of coordinates, even though the distance between any two points is the same regardless of the coordinate system.) This, in turn, suggests that we have reason to prefer these formulations. Physics prizes coordinate-independence, and with good reason.21 Since there is freedom in which coordinate system to use, any choice we do make will be arbitrary – a conventional choice made from among equally good descriptions. (Recall the different coordinate systems for the plane.) We can be misled into thinking that coordinate-dependent features, which rest on an arbitrary choice in description, refect genuine features of reality.22 A formulation that is independent of coordinates is then preferable, other things being equal, when it comes to fguring out what physics says about the world. So even if the equations of the three formulations are inter-derivable in some sense, there is also a sense in which the formulations are not mathematically on a par, a sense in which they are not completely equivalent. Some of them may more directly represent physical reality than others.23 We can go further. For the way in which the formulations differ in their reliance on coordinates suggests particular physical differences among them. Newtonian mechanics contains an implicit preference for Cartesian coordinates, the kind of coordinates in which its core equation takes the standard form. A preference for Cartesian coordinates, in turn, is indicative of a Euclidean metric structure. This suggests that the spatial structure of a Newtonian world is Euclidean. (Newton himself, of course, assumed such a structure.) Lagrangian mechanics, which allows for a wider range of coordinates in describing classical systems, does not constrain the spatial structure in the same way. This suggests that the physical space of a Lagrangian world has a “looser” metric structure. (I explore this difference, which will be refected in the theories’ statespace structures, in North (2021, ch.4).) Hamiltonian mechanics allows for even greater freedom of coordinates than that. (In particular, it allows for coordinate changes that mix up the ps and qs, whereas in Lagrangian mechanics, since q˙ is defned as the time derivative of q, there is no allowable transformation in which these coordinates “get intermingled” (Taylor, 2005, p. 538n10).24 ) As a result, the Hamiltonian formulation does not require a metric structure, but only a lesser type of structure akin to a volume measure. (I explore this difference in North (2009).) I’d go so far as to suggest that there is a hierarchy, in order of increasing mathematical structure, from Hamiltonian to Lagrangian to Newtonian mechanics – a mathematical inequivalence among the three. (In the above-mentioned writings, I argue that less such structure is in general a reason to prefer a theory.) If we take a theory’s mathematical structure seriously in telling us about the nature of the physical world, then this mathematical difference should refect a similar hierarchy in the physical structure of the world(s) each theory describes – a physical inequivalence among them. In other words, these may not be wholly equivalent formulations, neither mathematically nor physically, contrary to the standard view.25 28

Formulations of Classical Mechanics

2.4.2

Metaphysical Differences

Since the dynamical equations and basic quantities of the three formulations are inter-derivable in ways that physics books claim, you might want to conclude that the different formulations are simply “mutually supporting, compatible perspectives on the phenomena of mechanical motions” (Wilson, 2007, p. 179). That, once again, is the standard view.26 But there are other differences among the formulations, what I call here “metaphysical” ones, that could lead to a different conclusion. (Don’t let the term mislead you: these differences arguably matter to physics.) Although no theory wears its metaphysics on its sleeves, on a natural way of interpreting the formulations, they differ from one another in potentially significant ways. All assume a fundamental ontology of point-mass particles with relative positions. Beyond that, each one offers a fairly different picture of the world, given the different quantities that appear in their respective dynamical equations. (What follows are some initial suggestions; the metaphysics of the three formulations has not been much explored in the literature.) First compare Newtonian mechanics, on the one hand, with Lagrangian and Hamiltonian mechanics, on the other. Newtonian mechanics “describes the world in terms of forces and accelerations (as related by the second law)” (Taylor, 2005, p. 521), where “force is something primitive and irreducible” (Lanczos, 1970, p. 27). Lagrangian and Hamiltonian mechanics describe systems in terms of energy, with force being “a secondary quantity” derivable from the energy (Lanczos, 1970, p. 27). According to Newtonian mechanics, the world is fundamentally made up of particles that move around in response to the various forces between them. According to Lagrangian and Hamiltonian mechanics, particles move around and interact as a result of their energies. Although energy and force functions are inter-derivable in ways that physics books will show (albeit under certain contestable assumptions: note 10), these are nonetheless prima facie different pictures of the world, built up out of different fundamental quantities, with correspondingly different explanations of the phenomena. ¨ the Schrodinger and Heisenberg formulations of nonrelativistic quantum mechanics are generally considered inter-derivable, yet you might not want to regard them as wholly metaphysically equivalent even so; many philosophers take only the former to directly or perspicuously represent what is going on physically, for instance. (You might think that Lagrangian and Hamiltonian mechanics can be seen as fundamentally forced-based, given in terms of “generalized forces.” However, generalized forces are so-called by analogy to ordinary forces. It isn’t clear that they count as regular forces of the Newtonian kind.) There are potential metaphysical differences between the two energy-based approaches as well. In Lagrangian mechanics, generalized velocities are defined as the first time derivatives of the generalized positions. This suggests that positions are the only truly fundamental dynamical features of the particles, the velocities being defined in terms of them. In Hamiltonian mechanics, however, the canonical positions and momenta are both independent variables, neither being defined in terms of the other: both seem to be fundamental. (This, in turn, may amount to an “impetus” view in the medieval tradition, with further metaphysical repercussions: Arntzenius (2000, sec. 4). This assumes that the second equation of Hamiltonian mechanics is not a definition of the generalized momentum, as often claimed, but a further fundamental dynamical law.) Another difference is that the Hamiltonian is typically equal to the total energy of a system, whereas the Lagrangian is the difference between the kinetic and potential energy. Perhaps this, too, amounts to a genuine difference.27 In fact, there is a range of potential views on what’s fundamental to each of the formulations, and it is not clear which is correct. It is an open question whether, on any of them, ordinary three-dimensional space is fundamental, or whether what we usually think of as the merely abstract, high-dimensional statespace (or the configuration space) is. Relatedly, it is open whether particle features like positions and momenta are fundamentally defined on the low- or high-dimensional space. (Compare the debate in quantum mechanics over the fundamentality of the high-dimensional space

29

Jill North

of the wavefunction versus ordinary three-dimensional space.) Within energy-based approaches, it is open whether the energy function, L or H, is fundamental, or whether instead the potential and kinetic energies are; or indeed whether any energy quantity is fundamental, rather than the particle positions and velocities in terms of which the energy is standardly defned; or whether all of these might be fundamental. Analogous questions arise for Newtonian mechanics: are total forces or component forces fundamental?28 For that matter, can Newtonian mechanics be seen as a fundamentally energy-based theory, given the inter-derivability of the different quantities?29 Finally, are any of these genuinely distinct possibilities, or are they all equivalent – just different, equally legitimate ways of describing the same physical reality, analogous to the different coordinate-based descriptions of the plane? Although physics books generally assume the latter, certain metaphysical views will say that only one description gets at the real or fundamental properties (Lewis, 1983; Sider, 2011). In all, it seems very much an open question whether the three formulations of classical mechanics are genuinely equivalent, mere notational variants of a single theory, as usually thought. There is a case to be made that the differences are signifcant enough to render them more like distinct theories, with different accounts of what the physical world is like. All of this warrants further investigation.30

