DOCSLIB.ORG

Yvette Kosmann-Schwarzbach the Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Yvette Kosmann-Schwarzbach the Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E Sources and Studies in the History of Mathematics and Physical Sciences Yvette Kosmann-Schwarzbach The Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E. Schwarzbach ! "# $ # % "& & &%' & & & #&& %())% ) )*+*, Emmy Noether (1882–1935) (photograph courtesy of the Emmy Noether Foundation, Bar Ilan University) Yvette Kosmann-Schwarzbach The Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E. Schwarzbach 123 Yvette Kosmann-Schwarzbach Bertram E. Schwarzbach Centre de Mathématiques Laurent Schwartz (Translator) École Polytechnique 91128 Palaiseau France [email protected] "#$ %&'((*'&('&'+&(+ ("#$ %&'((*'&('&'+'(* ," &-%&'((*'&('&'+'(* $ . / 0 1 0 2 3 0 Mathematics Subject Classification (2010): 01-02, 22-03, 22E70, 49-03, 49S05, 70-03, 70H03, 70H33, 83-03 < =# 0 33 > 0 ? / 4 2 0 0 5 2 9 =# 0 33 ** $ . / $. * : ; 5 2 5 ; > 4 4 ( 4 5 5 5 0 > 0 5 24 0 0 4 / 5 0 > 0 5 200 ? 2 5 0 0 / > / 0 > 5 4 0 8 0 2 / ; 5 4 27 4 @ 0 0(5 5 =# 0 9 : Ö In memory of Yseult who liked science as well as history Preface What follows thus depends upon a combination of the methods of the formal calculus of variations and of Lie's theory of groups. Emmy Noether, 1918 This book is about a fundamental text containing two theorems and their converses which established the relation between symmetries and conservation laws for varia- tional problems. These theorems, whose importance remained obscure for decades, eventually acquired a considerable influence on the development of modern theo- retical physics, and their history is related to numerous questions in physics, in me- chanics and in mathematics. This text is the article “Invariante Variationsprobleme” by Emmy Noether, which was published in 1918 in the Gottinger¨ Nachrichten, and of which we present an English translation in Part I of this book. The translation of Noether’s article is followed, in Part II, by a detailed analysis of its inception, as well as an account of its reception in the scientific community. As the background to Noether’s research, we sketch some developments in the theory of invariants in the nineteeth century which culminated in the definition and study of differential invariants, we discuss several works in mechanics dating from the beginning of the twentieth century in which Sophus Lie’s infinitesimal methods in the theory of groups began to be applied, and we show that the immediate motiva- tion for her work was related to questions arising from Einstein’s general theory of relativity of 1915. We then summarize the contents of Noether’s article in modern language. In the subsequent chapters, we review the way in which Noether’s con- temporaries, the mathematicians Felix Klein, David Hilbert and Hermann Weyl, and the physicists Einstein and Wolfgang Pauli, acknowledged or failed to acknowledge her contribution; then we outline the quite different diffusions of her first and second theorems. Finally, we outline the genuine generalizations of Noether’s results that began to appear after 1970, in the field of the calculus of variations and in the theory of integrable systems. The present edition is based on the second edition of Les Theor´ emes` de Noether. Invariance et lois de conservation au XXe siecle` (Palaiseau: Editions´ de l’Ecole´ Polytechnique, 2006). For this English edition, the French text has been consider- ably revised and augmented, with much new information and additional references. Paris, July 2010 vii Acknowledgments It is a pleasure to thank the many colleagues and friends who have made the writing of this book possible. When the French version first appeared in 2004, it was at the insistance of Pierre Cartier whose subsequent support was unfailing, with Alain Guichardet’s generous help, and with the much appreciated contribution of Laurent Meersseman who had translated the “Invariante Variationsprobleme” into French and written a draft of the first section of a commentary. During the preparation of the French and English texts, I received information and assistance on various questions from Alain Albouy, Sophie Bade, Henri Besson, Katherine Brading, Harvey Brown, Leo Corry, Thibault Damour, Jean Eisenstaedt, Benjamin Enriquez, Robert Gergondey, Patrick Iglesias-Zemmour,´ Michel Janssen, Christoph Kopper, Franco Magri, Hartmann Romer,¨ Peter Roquette, Volodya Rubtsov, David Rowe, Erhard Scholz, Jean-Marie Souriau, Jim Stasheff, Alexander Vinogradov, Salomon Wald, Stefan Waldmann and Scott Walter. To all of them— and to any colleagues whose names I may have omitted inadvertently—I am very grateful. I