Relativity without Spacetime
In 1908, three years after Einstein first published his special theory of relativity, the mathematician Hermann Minkowski introduced his four-dimensional “spacetime” interpretation of the theory. Einstein initially dismissed Minkowski’s theory, remarking
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Relativity without Spacetime
Joseph K. Cosgrove
Relativity without Spacetime
Joseph K. Cosgrove Philosophy Providence College Providence, RI, USA
ISBN 978-3-319-72630-4 ISBN 978-3-319-72631-1 (eBook) https://doi.org/10.1007/978-3-319-72631-1 Library of Congress Control Number: 2018935932 © The Editor(s) (if applicable) and The Author(s) 2018 This work is subject to copyright. All rights are solely and exclusively licensed 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. Cover credit: Tetra Images / Getty Images Printed on acid-free paper This Palgrave Macmillan imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
For Natalia, a thinker
Contents
1 Introduction: A Critique of Minkowski Spacetime 1 Part I The Concept of Minkowski Spacetime 9 2 Minkowski’s “Space and Time” 11 2.1 Minkowski and Göttingen Science 11 2.2 “Space and Time,” Sections I and II 14 2.3 “Space and Time,” Section III 20 2.4 “Space and Time,” Section IV 24 3 Special Relativity and Spacetime 35 3.1 Spacetime and the Concept of a Continuum 35 3.2 The “Geometry of Spacetime”: Graphs and Images 39 3.3 Invariance and Special Relativity 48 3.3.1 Invariance and Objectivity 48 3.3.2 Invariance and the Clock Paradox 56 3.4 On the Arrow of Explanation 58 3.5 Conceptual Difficulties of Minkowski Spacetime: The Need for a Historical Approach 60
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Contents
Part II The Symbolic-Algebraic Constitution of the Concept of Spacetime 65 4 The Historical Sense-Structure of Symbolic Algebra 69 4.1 The Concept Number in Greek Mathematics 73 4.1.1 Arithmetical Operations in Euclid 73 4.1.2 The Concept of Ratio 74 4.1.3 Arithmetic and Geometry in Euclid 76 4.2 Algebraic Equations in Greek Mathematics: Diophantus of Alexandria 77 4.2.1 The Conce
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