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(Paris 7) Chasing Individuation: Mathematical Description

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(Paris 7) Chasing Individuation: Mathematical Description Université Sorbonne Paris Cité Université Paris.Diderot (Paris 7) ÉCOLE DOCTORALE : Savoirs scientifiques (ED 400) Laboratoire SPHERE, UMR 7219 Basic Research Community for Physics Projet ERC – Philosophie de la Gravitation Quantique Canonique Chasing Individuation: Mathematical Description of Physical Systems par Federico ZALAMEA Thèse de doctorat en Histoire et Philosophie des Sciences Dirigée par Gabriel CATREN Présentée et soutenue publiquement à l’Université Paris Diderot le 23 Novembre 2016 Président du jury: M. Olivier DARRIGOL, Directeur de recherche au CNRS Rapporteurs: M. James LADYMAN, Professeur à l’Université de Bristol M. Nicolaas P. LANDSMAN, Professeur à l’Université Radboud de Nijmegen Examinateurs: M. Marc LACHIÈZE-REY, Directeur de recherche au CNRS M. Thomas RYCKMAN, Professeur de l’Université de Stanford Directeur de thèse: M. Gabriel CATREN, Chargé de recherche au CNRS Université Sorbonne Paris Cité Université Paris.Diderot (Paris 7) ÉCOLE DOCTORALE : Savoirs scientifiques (ED 400) Laboratoire SPHERE, UMR 7219 Basic Research Community for Physics ERC Project – Philosophy of Canonical Quantum Gravity Chasing Individuation: Mathematical Description of Physical Systems by Federico ZALAMEA Ph.D. in Philosophy of Physics Under the supervision of Gabriel CATREN Presented and defended publicly at Paris Diderot University on November 23rd 2016 President of the jury: M. Olivier DARRIGOL, Directeur de recherche at CNRS Rapporteurs: M. James LADYMAN, Professor at the University of Bristol M. Nicolaas P. LANDSMAN, Professor at Radboud University Nijmegen Examinators: M. Marc LACHIÈZE-REY, Directeur de recherche at CNRS M. Thomas RYCKMAN, Professor at the University of Stanford Doctoral supervisor: M. Gabriel CATREN, Chargé de recherche at CNRS “All great insights and discoveries are not only usually thought by several people at the same time, they must also be re-thought in that unique effort to truly say the same thing about the same thing.” Martin Heidegger Acknowledgments The four years of research culminating with this dissertation were almost entirely funded by the European Research Council, under the European Community’s Seventh Framework Programme (Project Philosophy of Canonical Quantum Gravity, FP7/2007- 2013 Grant Agreement n° 263523). First, and foremost, I must thank my supervisor, Gabriel Catren, for providing me with the opportunity to be part of his project. During these past years, I have had the chance of evolving in the blurry borders between Physics, Mathematics and Philosophy. To be sure, the demand of navigating at ease within this triangle has oftentimes been a source of difficult challenges, and even of anguishing questions of identity. Butithas also significantly enlarged my curiosity and consolidated my thinking. I particularly appreciate the freedom I have enjoyed to explore many different fields and develop my own questions. I know this freedom to be the sign of a rare confidence in my work, for which I am the more grateful. I also would like to thank warmly James Ladyman and Klaas Landsman for ac- cepting to be the rapporteurs of my dissertation, for their careful reading and for the detailed comments which will greatly help me in improving my work; Olivier Darrigol, Marc Lachièze-Rey and Thomas Ryckman for accepting to be part of my jury, for their comments, suggestions and encouragements in the various occasions I have had the luck of meeting them. It is for me an honor to have my work evaluated by such a high-level group of thinkers whose several works I admire. Because my research would have been exceedingly more difficult without a vi- brant institution favoring exchanges and discussions, I am indebted to the laboratory i ii Acknowledgments SPHERE that hosted me during my graduate studies. In particular, I thank the former and current directors, David Rabouin and Pascal Crozet, as well as Roberto Angeloni for our discussions on the history of Quantum Mechanics, Karine Chemla for her inspir- ing and indefatigable enthusiasm, Nadine de Courtenay, Michel Paty and Jean-Jacques Szczeciniarz for their numerous advises and their generosity in sharing their valuable knowledge. The assistance of all the administrative staff has also been crucial all along the way. I thank Sandrine Pellé and Patricia Philippe for their help, Nad Fachard for her joyful presence and for her interest in all aspects of life, and Virginie Maouchi for her kindness and her unshakeable efficiency which rendered simple the otherwise labyrinthine administrative procedures. My colleagues of the ERC Project Philosophy of Canonical Quantum Gravity de- serve a special mention. In these four years, we spent countless hours together in informal discussions, meetings, seminars and workshops. My work would be signifi- cantly poorer had it not been confronted with their critical eye. I am extremely grateful to Alexandre Afgoustidis and Mathieu Anel for their patience and their generosity in guiding me through some regions of Mathematics which seemed inaccessible