Notes 1 Also charge, although there is a question whether electromagnetic features ought to be considered part of the domain of classical mechanics; see for instance note 10. 2 Of course, because of the previously mentioned troubles, it is not clear that classical mechanics can be a true fundamental theory of a world, but set that aside here. 3 A recent book-length exception: Sklar (2013). 4 There are other varieties I don’t discuss, such as formulations in terms of Poisson brackets, Hamilton-Jacobi theory, or four-dimensional spacetime geometry. 5 This is the fundamental ontology assumed here. Wilson (2013) discusses the classical mechanics of rigid bodies and continua and complications involved in trying to encompass all of these within a single theory. See Hall (2007, sec. 5.2), Esfeld et al. (2018), Allori (forthcoming) on the non-standard idea that particles don’t have fundamental intrinsic properties. 6 Alternatively, time can be included as an additional dimension of the statespace. 7 Another familiar version of the law, ordinarily seen as equivalent to the above, is given in terms of momentum: 6F = p˙ . See Hicks and Schaffer (2017) on whether these are equivalent. 8 Whether the theory really is deterministic is an interesting question. Apparent counterexamples are in Earman (1986) and Norton (2008); further discussion is in Malament (2008) and Wilson (2009). 9 Standard statespace constructions effectively assume the existence of physical space. See Belot (1999, 2000) on reconstructions that aim to do away with this assumption. 10 There are questions surrounding the further restrictions that forces be central and conservative. It is usually thought that nonconservative forces, like frictional ones depending on velocity, arise from fundamental conservative ones. As Feynman notably put it, “there are no nonconservative forces!” (2010, sec. 14.4). Newton himself did not restrict forces in this way; Feynman suggests it is an additional empirical posit. [The updated online version of this book no longer contains this sentence.] The restrictions are assumed in standard proofs of energy conservation and other theorems. (This is one place the question of electromagnetic features (note 1) comes into play. Consider the magnetic force on a moving charge, which does not satisfy these restrictions.) Concerns over the above have led the odd physicist to doubt the equivalence of the different formulations of classical mechanics: Lanczos (1970, 77 n1); Gallavotti (1983, ch. 3). See also Hertz (1899) and Wilson (2009, 2013, forthcoming) on these and other reasons to doubt their equivalence. 11 There are some mild constraints on generalized coordinates (Jose´ and Saletan, 1998, sec. 2.1.2). Wilson (2009) points out that the idea of generalized coordinates, as well as the requirements on them, is not as straightforward as usually assumed. 12 The number of degrees of freedom is the number of independent parameters “necessary and suffcient for a unique characterization” of the system (Lanczos, 1970, p. 10). 13 Standard examples in which it does not have this form come from outside the point-particle mechanics assumed here. See Jose´ and Saletan, (1998, sec. 2.2.4) and Goldstein et al. (2004, sec. 7.9) for examples from electromagnetism and special relativity.

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14 See Lanczos (1970) for discussion of these and other differences. See Butterfeld (2004) for an extended discussion of Lagrangian mechanics in particular. 15 See Brading and Castellani (2007, p. 1343). 16 A Hamiltonian statespace can in fact have a more general structure than this: North (2009). 17 See Goldstein et al. (2004), Taylor (2005, sec. 7.8) for conditions under which this holds. 18 The above is an instance of a Legendre transformation, which can be used to change back and forth between Hamiltonian and Lagrangian coordinates, energy functions, and statespaces: see Lanczos (1970, ch. 6), Arnold (1989, sec. 3.14), Jose´ and Saletan (1998, ch. 5). 19 Or so I assume here, setting aside reasons for hesitation on this point (note 10). 20 Examples: Arnold (1989); Marion and Thornton (1995); Hand and Finch (1998); Jose´ and Saletan (1998); Talman (2000); Goldstein et al. (2004); Taylor (2019); Baez and Wise (2019); see also Feynman (1965, ch. 2). 21 Lanczos notes of the Lagrangian equations that they “stand out as the frst example of that ‘principle of invariance’ [a kind of coordinate-independence] which was one of the leading ideas of 19th century mathematics, and which has become of dominant importance in contemporary physics” (1970, p. 117). 22 Einstein once said that the main reason it took him so long to develop general relativity is that “it is not so easy to free oneself from the idea that co-ordinates must have an immediate metrical meaning” (Schilpp, 1970, p. 67). 23 All that said, the role of coordinates in physics is more subtle and complicated than the above discussion might suggest: see North 2021. 24 There is a mathematical transformation between them (note 18), but even it “leads one to suspect that there actually is a nontrivial difference between L and q˙ on the one hand and H and p on the other” (Jose´ and Saletan, 1998, p. 217). 25 Opposition to this conclusion, for different reasons, can be found in Swanson and Halvorson (2012); Curiel (2014); Barrett (2015). Barrett (2019) points out how our judgments about the relationship between the theories will depend on what we take to be their core structures, with different views on their structures leading to different such judgments. 26 Following Coffey (2014), the standard view may more accurately be put as that Newtonian mechanics accurately represents classical mechanical reality, with Lagrangian and Hamiltonian mechanics being mere reformulations of it. 27 Baez and Wise (2019, ch. 1) tries to distinguish these physically. 28 Cartwright (1983, ch. 3) argues against the reality of component forces. 29 Wilson (2007) defends the existence of Newtonian forces against various objections. 30 Some further investigation is in North 2021, especially Chapters 4 and 7.

References Allori, V. (forthcoming). Fundamental objects without fundamental properties: A thin-object-oriented metaphysics grounded on structure. In D. Aerts, J. Arenhart, C. De Ronde, and G. Sergioli (eds.), Probing the Meaning and Structure of Quantum Mechanics. World Scientifc. Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics (2nd edition). New York: Springer. Translated by K. Vogtmann and A. Weinstein. Arntzenius, F. (2000). Are there really instantaneous velocities? The Monist, 83: 187–208. Baez, J.C. and Derek K. Wise (2019). Lectures on classical mechanics. Available at: http://math.ucr.edu/home/ baez/classical/texfles/2005/book/classical.pdf. Barrett, T.W. (2015). On the structure of classical mechanics. British Journal for the Philosophy of Science, 66: 801–828. Barrett, T.W. (2019). Equivalent and inequivalent formulations of classical mechanics. British Journal for the Philosophy of Science, 70: 1167–1199. Belot, G. (1999). Rehabilitating relationalism. International Studies in the Philosophy of Science, 13: 35–52. Belot, G. (2000). Geometry and motion. British Journal for the Philosophy of Science, 51: 561–595. Brading, K. and Castellani, E. (2007). Symmetries and invariances in classical physics. In J. Butterfeld and J. Earman (eds.), Handbook of the Philosophy of Science: Philosophy of Physics, Part B. Amsterdam: Elsevier, pp. 1331–1367. Butterfeld, J. (2004). Between laws and models: Some philosophical morals of Lagrangian mechanics. Unpublished manuscript. Available at: http://philsci-archive.pitt.edu/1937/. Cartwright, N. (1983). How the Laws of Physics Lie. Oxford: Oxford University Press. Coffey, K. (2014). Theoretical equivalence as interpretative equivalence. British Journal for the Philosophy of Science, 65: 821–844.

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Curiel, E. (2014). Classical mechanics is Lagrangian; it is not Hamiltonian. British Journal for the Philosophy of Science, 65: 269–321. Earman, J. (1986). A Primer on Determinism. Dordrecht: D. Reidel. Esfeld, M. and Deckert, D.-A. with Lazarovici, D., Oldofredi, A. and Vassallo, A. (2018). A Minimalist Ontology of the Natural World. New York and London: Routledge. Feynman, R. (1965). The Character of Physical Law. Cambridge, MA: MIT Press. Feynman, R.P., Leighton, R.B. and Sands, M. (2010). The Feynman Lectures on Physics: New Millennium Edition, volume 1. New York: Basic Books. Gallavotti, G. (1983). The Elements of Mechanics. New York: Springer-Verlag. Goldstein, H., Poole, C. and Safko, J. (2004). Classical Mechanics (3rd edition). Reading, MA: Pearson Education. Hall, N. (2007). Humean reductionism about laws of nature. Unpublished manuscript. Available at: http://philpapers.org/archive/HALHRA. Hand, L.N. and Finch, J.D. (1998). Analytical Mechanics. Cambridge: Cambridge University Press. Hertz, H. (1899). The Principles of Mechanics Presented in a New Form. London: Macmillan and Co. Hicks, M.T. and Schaffer, J. (2017). Derivative properties in fundamental laws. British Journal for the Philosophy of Science, 68: 411–450. ´ J.V. and Saletan, E.J. (1998). Classical Dynamics: A Contemporary Approach. Cambridge: Cambridge Jose, University Press. Lanczos, C. (1970). The Variational Principles of Mechanics (4th edition). New York: Dover. Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61: 343–377. Malament, D.B. (2008). Norton’s slippery slope. Philosophy of Science (Proceedings), 75: 799–816. Marion, J.B. and Thornton, S.T. (1995). Classical Dynamics of Particles and Systems (4th edition). Florida: Harcourt, Brace and Company. North, J. (2009). The ‘structure’ of physics: A case study. Journal of Philosophy, 106: 57–88. North, J. (2021). Physics, Structure, and Reality. Oxford: Oxford University Press. Norton, J.D. (2008). The dome: An unexpectedly simple failure of determinism. Philosophy of Science (Proceedings), 75: 786–798. Schilpp, P.A., editor. (1970). Albert Einstein: Philosopher-Scientist (3rd edition). La Salle, IL: Open Court. Sider, T. (2011). Writing the Book of the World. Oxford: Oxford University Press. Sklar, L. (2013). Philosophy and the Foundations of Dynamics. Cambridge: Cambridge University Press. Spivak, M. (2010). Physics for Mathematicians: Mechanics I. Publish or Perish, United States. Sussman, G.J. and Wisdom, J. (2014). Structure and Interpretation of Classical Mechanics (2nd edition). Cambridge, MA: MIT Press. Swanson, N and Halvorson, H. (2012). On North’s ‘the structure of physics’. Unpublished manuscript. Available at: http://philsci-archive.pitt.edu/9314/. Talman, R. (2000). Geometric Mechanics. New York: John Wiley & Sons. Taylor, J.R. (2005). Classical Mechanics. Sausalito, CA: University Science Books. Wilson, J. (2007). Newtonian forces. British Journal for the Philosophy of Science, 58: 173–205. Wilson, M. (2009). Determinism and the mystery of the missing physics. British Journal for the Philosophy of Science, 60: 173–193. Wilson, M. (2013). What is ‘classical mechanics’ anyway? In R. Batterman (ed.), The Oxford Handbook of Philosophy of Physics. Oxford: Oxford University Press, pp. 43–106. Wilson, M. (forthcoming). Newton in the pool hall: Subtleties of the third law. In C. Smeenk and E. Schliesser (eds.), The Oxford Handbook of Newton. Oxford: Oxford University Press.

