The Pythagorean World Why Mathematics is Unreasonably Effective in Physics Jane McDonnell The Pythagorean World Jane McDonnell The Pythagorean World Why Mathematics Is Unreasonably Effective In Physics Jane McDonnell Philosophy Department Clayton,Victoria Australia ISBN 978-3-319-40975-7 ISBN 978-3-319-40976-4 (eBook) DOI 10.1007/978-3-319-40976-4 Library of Congress Control Number: 2016957514 © Th e Editor(s) (if applicable) and Th e Author(s) 2017 Th is 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, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical way, and trans- mission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Th e use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Th e 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. Cover illustration: © Purcell Pictures, Inc. / Alamy Stock Photo Printed on acid-free paper Th is Palgrave Macmillan imprint is published by Springer Nature Th e registered company is Springer International Publishing AG Switzerland Th e registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Prefa ce Don’t be surprised then, Socrates, if it turns out repeatedly that we won’t be able to produce accounts on a great many subjects—on gods or the coming to be of the universe—that are completely and perfectly consistent and accurate. Instead, if we can come up with accounts no less likely than any, we ought to be content, keeping in mind that both I, the speaker, and you, the judges, are only human. Plato, T imaeus 29c v Acknowledgements Many people helped with the preparation of this book. Th e following deserve special recognition. Th anks to Graham Oppy for invaluable discussions, comments, sug- gestions and direction. Th anks to Monima Chadha for her encour- agement. Th anks to the students and staff of the Monash University Philosophy Department for general support and assistance. Th anks to Robert Griffi ths for patiently answering my questions about consistent histories quantum mechanics. Th anks to the examiners of my thesis, Eric Steinhart and Peter Forrest, for their suggested improvements. M y special thanks go to my husband, Michael McDonnell, for his critical insights. Without his support, this book would not have been possible. Any and all mistakes or omissions are mine alone. vii Abstract In this book, I argue that many problems in the philosophy of science and mathematics (in particular, the unreasonable eff ectiveness of mathematics in physics) can only be addressed within a broader meta- physical framework which provides a coherent world view. I attempt to develop such a framework and draw out its consequences. Th e attempt is in two parts: fi rstly, I develop a speculative framework based on an anal- ogy to set theory, then I combine elements of the framework with ideas from Leibnizian monadology and consistent histories quantum theory to introduce (what I call) quantum monadology. Th e two parts focus on dif- ferent aspects of the problem and should be viewed as stages on the way to a fi nal formulation. Th e inspiration for the book came from Plato’s Timaeus and Wigner’s comments on quantum mechanics. As it turned out, Leibniz’s M onadology became a third key source. Contents 1 Introduction 1 2 Th e Applicability of Mathematics 3 3 Th e Role of Mathematics in Fundamental Physics 69 4 One True Mathematics 125 5 What Mathematics Is About 179 6 Actuality from Potentiality 223 7 Conclusion 299 Appendix 1 331 ix x Contents Appendix 2 349 References 361 Index 383 1 Introduction In his article ‘Th e Unreasonable Eff ectiveness of Mathematics in the Natural Sciences’, the physicist Eugene Wigner asks “What is mathemat- ics?”, “What is physics?” and “Why is mathematics so unreasonably eff ec- tive in physics?”. Th ese are the key questions addressed in this book. Depending on which philosophical school one adheres to, mathematics might be a form of logic (logicism), a construction of the human mind (intu- itionism), a game played with symbols (formalism), a language describing the properties of real, abstract entities (platonism) or a language describing fi ctional entities (fi ctionalism). One thing which all schools have to explain is the eff ectiveness of mathematics in physics. It is a striking feature of math- ematics that it allows us to model and predict the behaviour of physical sys- tems to an amazing degree of accuracy. For example, the predicted magnetic moment of the electron agrees with the current experimental value to an accuracy of one part in a trillion. O ne of the oldest explanations of the eff ectiveness of mathematics in physics is that, in some profound way, the structure of the world is mathematical. Th e Pythagoreans believed that “everything is number”. If we interpret this explanation as saying that mathematical structure is © Th e Author(s) 2017 1 J. McDonnell, Th e Pythagorean World, DOI 10.1007/978-3-319-40976-4_1