especially thank Peter Olver who made many useful remarks on the manuscript of the first French edition, based on his thorough knowledge of the sub- ject, and again answered my questions when I was preparing the second edition in 2006. The three referees for the first edition of the French text made valuable rec- ommendations and suggestions, from which I benefitted. The help I received from David Schimmel in translating several difficult German texts remains a treasured memory of this learned friend who died in a road accident in 2007. Thanks are also due to Fred´ eric´ Zantonio at the Centre Polymedia´ de l’Ecole´ Polytechnique who helped me with the reproductions. The editors of Springer New York, Ann Kostant, Hans Koelsch, and Elizabeth Loew, extended their sup- port during the preparation of this book, for which I thank them cordially. I am grateful to the institutions which have allowed me to reproduce docu- ments in their collections: the Niedersachsische¨ Staats- und Universitatsbibliothek,¨ Gottingen,¨ the Einstein archive in Jerusalem, and the archive of Bryn Mawr College, and to Princeton University Press for permission to quote from several volumes of The Collected Papers of Albert Einstein. Last but not least, I thank the translator, Bertram E. Schwarzbach, for his work and for his advice on how to clarify and improve the French text. Yvette Kosmann-Schwarzbach July 2010 ix !"#"$% * * 66666 !" # $ % !" & ' ( ) ) * * + , - & ' ( ' & " - ." - ! , ( , + , / ! - $ ( 0 ! 1 $ & * ! 1 & ) && &) 2 3 4 & . 2 )# $ 2 + ,5 * ) xii Contents 3 The Noether Theorems as Seen by Contemporaries and by Historians of Science ........................................... 65 3.1 References to Noether in the Works of Klein, Hilbert and Weyl, and in Einstein’s Correspondence ............................. 65 3.2 The Eulogies of 1935 ....................................... 76 3.3 Personal Recollections ...................................... 79 3.4 The Introduction to Noether’s Gesammelte Abhandlungen / Collected Papers ........................................... 86 3.5 Translations of the Invariante Variationsprobleme ............... 87 3.6 Historical Analyses ......................................... 88 4 The Transmission of Noether’s Ideas, from Bessel-Hagen to Hill, 1921–1951 .............................................. 91 4.1 Bessel-Hagen and Symmetries up to Divergence. ................ 91 4.2 Pauli 1921 and 1941 ........................................ 93 4.3 Weitzenbock¨ 1923 . ......................................... 94 4.4 Courant and Hilbert 1924 . ................................... 95 4.5 In Quantum Mechanics . ..................................... 96 4.6 Negative Results ........................................... 98 4.7 Hill’s 1951 Article ..........................................101 5 The Reception of Noether’s First Theorem after 1950 ..............103 5.1 Symmetries and Conservation Laws in Classical Mechanics and Quantum Physics . ..........................................104 5.2 On Some Encyclopedia Articles ..............................104 5.3 Analysis of Several Works in Mathematics and Mechanics, 1950–1980 ................................................105 5.4 Analysis of Several Works in Physics, 1950–1980 ...............117 5.5 The Rediscoveries as Generalizations of “Noether’s Theorem” . ....121 6 The Reception of Noether’s Second Theorem after 1950 ............123 6.1 The Second Theorem and General Relativity . ...................123 6.2 The Second Theorem and Gauge Theories . .....................129 7 After 1970—Genuine Generalizations ............................133 7.1 Jet Bundles and Generalized Symmetries .......................134 7.2 Characteristics of Conservation Laws and the Converse of the First Theorem ........................................137 7.3 The Formal Calculus of Variations ............................138 7.4 Symmetries and Conservation Laws for Nonvariational Equations . 140 7.5 At the end of the twentieth century ............................144 Conclusion ........................................................145 Appendix I. Postcard from Noether to Klein, 15 February 1918 ..........149 Appendix II. Letter from Noether to Klein, 12 March 1918 ..............153 Contents xiii Apppendix III. Letter from Klein to Pauli, 8 March 1921 ...............159 Apppendix IV. Letter from Noether to Einstein, 7 January 1926 .........161 Apppendix V. Lectures at the Mathematical Society of Gottingen,¨ 1915-1919 .....................................................167
Load more

Recommended publications
  • Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
    BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein.