Further Reading from the Editors Lawrence Sklar’s Philosophy and the Foundations of Dynamics (Cambridge: Cambridge University Press 2013) is an excellent book-length treatment of classical mechanics which covers some of the issues discussed here. For a subtle approach to classical physics, rooted in a deep understanding of particular examples, look at Mark Wilson’s work, for example “What is ‘Classical Mechanics’ Anyway?” in Robert Batterman, ed., The Oxford Handbook of Philosophy of Physics (Oxford: Oxford University 2013). For two sides of the debate over whether Lagrangian or Hamiltonian mechanics is more fundamental, see Jill North’s “The ‘Structure’ of Physics: A Case Study.” (Journal of Philosophy 106, 57–88, 2009) and Erik Curiel’s “Classical Mechanics Is Lagrangian; It Is Not Hamiltonian.” (British Journal for the Philosophy of Science 65, 269–321, 2014).

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3 CLASSICAL SPACETIME STRUCTURE James Owen Weatherall

3.1 Introduction One often associates spacetime – a four-dimensional geometrical structure representing both space and time – with relativity theory, developed by Einstein and others in the early part of the 20th century.1 But soon after relativity theory appeared, several authors, such as Hermann Weyl (1952 ´ Cartan (1923, 1924), and Kurt Friedrichs (1927), began to study how the spatiotemporal [1918]), Elie structure presupposed by classical, i.e., Newtonian, physics could be re-cast using the methods of four-dimensional geometry developed in the context of relativity. These reformulations of classical physics were initially of interest for the insight they offered into the structure of relativity theory.2 But this changed in 1967, when Howard Stein proposed that the notion of a “classical” spacetime could provide important insight into Newtonian physics itself – including on issues that engaged Newton and his contemporaries regarding the status of “absolute” space and motion. On Stein’s reconstruction, Newton’s oft-decried “absolutism” about space amounted to the claim that the laws governing the motion of bodies presupposed that there are (unobservable) facts about which bodies are at rest. This idea can be reconstructed as a claim that space and time together have a certain geometrical structure, now known as Newtonian spacetime. Leibniz’s response, meanwhile, that there was no discernible difference between whether all of the bodies in the world were at rest or in uniform rectilinear motion can then be taken as an argument that Newton’s laws assume more structure than can be supported on metaphysical grounds; Leibniz, it seems, believed space and time had the structure of Leibnizian spacetime, which, in a certain precise sense, is less structure than Newtonian spacetime.3 Perhaps the most striking and infuential aspect of Stein’s paper was his argument that in fact the spatiotemporal structure presupposed by Newton’s laws of motion was somewhat less than Newton imagined – though somewhat more than Leibniz would have accepted. This intermediate structure has come to be known as Galilean spacetime.4 Remarkably, Galilean spacetime provides the resources needed both to avoid Leibniz’s famous shift argument,5 which exposes the unobservability of absolute uniform rectilinear motion, and to accept Newton’s famous bucket argument, which Newton and many others have taken to show that at least some absolute motions are empirically detectable. The remainder of this chapter will proceed as follows.6 I will begin by making some remarks about “space” in classical physics. I will then introduce the notion of Galilean spacetime and discuss several senses in which it is the spacetime structure presupposed by Newtonian physics. Next I will discuss Newtonian spacetime and Leibnizian spacetime. I will conclude by briefy discussing some

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other ideas concerning classical spacetime structure, including the recent proposal by Saunders (2013) that one should take a structure strictly intermediate between Galilean and Leibnizian spacetimes to be what is really presupposed by Newtonian physics.

3.2 Space Before discussing classical spacetime, we frst consider the geometry of space – something that all of the structures I will discuss below agree on. Space, in what follows, will be understood as a collection of (infnitesimally) small places, i.e., locations where small bodies may be situated. We will call these locations the points of space.7 (Since most physical objects are not vanishingly small, their locations cannot be represented by single points of space; in general, bodies occupy regions of space, i.e., collections of contiguous points.) This collection of points is understood to be structured, in the sense that there are various further relations defned on them. It is these relations that one aims to characterize when one speaks of “spatial structure” – or, mutatis mutandis, “spatiotemporal structure.” For example, space is three-dimensional. We make this idea precise by introducing the notion of an arrow between pairs of points of space.8 (See Figure 3.1a.) First, pick any two points of space, p and q. We suppose one can always draw a (unique) arrow whose tail begins at p and whose head lands at q, and likewise, one can always draw an arrow whose tail begins at q and whose head ends at p. Conversely, we suppose that any arrow whose tail begins at p ends somewhere in space, i.e., at some point. This means that we can think of any point and an arrow originating at that point as uniquely picking out another point of space; and we can think of any ordered pair of points as uniquely determining an arrow. We will call arrows that originate at a point p arrows at p. We suppose that given an arrow at p, one can identify the “same” arrow – i.e., an arrow with the same length and orientation – at any other point. Thus, it makes sense to speak of an arrow without mentioning the point at which it is based. Now pick any point of space, p. We defne two basic ways of manipulating the arrows at p. For one, we assume we can always “scale” any arrow. (See Figure 3.1b.) That is, given any arrow u (say), one can uniquely identify an arrow that points in the same direction as u, but which is twice as long (say). We will call this arrow 2u. Likewise, one can identify an arrow that points in precisely the opposite direction as u; we will call this −u, short for (−1)u. Given any arrow u at p, 0u is the (unique) arrow from p to itself. The second operation one can perform on arrows at a point is to add them to one another. Consider two arrows u and v at p. We defne the arrow u + v to be the unique arrow from p to the r 2u u

q

u

v

q u+v

-u

q

p

p

u p (a)

(b)

(c)

Figure 3.1 (a) An arrow u relating two points p and q of space. (b) “Scaling” an arrow u at a point p by various amounts, including fipping its direction. (c) Adding arrows u and v: here u + v is the arrow relating p and r, where r is the point determined by q and v, and q is the arrow determined by p and u.

