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Symmetry and Mathematics A Conceptual History of Space and Symmetry Pietro Giuseppe Fré A Conceptual History of Space and Symmetry From Plato to the Superworld 123 Pietro Giuseppe Fré University of Torino Turin, Italy ISBN 978-3-319-98022-5 ISBN 978-3-319-98023-2 (eBook) https://doi.org/10.1007/978-3-319-98023-2 Library of Congress Control Number: 2018950942 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This book is dedicated to my family, namely to my beloved daughter Laura, to my darling wife Olga, to my young son Vladimir and to my former wife Tiziana, Laura’s mother, with whom, notwithstanding our divorce, the ties due to a common daughter and to more than twenty five years spent together remain strong. This work is also dedicated to my dear Franca, to whom, since my mother’s death, I am tied by a profound filial affection and by the sharing of deep intellectual interests. My feelings of love are hereby transmitted to all of them, with all the hopes and the worries of an old father who stubbornly believes that culture and science are just one thing and encode, for mankind, the unique escape route from global disasters. Preface This book forms a twin pair with another book [90] by the same author, which is of a different, thoroughly mathematical character, while the character of the present volume is historically and philosophically oriented. These two pieces of work constitute a twin pair since, notwithstanding their different profiles and contents, they arise from the same vision and pursue complementary goals. The vision, extensively discussed in the first chapter of the present book and throughout several other chapters, consists of the following main conceptual assessments: 1. Our current understanding of the Fundamental Laws of Nature is based on a coherent, yet provisional, set of five meta-theoretical principles, listed by me as A)-E) and dubbed the current episteme. This episteme is of genuine geometrical nature and can be viewed as the current evolutionary state of Einstein’s ideas concerning the geometrization of physics. 2. Geometry and Symmetry are inextricably entangled, and their current conception is the result of a long process of abstraction, traced back in the present work, which was historically determined and makes sense only within the Analytic System of Thought of Western Civilization, started by the ancient Greeks. 3. The evolution of Geometry and Symmetry Theory in the last 40 years has been deeply and very much constructively influenced by Supersymmetry/ Supergravity and the allied constructions of Strings and Branes. My reader is not supposed to know what supersymmetric field theories and supergravity are, since this book has the ambition to be written for a much more general public than that formed by theoretical physicists specialized in the super-world.It suffices to be alerted that, since the seventies of the twentieth century, an entire new theoretical world has opened up by the introduction of a new symmetry principle, indeed dubbed supersymmetry, that can be enforced on field theories, in particular on Einstein Gravity, at the price of adding to every physical field a 1 new copy of the same, named its superpartner with a spin shifted by 2. This entrains that the superpartners of bosons are fermions and vice versa. In a vii viii Preface nutshell, supersymmetry is the invariance of such new type of field theories under the exchange of bosons with fermions. 4. Further advances in Theoretical Physics cannot be based simply on the Galilean Method of Interrogating first Nature and then formulating a testable theory that explains the observed phenomena. As stated later on in the present work, one ought to Interrogate also Human Thought, by this meaning frontier-line math- ematics concerned with geometry and symmetry in order to find there the threads of so far unobserved correspondences, reinterpretations and renewed conceptions. The complementary pursued goals are as follows: (a) In the case of the present book • the historical and conceptual analysis of the process mentioned in point (2). of the above list which led to the current episteme. • the philosophical argumentation, on historical basis, of the assessment made in point (4). of the above list. (b) In the case of book [90], the mathematical full-fledged illustration of the main developments in geometry and symmetry theory that occurred under the fer- tilizing influence of Supersymmetry/Supergravity and that would be incon- ceivable without the latter. Repeating in a slightly different from the arguments advocated particularly in Chapter One of the present work, I think that what is currently practiced in the whole world as Fundamental Physics or Mathematics is based on the Greek view of the episteme and it is meaningful only inside the Analytic System of Thought founded by the Ancient Greeks. To recuperate a full-conscience of this fact is mandatory in order to continue on the difficult but exciting path we are confronted with. The twin pair of which this book is a member is viewed by the author as his limited, humble contribution to the promotion of a new season of more scholarly teaching of physical-mathematics. TO MY READER Ideally, this book has been written for an audience wider than that composed by mathematicians and theoretical physicists. My ambitious aim was that of attracting the attention of all finely educated persons with an interest in the history of Ideas, mathematical conceptions being an integral part of the general episteme. Yet, mathematics is also technical and it is very difficult to single out the main ideas and tell the story of their evolution without mentioning some fundamental definitions and some formulae. Coping with this essential difficulty, well known to everyone who tries to explain the theories of the world to wider audiences, I have tried my best, where formulae and definitions were unavoidable, to convey the underlying Preface ix concepts by means of wordly explanations and with the help of numerous ad hoc created images. I have tried to intertwine storytelling and philosophic argumenta- tions with some essential technical material introduced according to the above- mentioned method, hoping to keep the attention of all my dear readers alive throughout the chapters and up to my epilogue that expresses the results of long self-debating meditations. Spes, ultima dea. Turin, Italy Pietro Giuseppe Fré June 2018 Acknowledgements With great pleasure, I would like to thank my collaborators and colleagues Pietro Antonio Grassi, and Mario Trigiante for the many suggestions and discussions we had during the writing of the present book. Similarly, I express my gratitude to the Editors of Springer-Verlag, in particular to Maria Bellantone, for their continuous assistance, constructive criticism and suggestions. My thoughts, while finishing the writing of this long essay, that mostly occurred in Moscow in my last 2 years (2016–2017) of service as Scientific Counsellor of the Italian Embassy in Russia, were frequently directed to my late parents, whom I miss very much and I will never forget. To them I also express my gratitude for all what they taught me in their life, in particular to my father who, with his own example, introduced me, since my childhood, to the great satisfaction and deep suffering of writing books. Furthermore, it is my pleasure to thank my very close friend and collaborator Aleksander Sorin for his continuous encouragement and for many precious consultations. xi Contents 1 The Episteme ......................................... 1 2 Symmetry and Mathematics .............................. 9 2.1 Setting the Stage: Symmetry and Beauty ................. 9 2.2 Symmetry Principles ................................ 13 2.3 Geometry and Algebra up to the Birth of Group Theory ...... 14 2.4 Galois and the Advent of Group Theory ................. 21 2.5 A New Conception of Symmetry ....................... 23 2.5.1 Conceptual Analysis of Galois Results............. 27 2.5.2 Symmetry After Galois ........................ 28 2.6 Symmetry, Geometry and Space ....................... 29 3 How Group Theory Came into Being ....................... 33 3.1 The Essentials of Group Theory in Modern Parlance ........ 33 3.1.1 Examples .................................. 35 3.1.2 Groups as Transformation Groups ................ 35 3.1.3 Representations of a Group ..................... 37 3.2 From Cayley and Sylvester’s Matrices to Vector Spaces and Groups: A Long Gestation ........................ 41 3.2.1 Cayley and Sylvester: A Short Account of Their Lives .............................. 41 3.2.2 Matrices in the Middle 1850s ................... 42 3.2.3 Cayley and Sylvester: The Invariant Twins ......... 44 3.2.4 The Calculus of Operations: Cayley and George Boole ...................................
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