    [Show full text]
  • Abstract Book
    14th International Geometry Symposium 25-28 May 2016 ABSTRACT BOOK Pamukkale University Denizli - TURKEY 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 14th International Geometry Symposium ABSTRACT BOOK 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ E-Published By: Pamukkale University Department of Mathematics Denizli, TURKEY All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected] 2 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium May 25-28, 2016 Denizli, Turkey. Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey 3 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications.
    [Show full text]
  • Karen Uhlenbeck Awarded the 2019 Abel Prize
    RESEARCH NEWS Karen Uhlenbeck While she was in Urbana-Champagne (Uni- versity of Illinois), Karen Uhlenbeck worked Awarded the 2019 Abel with a postdoctoral fellow, Jonathan Sacks, Prize∗ on singularities of harmonic maps on 2D sur- faces. This was the beginning of a long journey in geometric analysis. In gauge the- Rukmini Dey ory, Uhlenbeck, in her remarkable ‘removable singularity theorem’, proved the existence of smooth local solutions to Yang–Mills equa- tions. The Fields medallist Simon Donaldson was very much influenced by her work. Sem- inal results of Donaldson and Uhlenbeck–Yau (amongst others) helped in establishing gauge theory on a firm mathematical footing. Uhlen- beck’s work with Terng on integrable systems is also very influential in the field. Karen Uhlenbeck is a professor emeritus of mathematics at the University of Texas at Austin, where she holds Sid W. Richardson Foundation Chair (since 1988). She is cur- Karen Uhlenbeck (Source: Wikimedia) rently a visiting associate at the Institute for Advanced Study, Princeton and a visiting se- nior research scholar at Princeton University. The 2019 Abel prize for lifetime achievements She has enthused many young women to take in mathematics was awarded for the first time up mathematics and runs a mentorship pro- to a woman mathematician, Professor Karen gram for women in mathematics at Princeton. Uhlenbeck. She is famous for her work in ge- Karen loves gardening and nature hikes. Hav- ometry, analysis and gauge theory. She has ing known her personally, I found she is one of proved very important (and hard) theorems in the most kind-hearted mathematicians I have analysis and applied them to geometry and ever known.
    [Show full text]
  • Ernst Zermelo Heinz-Dieter Ebbinghaus in Cooperation with Volker Peckhaus Ernst Zermelo
    Ernst Zermelo Heinz-Dieter Ebbinghaus In Cooperation with Volker Peckhaus Ernst Zermelo An Approach to His Life and Work With 42 Illustrations 123 Heinz-Dieter Ebbinghaus Mathematisches Institut Abteilung für Mathematische Logik Universität Freiburg Eckerstraße 1 79104 Freiburg, Germany E-mail: [email protected] Volker Peckhaus Kulturwissenschaftliche Fakultät Fach Philosophie Universität Paderborn War burger St raße 100 33098 Paderborn, Germany E-mail: [email protected] Library of Congress Control Number: 2007921876 Mathematics Subject Classification (2000): 01A70, 03-03, 03E25, 03E30, 49-03, 76-03, 82-03, 91-03 ISBN 978-3-540-49551-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting by the author using a Springer TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 46/3100/YL - 5 4 3 2 1 0 To the memory of Gertrud Zermelo (1902–2003) Preface Ernst Zermelo is best-known for the explicit statement of the axiom of choice and his axiomatization of set theory.