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point r, where r is defned as follows: frst, consider the point q determined by p and u; then take v, and let r be the point determined by q and v. (See Figure 3.1c.) In other words, we understand u + v to be the arrow taking us from p to the point we would reach by frst following u and then following v. We assume that for all u and v, u + v = v + u – i.e., following v and then u leads to the same place as following u and then v.9 I can now say what it means for space to be three-dimensional. Space is three-dimensional just in case at any point p, there exist three arrows, x, y, and z, which are such that (1) none of these three can be constructed by any process of scaling or adding the other two; and (2) one can construct any arrow v at p just by scaling and/or adding together x, y, and z. The important point, here, is that for ordinary space, any collection of arrows with these two properties always has exactly the same number of elements: namely, three. There is a bit more structure that we will take space to have. First, given any two points p and q, we assume we can say how far apart they are: that is, we have a notion of spatial distance.10 In other words, we assign a length – a non-negative number, which is zero only for the arrow taking a point to itself – to the arrow v between p and q, which we will write ||v||. Likewise, given two arrows u and v, we can assign an angle – a number between 0 and 2π – to them, where an angle of 0 (or 2π) means the arrows point in the same direction; an angle of π means they point in opposite directions; and an angle of π/2 or 3π/2 means they are orthogonal, i.e., the angle between u and v is precisely the same as the angle between u and −v (or, equivalently, between −u and v). Finally, we assume that this notion of length satisfes the following two conditions: (a) for any real number a and any arrow u, ||au|| = |a| · ||u||, where |a| is the absolute value of a; and (b) if u and v are orthogonal, then ||u + v||2 = ||u||2 + ||v||2 .

3.3 Galilean Spacetime We now turn to our frst spacetime structure: Galilean spacetime. To characterize Galilean spacetime, we begin much as in our discussion of space: Galilean spacetime consists in a collection of points, now understood as locations not only in space, but also in time. Rather than “places,” these locations are events, in the sense that they represent occurrences in a small region of space for an instant of time. They are not necessarily signifcant events: an event might be a speck of dust existing at a moment, or even the occurrence of nothing at all. Just as ordinary extended objects are not located at single points of space, neither are objects that are extended in space – say, a rope – nor objects that persist through time – say, a particle or a person – represented by single points in spacetime. Instead, these are represented by various sorts of curves and surfaces, as described below. This collection of events is once again structured. In particular, Galilean spacetime is a fourdimensional “space” of events, in much the same way that space is three-dimensional. That is, we suppose that any pair of spacetime points p and q are uniquely related by an arrow; and that given any point p, and any arrow v, there is a unique point q such that v is the arrow from p to q. Now, though, we suppose that at any point p of Galilean spacetime, one can fnd four arrows – t, x, y, and z – with the properties that (1) none of these four can be constructed by any process of scaling or adding the other three; and (2) one can construct any other arrow by scaling and/or adding together t, x, y, and z.11 Now consider any two points p and q in Galilean spacetime. There are several additional relations that hold between them. One such relation is temporal distance, t, which assigns a real number to any ordered pair of events – or, equivalently, a temporal length to any arrow relating events. This number represents the duration between those two events. Temporal distance has the following properties. First, if u and v are arrows, then t(u + v) = t(u) + t(v), i.e., if p and q are related by u, and q and r are related by v, then the temporal distance between p and r is the sum of the temporal distance between 35

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p and q and the temporal distance between q and r. Likewise, given any real number a and any vector u, the temporal distance satisfes t(au) = at(u), where on the left-hand side of this equation we are applying the scaling operation, and on the right-hand side we are just multiplying real numbers.12 Finally, we can always fnd a (non-unique) collection of three arrows, x, y, and z, with the properties that (1) none of these three can be constructed by any process of scaling or adding the other two; and (2) the temporal distance assigned to each of these arrows, and thus all other arrows that can be constructed by scaling and adding them, is zero. If the temporal distance from p and q is positive, we say that q is in the future of p; if it is negative, we say it is in the past of p; if it is zero, then p and q are simultaneous. Now let p be a point, and let x, y, and z be three arrows satisfying (1) and (2) in the previous paragraph. Then any point q related to p by an arrow that can be constructed by scaling or adding x, y, and z will be simultaneous with p, and all events simultaneous with p can be found in this way. Thus, the events simultaneous with p form a three-dimensional “space.” We take such collections to represent space at a time. Finally, given space at any time, Galilean spacetime includes a notion of spatial distance between those events, and a notion of angle between arrows relating those events, that satisfy all of the conditions we placed on spatial distance above. Note, however, that we do not have any notion of spatial distance between points that are not simultaneous – a caveat that will turn out to be important in what follows. Summing up, we can think of Galilean spacetime as an infnite collection of copies of three-dimensional space, stacked on top of one another to form a four-dimensional structure.13 (See Figure 3.2a.) Each copy of space has all of the structure we described in the previous section, including the relations of spatial distance and angle. Moreover between any two slices, we have a notion of duration. It is in this sense that Galilean spacetime may be taken to represent space and time – i.e., to deserve the name “spacetime.” Having described this structure, we can now say the sense in which it is the “right” structure for Newtonian physics. The key is to understand what Galilean spacetime allows one to say about motion. Consider a particle – i.e., some vanishingly small body that we understand to “persist” in the sense of existing over time. We represent this particle by a collection of events – specifcally, by a curve through spacetime with the property that it intersects each spatial slice no more than once.14 Such

r t

p

q

(a)

(b)

Figure 3.2 Galilean spacetime may be thought of as consisting of copies of three-dimensional space stacked on top of one another to form a four-dimensional structure. (a) Here points p and q are simultaneous, and so the arrow between them has temporal length zero, but non-zero spatial length; the point r is not simultaneous with p or q and so it has a non-zero temporal distance from both, and its spatial distance to p and q is not defned. (b) Here we depict the straight line through p determined by a given arrow of unit temporal length.

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a curve is called a “trajectory” or a “world-line.” The idea is that the (single) point of intersection between the particle’s world-line and any spatial slice represents the location of the particle at that time. The whole world-line, then, represents the history of the particle over time: it consists in the collection of places, at successive times, that are occupied by the particle. In other words, it represents the motion of the particle through space. Extended bodies are represented, similarly, by “world-tubes” that are bounded in each slice of space; the intersection of the world-tube with each slice represents the confguration of the body in space at that instant. In what follows, I will focus on the particle case for simplicity, though much of what is said carries over because one can generally associate a “center of mass” curve with extended bodies, which characterizes their mass-averaged motion. We now turn to Newton’s laws of motion. We frst remark that, in Galilean spacetime, we have the resources to characterize a special kind of trajectory: namely, a “straight line through spacetime.” (See Figure 3.2b.) Given any point p and any arrow v with temporal length one, we defne the straight-line trajectory through p with 4-velocity v to consist of all the points one can reach from p by scaling v. (We say 4-velocity because the arrow v relates points in four-dimensional spacetime.) These trajectories describe motions through spacetime wherein a body moves in some fxed direction at a constant velocity. Such trajectories play an important role in Newton’s theory, encapsulated in his frst law of motion. Newton’s First Law: In the absence of any external force, massive bodies will follow straight-line trajectories. This law establishes a tight connection between two classes of curves: the mathematically privileged curves picked out by the structure of Galilean spacetime, and the physically privileged curves picked out by the “default,” force-free motions of bodies. These default trajectories are known as inertial trajectories.15 Before proceeding, let us briefy comment on the role of 4-velocity here. We usually understand velocity as a quantity with some direction (and magnitude) in space; 4-velocity, meanwhile, has direction in spacetime. These are different. To recover velocity as we usually conceive of it – i.e., 3-velocity – we need to introduce an observer, O, which is an idealized measuring apparatus, situated somewhere in space and with its own inertial state of motion (represented by some trajectory with 4-velocity, which we also denote O). 3-velocity at a time as determined by this observer is given by u − O,

O

O

u

u

O’

V’ p

p

V

V

(b)

(a)

Figure 3.3 The motion of a body is described by its 4-velocity, which is an arrow u with temporal length one. (a) To recover ordinary “3-velocity” one introduces an observer O with its own 4-velocity; the (relative) 3-velocity of the body at p is given by V = u − O, which is an arrow with temporal length zero. (b) Two different observers, O and O0 , with different states of motion, will determine different 3-velocities, V and V 0 . It is in this sense that (3-)velocity is relative in Galilean spacetime.