to me, to Christine Cachot for the many conceptual confusions she helped me dissolving and for her meticulous correction of the various drafts of my dissertation, to Julien Page for all the roads he has been suggesting me in order to broaden my investigations, and to Dimitri Vey for his encouragements and friendship in the most difficult times. Of course, I also think of all my fellow companions who shared with me the vicis- situdes of the daily life of a graduate student. It has been a truly rewarding experience to share so many lunches and coffees discussing our widely different experiences. In particular, I would like to thank: Nacéra Bensaou, Pascal Bertin, Pierre Chaigneau, João Cortese, Vincent Daudon, Simon Decaens, Sarah Erman, Fernando Gálvez, Fabien Grégis, Roberto Hastenreiter-Cruz, Zeinab Karimian, Ramzi Kebaili, Jonathan Regier, Eleonora Sammarchi, Philippe Stamenkovic, Sergio Valencia and Xiaofei Wang. During my PhD, I have had the chance of traveling and participating in several workshops which have lead to important encounters. Particularly inspiring for me was the meeting of Johannes Kleiner, Robin Lorenz and the Basic Research Community Acknowledgments iii for Physics of which I am now a proud member. I also thank Charles Alunni and Jean- Pierre Marquis for their interest in my work and their support, and Michael Wright for his help with my babbling English. Finally, I am beholden to the whole stackexchange community without which the present document would simply not exist. Last, my thoughts go to my closest friends—Alexandre, Ivan, Philippe, Lou and Vera—which laughed with me about my obstinate existential doubts, and to my par- ents. They have managed to create a serene atmosphere of insatiable curiosity and openness towards all regions of knowledge which has deeply influenced me. Their con- stant support and feedback during these four years, and beyond, has been invaluable. And, with all my heart, I thank Camila, who accompanied me in every step of this long journey, silently feeling helpless next to me without realizing she brought me the most essential facet of them all: beauty. Contents List of Figures viii List of Tables ix Introduction 1 I Mathematical Description of Physical Systems 11 I.1 A case study: the birth of Quantum Mechanics (1925–1932) . 18 I.1.1 Matrix and Wave Mechanics ....................... 19 I.1.1.a Matrix Mechanics .......................... 20 I.1.1.b Wave Mechanics ........................... 26 I.1.2 Extracting the physics (1): Schrödinger’s mathematical equivalence . 30 I.1.3 Extracting the physics (2): Dirac’s transformation theory . 40 I.1.4 Extracting the physics (3): von Neumann’s process of abstraction . 48 I.2 Abstract mathematical entities ........................ 58 I.2.1 Ontology: the abstract/particular as hierarchy of objects . 60 I.2.2 Epistemology: the abstract/particular as hierarchy of information . 68 I.2.3 Mixture: the abstract/particular as hierarchy of identities . 79 I.3 Abstract mathematical structures ...................... 88 I.3.1 Mathematical structuralism ....................... 90 I.3.1.a Characterizing structuralism .................... 91 I.3.1.b Eliminative vs. ante rem structuralism . 99 I.3.2 Identity within an abstract mathematical structure . 105 I.3.2.a The problem of the identity of indiscernibles . 105 I.3.2.b First attempts at a solution . 109 I.3.2.c The solution: primitive typed identity . 115 I.3.3 Individuation within an abstract mathematical structure . 119 v vi Contents I.3.3.a Abstract structures as structured types . 120 I.3.3.b Abstract sets vs. abstract collections . 126 I.3.3.c Four grades of discernibility . 130 I.4 Conclusion ................................... 135 II The Classical and Quantum Kinematical Arenas 139 II.1 The double role of properties in standard Kinematics . 143 II.1.1 Standard Classical Kinematics . 147 II.1.2 Standard Quantum Kinematics . 155 II.2 The Quantum seen from the Classical: the geometric formulation . 165 II.2.1 The quantum space of states as a classical space of states . 171 II.2.2 The additional geometric structure of the Quantum . 178 II.2.3 The geometrical formulation of the superposition principle . 187 II.3 The Classical seen from the Quantum: the algebraic formulation . 192 II.3.1 The grand algebraic analogy . 198 II.3.2 States and representations of algebras . 205 II.3.3 Reconstructing the algebra of properties from the pure state space . 214 II.3.4 New look into the Classical and characterization of the Quantum . 224 II.4 Conclusion ................................... 230 III Constructing the Mathematical Description of a Physical System 235 III.1 The general strategy for introducing discernibility . 236 III.2 Introducing discernibility through groups (1): the representation problem 243 III.2.1 In Classical Kinematics . 245 III.2.1.a Importance of (strongly) Hamiltonian actions . 249 III.2.1.b Hamiltonian vs. strongly Hamiltonian actions . 255 III.2.2 In Quantum Kinematics . 262 III.2.2.a The quantum analogue of (strongly) Hamiltonian actions . 264 III.2.2.b Unitary vs. projective representations . 268 III.3 Introducing discernibility through groups (2): the individuation problem .
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