    [Show full text]
  • Covariant Momentum Map for Non-Abelian Topological BF Field
    Covariant momentum map for non-Abelian topological BF field theory Alberto Molgado1,2 and Angel´ Rodr´ıguez–L´opez1 1 Facultad de Ciencias, Universidad Autonoma de San Luis Potosi Campus Pedregal, Av. Parque Chapultepec 1610, Col. Privadas del Pedregal, San Luis Potosi, SLP, 78217, Mexico 2 Dual CP Institute of High Energy Physics, Colima, Col, 28045, Mexico E-mail: [email protected], [email protected] Abstract. We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.
    [Show full text]
  • Witold Roter (1932-2015)
    COLLOQUIUMMATHEMATICUM Online First version WITOLD ROTER (1932–2015) BY ANDRZEJ DERDZIŃSKI (Columbus, OH) Witold Roter was born on September 20th, 1932 and died on June 19th, 2015. He authored or co-authored 40 papers in differential geometry, pub- lished between 1961 and 2010. His nine Ph.D. advisees, listed here along with the year of receiving the degree, are: Czesław Konopka (1972), Edward Głodek (1973), Andrzej Gębarowski (1975), Andrzej Derdziński (1976), Zbig- niew Olszak (1978), Ryszard Deszcz (1980), Marian Hotloś (1980), Wiesław Grycak (1984), and Marek Lewkowicz (1989). Also, for many years, he ran the Wrocław seminar on differential geome- try, first started by Władysław Ślebodziński, who had been Witold Roter’s Ph.D. advisor. For more biographical information (in Polish), see Zbigniew Olszak’s article [44]. This is a brief summary of Witold Roter’s selected results, divided into four sections devoted to separate topics. Since he repeatedly returned to questions he had worked on earlier, our presentation is not chronological. 1. Parallel Weyl tensor in the Riemannian case. The curvature tensor R of a given n-dimensional pseudo-Riemannian manifold (M; g) is naturally decomposed into the sum R = S + E + W of its irreducible com- ponents [41, p. 47]. The first two correspond to the scalar curvature and Einstein tensor (the traceless part of the Ricci tensor). The third component is the Weyl tensor W , also known as the conformal curvature tensor, which is of interest only in dimensions n ≥ 4 since, for algebraic reasons, W = 0 whenever n ≤ 3. Viewed as a (1; 3) tensor field, so that it sends three vector fields trilin- early to a vector field, W is a conformal invariant: it remains unchanged when the metric g is replaced by the product φg, where φ is any smooth positive function.
    [Show full text]
  • Twenty Female Mathematicians Hollis Williams
    Twenty Female Mathematicians Hollis Williams Acknowledgements The author would like to thank Alba Carballo González for support and encouragement. 1 Table of Contents Sofia Kovalevskaya ................................................................................................................................. 4 Emmy Noether ..................................................................................................................................... 16 Mary Cartwright ................................................................................................................................... 26 Julia Robinson ....................................................................................................................................... 36 Olga Ladyzhenskaya ............................................................................................................................. 46 Yvonne Choquet-Bruhat ....................................................................................................................... 56 Olga Oleinik .......................................................................................................................................... 67 Charlotte Fischer .................................................................................................................................. 77 Karen Uhlenbeck .................................................................................................................................. 87 Krystyna Kuperberg .............................................................................................................................