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where u is the 4-velocity of the body at that time. (See Figure 3.3a.) This will always be a vector with temporal length zero (since O and u both have temporal length one), and so 3-velocity relative to any observer has spatial length, representing the “speed” of motion. The important point about 3-velocity is that it is an essentially relative notion: that is, its value at a time depends on the state of motion of the observer. Different observers, with different states of motion, would attribute different 3-velocities to a body. (See Figure 3.3b.) Indeed, in Galilean spacetime, the notion of an “absolute,” i.e., non-relative, 3-velocity does not make sense. But this does not mean that all measures of motion are relative. In particular, 4-velocity may be characterized independently of any observer. Thus far, we have focused on straight-line trajectories. But not all trajectories are straight: in general, the motion of a body will change over time. We capture this by associating an instantaneous 4-velocity with a body at each instant. Given two (necessarily non-simultaneous) points p and q on a body’s trajectory, we can defne the change in 4-velocity from p to q to be the difference v(p) − v(q), where v(p) is the 4-velocity at p and v(q) is the 4-velocity at q; by considering the change in 4velocity between pairs of points that are ever closer together, and scaling appropriately, we can defne the instantaneous rate of change of the particle’s 4-velocity, or its acceleration, at each point of its world-line. The acceleration will itself be an arrow at p with temporal length zero. Acceleration is important in Newton’s second law of motion. Newton’s Second Law: If a body of mass m is subject to an external force F at a point, then the acceleration a of the body at that point will satisfy F = ma. Thus, we see that an impressed force, represented by an arrow F, causes a body to deviate from inertial motion, i.e., to accelerate in the direction of the force. The mass of the body is what determines the magnitude of the acceleration. It is important to emphasize that acceleration as just defned in Galilean spacetime is an absolute quantity – it is defned without reference to any observer – which is crucial to Newton’s second law because force is meant to be an objective, absolute quantity (i.e., not observer dependent), and it would not make sense to have a law that equates a relative quantity with an absolute quantity. For completeness, we include Newton’s third law, although it is not essential for the discussion (and we suppress commentary for space reasons). Newton’s Third Law: If body 1 exerts a force F on body 2, then body 2 exerts a force −F on body 1. Finally, although it is not strictly part of Newtonian mechanics, it is worth noting that Newton’s laws provide a framework for detailed studies of particular forces. Newton, for instance, focused on gravitation. Newton’s Law of Universal Gravitation: A body of mass m1 located at point p will exert on a body of mass m2 located at (simultaneous) point q a force given by F=

Gm1 m2 r ||r||3

where r is the arrow from q to p, ||r|| is the spatial length of that arrow, and G is Newton’s constant. It is worth emphasizing that this law invokes temporal length (to defne simultaneity) and spatial distance (to defne distance between the bodies). To further relate this force to the acceleration of a body, of course, one also requires the full four-dimensional arrow structure of spacetime, as discussed above in connection with Newton’s second law. Thus, we see that all of the structure of Galilean spacetime enters into Newton’s laws of motion and his law of universal gravitation. 38

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3.4 Newtonian and Leibnizian Spacetimes Although the language may be anachronistic, Newton, in the Principia and elsewhere,16 took for granted the structure of Galilean spacetime. But as Stein (1967) convincingly argues, Newton also believed that spacetime had further structure: namely, what Newton called absolute space, which is the structure needed to say whether two places, at different times, are “the same.” In other words, Newton believed there was a basic matter of fact about whether any given object remains in one place over time – i.e., at rest – or whether it moves. As we have just seen, this is precisely what Galilean spacetime does not provide. There, one can say whether a body is accelerating, i.e., if its velocity is changing over time. But one cannot say that the body is at rest or moving (at a constant velocity). To represent this further structure, we need a way of identifying points in space at different times: that is, a way of saying that “here” now is the same as “here” fve minutes ago. We do so by choosing some arrow ξ , of unit temporal length, as special. This arrow represents Newton’s absolute space as follows: two events p and q are at the same place (at different times) if and only if the arrow from p to q is some multiple (possibly zero) of ξ . Likewise, a particle is at rest (relative to absolute space) if its world-line is the straight line determined by ξ . More generally, a particle with constant velocity (in the sense of Galilean spacetime) has absolute velocity ||ξ − u||, where u is the arrow of temporal length one determining that particle’s world-line. If we take Galilean spacetime, and we add to it this further structure of a privileged arrow ξ , we arrive at Newtonian spacetime.17 There are several reasons why someone – even someone who accepted all of Newton’s physics – might worry that the world does not exhibit the full structure of Newtonian spacetime. One particularly infuential reason for skepticism, most famously associated with Leibniz’s arguments in his correspondence with Clarke (Leibniz and Clarke, 2000, pp. 18f.), is that the structure of absolute space, as characterized by ξ , has no observable consequences. In particular, one could take any system of bodies and imagine setting the entire system in motion, at some constant speed (relative to ξ ) in some fxed direction, and show that, by the lights of Newton’s own theory, the relative motions of the bodies would be unchanged. Newton fully understood this, of course: Corollary 5 to the Laws of Motion (Newton, 1999 [1687 / 1713 / 1726], p. 423) demonstrates precisely this fact. But Leibniz took it to have great signifcance. He concluded, on its basis, that space and time could not have the structure of absolute space. But what structure would Leibniz attribute to space and time? It is not entirely clear that a satisfactory or complete answer is to be extracted from Leibniz’s writings. But as Stein (1977) argues, and Earman (1977, 1979, 1986, 1989a,b) develops, one might begin with Leibniz’s claim, in various places, that all motion is relative – i.e., there are facts only about how the relative confgurations of bodies change over time.18 In other words, it would not make sense to say anything more about the motion of a given body than to describe the rates at which its distance from other bodies is changing over time. Both Newtonian spacetime and Galilean spacetime provide the resources to describe these changes. One can consider, for instance, at each time a collection of point-like particles in space, whose relative confguration is represented by the places at which they are located (and the arrows between those places); and one can describe, by word-lines through spacetime, the changes in relative position of these bodies over time. But one can also say more than this: for instance, in both of these structures, there is a fact about each particle’s acceleration – not as a relative matter of the rate of change of the relative velocity of one particle with respect to another, but as an absolute matter. Moreover, this is precisely what someone who thinks all motion is relative would deny. Thus, to characterize Leibniz’s views, we need to excise from Galilean spacetime the structure that allowed us to distinguish a special class of motions – namely, the arrows relating non-simultaneous points. We are left with Leibnizian spacetime, which is a collection of events with a notion of temporal 39

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distance between pairs of events, such that the collection of all points simultaneous with any given point has the structure of three-dimensional space as described in Section 3.2, but where we have no arrows between non-simultaneous events.19 We saw above that Galilean spacetime – and thus, Newtonian spacetime – provided the resources needed for Newtonian physics. Can we say the same of Leibnizian spacetime? In short, no – for reasons Newton himself pointed out.20 Indeed, Newton offered at least one case in which absolute motion would be empirically testable. Imagine a bucket partially flled with water. Newton argued that if the bucket is rotating (absolutely) then the surface of the water will appear curved; if the bucket is at rest, then the surface of the water will appear fat. Newton argued that this would be the case, according to his theory, even if there were nothing in the universe but the bucket and the water, and so the behavior of the water in the bucket could not have anything to do with merely relative motion. This thought experiment has been very infuential, and Newton himself believed it settled the issue in favor of absolute space – i.e., Newtonian spacetime. But it is not quite what it seems. The key to understanding the bucket experiment is to realize that any rotating object is accelerating. In particular, each little bit of the bucket is constantly changing velocity, because it is changing the direction in which it is moving. Thus, Newton’s thought experiment is not a way of measuring absolute motion in the sense of measuring absolute velocity. Instead, it is an experiment to determine (one kind of) absolute acceleration. In other words, the bucket experiment is an argument that we need at least the structure of Galilean spacetime to make sense of Newtonian physics; meanwhile, Leibniz’s shift argument against Newtonian spacetime, that there are no empirical tests of absolute velocity in Newtonian physics, still stands – as it must, in light of Corollary 5. Taken together, then, we are pushed to Galilean spacetime as a spacetime structure intermediate between Newtonian and Leibnizian spacetimes.

3.5

Maxwell-Huygens and Newton-Cartan Spacetimes

At the end of the previous section, I observed that in order to accommodate Newton’s bucket thought experiment, one needs more structure than Leibnizian spacetime provides. We went from this remark to the conclusion that one needs the structure of Galilean spacetime (at least) to support Newtonian physics. But one might worry that this conclusion is too fast: Galilean spacetime makes all acceleration absolute, whereas the bucket thought experiment concerns only a very special kind of accelerated motion: namely, rotation. Perhaps rotation must be absolute to accommodate Newtonian physics, but does it follow that all acceleration is absolute? This worry is buttressed by the fact that, immediately after Corollary 5 to the Laws of Motion, Newton proves another result: Corollary 6 to the Laws of Motion establishes that if a system of bodies is undergoing uniform linear acceleration – i.e., all of the bodies have, in addition to their other motions, some fxed acceleration in a given spatial direction – then their motions relative to one another would be indistinguishable from the case in which that acceleration was absent (Newton, 1999 [1687 / 1713 / 1726], p. 423ff). Corollary 6 makes precise a certain sense in which absolute linear acceleration has the same status as absolute velocity. Very recently, a number of authors have taken up the question of whether Corollary 6 motivated adopting some alternative spacetime structure, intermediate between Leibnizian spacetime and Galilean spacetime, as the structure presupposed by Newtonian physics.21 For instance, Simon Saunders (2013) argues that a spacetime structure that he calls Newton-Huygens spacetime is the proper setting for Newtonian physics.22 Newton-Huygens spacetime is like Leibnizian spacetime – i.e., there are notions of spatial and temporal distance, but no arrows between non-simultaneous events – except one has, in addition, a standard for whether a system of bodies is rotating over time.23 But one does not, in general, have a notion of absolute acceleration. 40

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There are reasons to be cautious about accepting this move, however. For one, it is more radical than it may at frst appear. In Newtonian physics, forces are not relative: either a body experiences a force or it does not.24 But then, by dint of Newton’s second law, acceleration must also not be relative, since it is incoherent to say that an absolute quantity is proportional to a relative quantity.25 It follows that to accept Newton-Huygens spacetime, one needs to revise both the conceptual and mathematical foundations of Newtonian physics.26 How signifcant a revision this amounts to, and whether it can succeed, is a topic of ongoing debate. A second reason to be cautious is that another response to Corollary 6 is available. In particular, Eleanor Knox (2011) has argued that Corollary 6 supports a move to a different theory of gravitation, sometimes known as geometrized Newtonian gravitation or Newton-Cartan theory, developed by Cartan (1923, 1924) and Friedrichs (1927).27 Geometrized Newtonian gravitation is a theory with the same empirical consequences as Newtonian gravitation, but set in a spacetime structure importantly different from any of those discussed thus far: it is a theory in which spacetime is curved by the distribution of matter in the universe, and where the motion of bodies in spacetime is infuenced by that curvature.28 The details of geometrized Newtonian gravitation are beyond the scope of this chapter, but one point is worth emphasizing. In geometrized Newtonian gravitation, while there are (in general) no arrows between non-simultaneous events, there is nonetheless a standard of absolute acceleration.

Acknowledgments This chapter is partially based upon work supported by the National Science Foundation under Grant No. 1331126. I am grateful to David Malament for helpful comments on an earlier draft.

Notes 1 The idea of “spacetime” was actually introduced by Minkowski (2013 [1911]), several years after Einstein frst introduced relativity theory. 2 And indeed, they remain of interest for this reason: see, for instance, Friedman (1983), Malament (1986a,b, 2012), Earman (1989b), Fletcher (2014), Barrett (2015), Weatherall (2011a,b, 2014, 2017c,d), and Weatherall (2018) for philosophical discussions of the relationship between relativity theory and Newtonian physics that make heavy use of this formalism. 3 For more on the notion of “structure” being used here, see Barrett (2015) and Weatherall (2017b). Newtonian spacetime was introduced in Stein’s original 1967 paper; Leibnizian spacetime was introduced by Stein somewhat later (Stein, 1977), and then developed by Earman (1977, 1979, 1986, 1989a,b). Note, however, that there are subtle differences in Earman and Stein’s understanding of what Leibnizian spacetime captures (Weatherall, 2020), and that Earman, in particular, believed that Leibnizian spacetime includes an implicit commitment to some form of “substantivalism” (see endnote 7) that Leibniz would have denied; he suggested that one should move to an algebraic framework to better refect Leibniz’s views. For replies to this proposal, see Rynasiewicz (1992) and Rosenstock et al. (2015). 4 What we now call Galilean spacetime was frst discussed by Weyl (1952 [1918]). It is not clear whether any of Newton’s contemporaries were in a position to recognize this intermediate structure, though Stein (1967, 1977) and others have suggested that Christian Huygens came closest. See Stan (2016) for a recent discussion of Huygens’ views on rotation, which is the context in which he most closely approached the idea of Galilean spacetime. 5 What I call the shift argument concerns setting the whole world in motion at a constant velocity; it is sometimes known as the “kinematic” shift argument, to distinguish it from the “static” shift argument, wherein one considers shifting the entire universe by some fxed amount (Pooley, 2013); see also endnote 7. 6 My presentation will not be particularly historically sensitive; nor will it be technical. For more historical detail, but with the same basic perspective, see Weatherall (2016c) and (especially) references therein; for technical details, see Weatherall (2016b). For a treatment at a level comparable to the one attempted here, which also develops ideas from relativity theory, see Geroch (1981); a more philosophical perspective is offered by Maudlin (2012).

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7 There is a subtle metaphysical issue here, concerning whether we understand these “points” or “places” to themselves be physical objects, existing (metaphysically) prior to and independently of bodies (substantivalism); or if instead they characterize something about relations between bodies (relationism). Nothing I say here should be understood to be taking one or the other of these positions for granted. Perhaps more importantly, the classical spacetime structures I describe here are supposed to provide insight on a different issue, concerning the character of motion, rather than, in the frst instance anyway, the metaphysics of space or time. See Weatherall (2018, 2020) for more on this perspective; for further discussion and other perspectives, see Earman (1989b), Belot (2000), and Pooley (2013). 8 These arrows will make space a (three-dimensional) affne space. See Malament (2009) or Weatherall (2016b) for details. 9 The arrows at each point are to be understood as forming a vector space; and there is a canonical isomorphism between the vector spaces at each point. Vector spaces are required to satisfy some additional conditions that I do not mention, but these are all met by the mental model meant to be invoked by this description. See the texts already cited for details. 10 This notion of spatial distance consists in a Euclidean metric on the vector space of arrows between points. 11 In other words, Galilean spacetime consists in a four-dimensional affne space of events, with further structure to be described. 12 That is, temporal duration is a linear functional acting on the vector space associated with Galilean spacetime. 13 More precisely, Galilean spacetime is a four-dimensional affne space endowed with (1) a non-vanishing linear functional t on its associated vector space; and (2) on each affne subspace by the subspace of vectors to which t assigns 0, a Euclidean metric h. 14 We intend by “curve” a map from (some open subset of) the real numbers into Galilean spacetime that is continuous and at least twice differentiable relative to a topological and differential structure canonically determined by the affne space structure. For details, see Weatherall (2016b). 15 See Earman and Friedman (1973) for a discussion of the status of Newton’s frst law of motion; see DiSalle (2016) for a discussion of inertial frames more generally. 16 See, especially, the Scholium to the Defnition, i.e., the Scholium on Space, Time, Place, and Motion (Newton, 1999 [1687 / 1713 / 1726], pp. 408–415). 17 In other words, Newtonian spacetime is Galilean spacetime as described in endnote 13, with a privileged vector of unit temporal length. 18 It is not universally accepted that Leibniz was committed to Leibnizian spacetime: see, for instance, Roberts (2003); see also endnote 3. For more background on Leibniz’s views on physics (and philosophy of physics), see, for instance, Garber (1995), McDonough (2014), and references therein. 19 More precisely, Leibnizian spacetime is a three-dimensional affne bundle over a one-dimensional affne space, where each fber of the bundle is endowed with a Euclidean metric h and there is a one form t on the base space representing temporal distance. 20 Again, see the discussion in the Scholium to the Defnitions (Newton, 1999 [1687 / 1713 / 1726], pp. 408–415). 21 Questions about classical spacetime structure in light of Corollary 6 are hardly new: see, for instance, Stein (1977) and DiSalle (2008); see also Malament (1995) and Norton (1995) for an older discussion of the “relativity of acceleration” in Newtonian physics in a different context. But whether Corollary 6 provides an argument against Galilean spacetime has been of particular interest recently (Saunders, 2013; Knox, 2014; Weatherall, 2016a, 2017a; Wallace, 2016, 2017; Teh, 2018; Dewar, 2018). 22 Newton-Huygens spacetime was frst introduced by Earman (1989b) under the moniker Maxwellian spacetime (Earman, 1989b); Weatherall (2016a) and others have called it Maxwell-Huygens spacetime. 23 In other words, Newton-Huygens spacetime is Leibnizian spacetime endowed with a standard of rotation (Weatherall, 2017a). 24 Curiously, Leibniz, too, seems to have taken force to be absolute in this sense, which may raise issues for his views on space and time as reconstructed here. See Roberts (2003), Garber (2012), and McDonough (2014) for more on Leibniz on force. 25 As Stein (1977, pp. 19–20) puts it in a very nice discussion of precisely these issues, absolute acceleration is a vera causa in Newtonian physics. 26 This is a project that Saunders (2013) and Dewar (2018) have undertaken. 27 For background on geometrized Newtonian gravitation, see Trautman (1965) and Malament (2012, Ch. 4). For more on the relationship between geometrized Newtonian gravitation and ordinary Newtonian gravitation, see, in addition to those resources, Glymour (1980), Knox (2014), and Weatherall (2016d); for discussions of the relationship between geometrized Newtonian gravitation and gravitation in NewtonHuygens spacetime, see Weatherall (2016a), Wallace (2016), and Dewar (2018).