    [Show full text]
  • Harmonic and Complex Analysis and Its Applications
    Trends in Mathematics Alexander Vasil’ev Editor Harmonic and Complex Analysis and its Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961 Harmonic and Complex Analysis and its Applications Alexander Vasil’ev Editor Editor Alexander Vasil’ev Department of Mathematics University of Bergen Bergen Norway ISBN 978-3-319-01805-8 ISBN 978-3-319-01806-5 (eBook) DOI 10.1007/978-3-319-01806-5 Springer Cham Heidelberg New York Dordrecht London Mathematics Subject Classification (2010): 13P15, 17B68, 17B80, 30C35, 30E05, 31A05, 31B05, 42C40, 46E15, 70H06, 76D27, 81R10 c Springer International Publishing Switzerland 2014 This work is subject to copyright.
    [Show full text]
  • A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
    Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge­ bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe­ maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe­ matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members.
    [Show full text]
  • President's Report
    AWM ASSOCIATION FOR WOMEN IN MATHE MATICS Volume 36, Number l NEWSLETTER March-April 2006 President's Report Hidden Help TheAWM election results are in, and it is a pleasure to welcome Cathy Kessel, who became President-Elect on February 1, and Dawn Lott, Alice Silverberg, Abigail Thompson, and Betsy Yanik, the new Members-at-Large of the Executive Committee. Also elected for a second term as Clerk is Maura Mast.AWM is also pleased to announce that appointed members BettyeAnne Case (Meetings Coordi­ nator), Holly Gaff (Web Editor) andAnne Leggett (Newsletter Editor) have agreed to be re-appointed, while Fern Hunt and Helen Moore have accepted an extension of their terms as Member-at-Large, to join continuing members Krystyna Kuperberg andAnn Tr enk in completing the enlarged Executive Committee. I look IN THIS ISSUE forward to working with this wonderful group of people during the coming year. 5 AWM ar the San Antonio In SanAntonio in January 2006, theAssociation for Women in Mathematics Joint Mathematics Meetings was, as usual, very much in evidence at the Joint Mathematics Meetings: from 22 Girls Just Want to Have Sums the outstanding mathematical presentations by women senior and junior, in the Noerher Lecture and the Workshop; through the Special Session on Learning Theory 24 Education Column thatAWM co-sponsored withAMS and MAA in conjunction with the Noether Lecture; to the two panel discussions thatAWM sponsored/co-sponsored.AWM 26 Book Review also ran two social events that were open to the whole community: a reception following the Gibbs lecture, with refreshments and music that was just right for 28 In Memoriam a networking event, and a lunch for Noether lecturer Ingrid Daubechies.
    [Show full text]
  • Kant, Schlick and Friedman on Space, Time and Gravity in Light of Three Lessons from Particle Physics
    Erkenn DOI 10.1007/s10670-017-9883-5 ORIGINAL RESEARCH Kant, Schlick and Friedman on Space, Time and Gravity in Light of Three Lessons from Particle Physics J. Brian Pitts1 Received: 28 March 2016 / Accepted: 21 January 2017 Ó The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Kantian philosophy of space, time and gravity is significantly affected in three ways by particle physics. First, particle physics deflects Schlick’s General Relativity-based critique of synthetic a priori knowledge. Schlick argued that since geometry was not synthetic a priori, nothing was—a key step toward logical empiricism. Particle physics suggests a Kant-friendlier theory of space-time and gravity presumably approximating General Relativity arbitrarily well, massive spin- 2 gravity, while retaining a flat space-time geometry that is indirectly observable at large distances. The theory’s roots include Seeliger and Neumann in the 1890s and Einstein in 1917 as well as 1920s–1930s physics. Such theories have seen renewed scientific attention since 2000 and especially since 2010 due to breakthroughs addressing early 1970s technical difficulties. Second, particle physics casts addi- tional doubt on Friedman’s constitutive a priori role for the principle of equivalence. Massive spin-2 gravity presumably should have nearly the same empirical content as General Relativity while differing radically on foundational issues. Empirical content even in General Relativity resides in partial differential equations, not in an additional principle identifying gravity and inertia. Third, Kant’s apparent claim that Newton’s results could be known a priori is undermined by an alternate gravitational equation.
    [Show full text]
  • Mathematicians Fleeing from Nazi Germany
    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
    [Show full text]