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28 In this, it is like general relativity. See Wald (1984) or Malament (2012) for textbook treatments of general relativity; see also the chapter of this volume on relativistic spacetime.

References Barrett, T. (2015). Spacetime structure. Studies in History and Philosophy of Modern Physics, 51: 37–43. Belot, G. (2000). Geometry and motion. British Journal for the Philosophy of Science, 51(4): 561–595. ´ ´ ´ (premier ´ es ´ a` connexion affne, et la theorie ´ ` e partie). de la relativite´ generalisee Cartan, E. (1923). Sur les variet ´ ´ ieure, 40: 325–412. Annales scientifques de l’Ecole Normale Super ´ es ´ a` connexion affne, et la theorie ´ ´ ´ ´ (premier ` e partie) Cartan, E. (1924). Sur les variet de la relativite´ generalisee ´ ´ ieure, 41: 1–25. (suite). Annales scientifques de l’Ecole Normale Super Dewar, N. (2018). Maxwell gravitation. Philosophy of Science, 85(2): 249-270. Dewar, N. and Weatherall, J.O. (2018). On gravitational Energy in Newtonian Theories. Foundations of Physics, 48: 558–578. Available at: arXiv:1707.00563 [physics.hist-ph]. DiSalle, R. (2008). Understanding Space-Time. New York: Cambridge University Press. DiSalle, R. (2016). Space and time: Inertial frames. In E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Winter 2016 edition. Available at: https://plato.stanford.edu/archives/win2016/entries/spacetime-iframes/. Earman, J. (1977). Leibnizian space-times and Leibnizian algebras. In R.E. Butts, J. Hintikka (eds.), Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science. Dordrecht: Reidel, pp. 93–112. Earman, J. (1979). Was Leibniz a relationist? Midwest Studies in Philosophy, 4(1): 263–276. Earman, J. (1986). Why space is not a substance (at least not to frst degree). Pacifc Philosophical Quarterly, 67(4): 225–244. Earman, J. (1989a). Leibniz and the absolute vs. relational dispute. In N. Rescher (ed.), Leibnizian Inquiries. A Group of Essays. Lanham, MD: University Press of America, pp. 9–22. Earman, J. (1989b). World Enough and Space-Time. Cambridge, MA: The MIT Press. Earman, J. and Friedman, M. (1973). The meaning and status of Newton’s law of inertia and the nature of gravitational forces. Philosophy of Science, 40: 329. Fletcher, S.C. (2014). On the reduction of general relativity to Newtonian gravitation. Unpublished manuscript. Friedman, M. (1983). Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science. Princeton University Press, Princeton, NJ. Friedrichs, K.O. (1927). Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und der ¨ Grenzuberganges vom Einsteinschen zum Newtonschen Gesetz. Mathematische Annalen, 98: 566–575. Garber, D. (1995). Leibniz: Physics and philosophy. In N. Jolley (ed.), The Cambridge Companion to Leibniz. Cambridge: Cambridge University Press, pp. 270–352. Garber, D. (2012). Leibniz, Newton, and force. In A. Janiak and E. Schliesser (eds.), Interpreting Newton. Cambridge: Cambridge University Press, pp. 33–47. Geroch, R. (1981). General Relativity from A to B. Chicago, IL: University of Chicago Press. Glymour, C. (1980). Theory and Evidence. Princeton, NJ: Princeton University Press. Knox, E. (2011). Newton-Cartan theory and teleparallel gravity: The force of a formulation. Studies in History and Philosophy of Modern Physics, 42(4): 264–275. Knox, E. (2014). Newtonian spacetime structure in light of the equivalence principle. The British Journal for the Philosophy of Science, 65(4): 863–888. Leibniz, G.W. and Clarke, S. (2000). Correspondence, trans. and ed. by Roger Ariew. Indianapolis, IN: Hackett Publishing Co. Malament, D. (1986a). Gravity and spatial geometry. In R.B. Marcus, G. Dorn and P. Weingartner (eds.), Logic, Methodology and Philosophy of Science. Vol. VII. New York: Elsevier Science Publishers, pp. 405–411. Malament, D. (1986b). Newtonian gravity, limits, and the geometry of space. In R. Colodny (ed.), From Quarks to Quasars. Pittsburgh: University of Pittsburgh Press, pp. 181–201. Malament, D. (1995). Is Newtonian cosmology really inconsistent? Philosophy of Science, 62(4): 489–510. Malament, D.B. (2009). Notes on geometry and spacetime. Unpublished lecture notes. Available at: http://www.socsci.uci.edu/ dmalamen/courses/geometryspacetimedocs/GST.pdf. Malament, D.B. (2012). Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. Chicago: University of Chicago Press. Maudlin, T. (2012). Philosophy of Physics: Space and Time. Princeton, NJ: Princeton University Press. McDonough, J.K. (2014). Leibniz’s philosophy of physics. In E.N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2014 edition). Available at: https://plato.stanford.edu/archives/spr2014/entries/leibniz-physics/. Minkowski, H. (2013) [1911]. Space and time. In F. Lewertoff and V. Petkov (eds.), Space and Time: Minkowski’s Papers on Relativity. Montreal: Minkowski Institute Press, pp. 111–125.

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Newton, I. (1999) [1687 / 1713 / 1726]. The Principia: Mathematical Principles of Natural Philosophy, edited and trans. by I. Bernard Cohen and Anne Whitman. Berkeley, CA: University of California Press. Norton, J.D. (1995). The force of Newtonian cosmology: Acceleration is relative. Philosophy of Science, 62(4): 511–522. Pooley, O. (2013). Substantivalist and relationalist approaches to spacetime. In R. Batterman (ed.), The Oxford Handbook of Philosophy of Physics. Oxford: Oxford University Press, pp. 522–586. Roberts, J.T. (2003). Leibniz on force and absolute motion. Philosophy of Science, 70(3): 553–573. Rosenstock, S., Barrett, T. and Weatherall, J.O. (2015). On einstein algebras and relativistic spacetimes. Studies in History and Philosophy of Modern Physics, 52B: 309–316. Rynasiewicz, R. (1992). Rings, holes and substantivalism: On the program of Leibniz algebras. Philosophy of Science, 59(4): 572–589. Saunders, S. (2013). Rethinking Newton’s Principia. Philosophy of Science, 80(1): 22–48. Stan, M. (2016). Huygens on inertial structure and relativity. Philosophy of Science, 83(2): 277–298. Stein, H. (1967). Newtonian space-time. The Texas Quarterly, 10: 174–200. Stein, H. (1977). Some philosophical prehistory of general relativity. In J. Earman, C. Glymour and J. Stachel (eds.), Foundations of Space-Time Theories. Minneapolis: University of Minnesota Press, pp. 3–49. Teh, N. (2018). Recovering recovery: On the relationship between gauge symmetry and trautman recovery. Philosophy of Science , 85(2): 201–224. Trautman, A. (1965). Foundations and current problem of general relativity. In S. Deser and K.W. Ford (eds.), Lectures on General Relativity. Englewood Cliffs, NJ: Prentice-Hall, pp. 1–248. Wald, R. (1984). General Relativity. Chicago: University of Chicago Press. Wallace, D. (2016). Fundamental and emergent geometry in Newtonian physics. Available at: http://philsciarchive.pitt.edu/12497/. Wallace, D. (2017). More problems for Newtonian cosmology. Studies in History and Philosophy of Modern Physics, 57: 35–40. Weatherall, J.O. (2011a). On (some) explanations in physics. Philosophy of Science, 78(3): 421–447. Weatherall, J.O. (2011b). On the status of the geodesic principle in Newtonian and relativistic physics. Studies in the History and Philosophy of Modern Physics, 42(4): 276–281. Weatherall, J.O. (2014). What is a singularity in geometrized Newtonian gravitation? Philosophy of Science, 81(5): 1077–1089. Weatherall, J.O. (2016d). Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent? Erkenntnis, 81: 1073–1091. ErkenntnisPublished online. doi:10.1007/s10670-015-9783-5. Weatherall, J.O. (2016a). Maxwell-Huygens, Newton-Cartan, and Saunders-Knox spacetimes. Philosophy of Science, 83(1): 82–92. Weatherall, J.O. (2016b). Space, time, and geometry from Newton to Einstein, feat. Maxwell, lecture notes from the 2016 MCMP summer school in mathematical philosophy; available on request. Weatherall, J.O. (2016c). Void: The Strange Physics of Nothing. New Haven, CT: Yale University Press. Weatherall, J.O. (2017a). A brief comment on Maxwell(/Newton)[-Huygens] spacetime. Available at: arXiv:1707.02393 [physics.hist-ph]. Weatherall, J.O. (2017b). Categories and the foundations of classical feld theories. In E. Landry (ed.), Categories for the Working Philosopher. Oxford: Oxford University Press. Available at: arXiv:1505.07084 [physics.hist-ph]. Weatherall, J.O. (2017c). Conservation, inertia, and spacetime geometry. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 67: 144–159. Weatherall, J.O. (2017d). Inertial motion, explanation, and the foundations of classical space-time theories. In D. ¨ Lehmkuhl, G. Schiemann and E. Scholz (eds.), Towards a Theory of Spacetime Theories. Boston, MA: Birkhauser, pp. 13–42. Available at: arXiv:1206.2980 [physics.hist-ph]. Weatherall, J.O. (2018). Regarding the ‘Hole Argument’. The British Journal for Philosophy of Science, 69(2): 329–350 Available at: arXiv:1412.0303 [physics.hist-ph]. Weatherall, J.O. (2020). Some philosophical prehistory of the Earman-Norton hole argument. Studies in History and Philosophy of Modern Physics, 70: 79–87. Weyl, H. (1952) [1918]. Space Time Matter. Mineola, NY: Dover Publications.

Further Reading from the Editors The classic paper which introduced the idea of Newtonian spacetime is Howard Stein’s “Newtonian space-time” (The Texas Quarterly 10, 174–200, 1967). John Earman’s World Enough and Space-Time (The MIT Press, Cambridge, MA. 1989) surveys many of the arguments about classical spacetime. More recent interest in Newton-

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Cartan spacetime can be traced back to (among others) David Malament’s “Is Newtonian cosmology really inconsistent?” (Philosophy of Science 62 (4), 489–510, 1995). Saunders’ “Rethinking Newton’s Principia” (Philosophy of Science 80 (1), 22–48, 2013) and Knox’s “Newtonian spacetime structure in light of the equivalence principle” (The British Journal for the Philosophy of Science 65 (4), 863–888, 2014) give different perspectives on how to think about Newtonian spacetime structure in the light of accelerative symmetries.

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4 RELATIONISM IN CLASSICAL DYNAMICS Julian Barbour

4.1 Relational Arenas Mach (1960) said that the universe is given “only once, with its relative motions alone determinable.” He also said, “It is utterly impossible to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive by means of the changes of things.” This chapter, based on work with collaborators, will propose a concrete relational theory that refects these aphorisms. For this we need a framework in which the universe is “given.” Mach was reacting to Newton’s concepts and beliefs, which included infnite extent of the material universe. That cannot be concretely “given,” but N self-gravitating mass points in Euclidean space can. They form a dynamically closed system with counterpart in general relativity: Pa spatially closed universe. Besides relative masses m ¯ a = ma /mtot , mtot = a ma , a = 1, . . . , N, and ratios of the distances between them, nothing but the symmetries of Euclidean space is used. The extra structures that Newton introduced with absolute space and time play no role in the universe’s dynamics but emerge for its subsystems, expressing their relation to the whole: the universe and its evolution. The relational principles developed for Euclidean space can be carried over with little change to Riemannian geometry. Euclidean geometry is based on the similarity group Sim. Suppose the (Cartesian) coordinates r1a , ra2 , a = 1, . . . , N, specify two sets of particle positions. Hold the coordinates of set 1 fxed. If, using the translations, rotations and scaling (dilatation) transformations of Sim set 2 can be brought to “overlap” with set 1, r1a = r2a for all a, then the two fgures are similar. If the condition can be achieved using translations and rotations alone, then the two fgures are congruent. In absolute space, two congruent or similar fgures not in overlap are different states. Leibniz Alexander (1956) argued that, since nothing intrinsically observable differs, absolute space is a fction. We formalize this intuition by comparison of N-body spaces. First, the standard 3N-dimensional Newtonian confguration space Q, which is N copies of Euclidean space equipped (for convenience) with Cartesian coordinates. Identifying all the confgurations q ∈ Q that translations and rotations make “Leibniz-identical,” we obtain the (3N −6)dimensional relative confguration space R Barbour (1974). Taking the process further by identifying confgurations related by dilatations, we come to the (3N − 7)-dimensional space S of possible shapes s ∈ S of the universe: shape space (introduced in Barbour (1999) and used in Barbour (2003)). A law of relational dynamics in either R or S requires at least three particles: at least one dependent variable and an independent variable. For it Newton introduced absolute time t. But Mach’s aphorisms require us to extract time from relative change. In a universe of a single particle, no relative

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change can occur. In the two-particle R, an assumed external scale allows an interparticle separation but nothing to serve as independent variable. In S, for which no external scale is defned, we do not even have a dependent variable with only two particles; they either do or don’t coincide. But the separations of three particles fx two shape-defning internal angles of a triangle, the bare minimum for shape dynamics. I do not see how, at our present level of understanding, the elimination of absolute elements could be taken beyond Sim, for which angles are defned, to the general linear group (for which they are not). In S all quantities are dimensionless:mass ratios and ratios of distances. Because all measurements yield ratios,1 one may call S the empirical space. In Newtonian dynamics, a solution is fxed by specifying initial coordinates ra and velocities r˙ a . This uses the extra Newtonian structure not present in S and absolute time. To eliminate all trace of Newton’s absolute elements, we propose: The Fundamental Postulate of Relational Dynamics. An initial point and an undirected line through it in S should uniquely determine a solution.2 Here “undirected line” eliminates the notion of velocity, which has magnitude and direction. The two directions from a point on a line have equal status and no magnitude.

4.2 Newtonian Dynamics in Shape Space Suppose the law of the universe is what we have in the Newtonian N-body problem, N ≥ 3. To what extent would it satisfy the fundamental postulate? Of course, even if it does not it must not come into confict with observation. As for all scientifc theories, empirical adequacy is also required in relational dynamics. The N-body problem is usually defned by 3N mass-weighted Cartesian coordinates √ ma xai , i = x, y, z. However, these hide heterogeneity that is at the heart of relationism: 3N − 7 degrees of freedom (dofs) defne the instantaneous shape of the system, three fx the center of mass, three fx the orientation, and one that turns out to be exceptionally important the overall scale ¯ a = ma /mtot : defned by the root-mean-square length `rms using the dimensionless masses m sX 1 X 2 ≡ 1 I `rms = m ¯ am ¯ b rab ma racm · racm , (4.1) cm = m tot m tot a a