Doing Mathematics : Convention, Subject, Calculation, Analogy

492 Pages·2015·4.96 MB·English

by  Martin H Krieger

d o i n g m a t h e m a t i c s
Convention, Subject, Calculation, Analogy
Second Edition
FIGURE P.1: A Riemann surface for z1/2 (Neumann, Vorlesungen über Riemann’s
Theorie der Abel’schen Integrale, 1884)
d o i n g m a t h e m a t i c s
Convention, Subject, Calculation, Analogy
Second Edition
Martin H Krieger
University of Southern California, USA
World Scientific
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Portions of this book, reprinted herein with permission of the publishers, appeared in somewhat different form in:
Martin H. Krieger, “Making a Paradigmatic Convention Normal: Entrenching Means and Variances as Statistics,”
Science in Context 9 (Winter 1996): 487–509 (Cambridge University Press); Martin H. Krieger, “Apocalypticism,
One-Time Events, and Scientific Rationality,” in, S.S. Hecker and G.-C. Rota, eds., Essays on the Future, In
Honor of Nick Metropolis, Boston: Birkhäuser, 2000, pp. 135–151, ©2000 Springer-Verlag; Martin H. Krieger, “The
Mathematics of the Ideal City in the Nineteenth Century,” in, M. Leary, B. Dermody, and A. Markusen, eds.,
Envisioning Cities and Regions: A Celebratory Anthology on Planning by University of Minnesota Faculty,
Minneapolis: Hubert H. Humphrey Institute of Public Affairs, University of Minnesota, 2000, pp. 309–316.
C. N. Yang, “The Spontaneous Magnetization of a Two-Dimensional Ising Model,” Physical Review 85 (1952): 808–
816, ©1952 American Physical Society. Excerpt from “On the Dirac and Schwinger Corrections to the Ground-State
Energy of an Atom” by C. Fefferman and L.A. Seco in Advances in Mathematics, Volume 107, 1–185, copyright 1994,
Elsevier Scientific (USA), reproduced with permission from the publisher. A. Weil, Oeuvres Scientifiques, volume 1,
pp. 244–255, ©1978 Springer-Verlag; J. Leray, Selected Papers, volume 1, pp. 61–64, ©1998 Springer-Verlag.
Figures 1.1, 3.11, courtesy of Springer Verlag; 1.3, 3.5, Koninklijke Nederlandse Akademie van Wetenschappen and
the Brouwer Archive; 3.3, 3.7, 3.13, 5.4, Dover Publishing Company; 3.4, Heldermann Verlag; 3.12, A.K. Peters,
Ltd.; 3.14, American Mathematical Society; 4.1, N.2, N.3, American Physical Society.
Library of Congress Cataloging-in-Publication Data
Krieger, Martin H.
Doing mathematics : convention, subject, calculation, analogy / by Martin H Krieger (University of Southern
California, USA). -- 2nd edition.
pages cm
Includes bibliographical references and index.
ISBN 978-981-4571-83-8 (hardcover : alk. paper) -- ISBN 978-981-4571-84-5 (pbk. : alk. paper)
1. Mathematics--Research. I. Title.
QA11.2.K75 2015
510--dc23
2014043918
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Cover illustration is a montage of Figures 2 and 3 of L. E. J. Brouwer, “Zur Analysis Situs,” Mathematische Annalen
68 (1910): 422–434.
Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
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Printed in Singapore
To the memory of my parents, Louis and Shirley Krieger,

my dear friend Miriam Brien and my cousin Marion Herbst,

and the memory of Abraham Polonsky and Gian-Carlo Rota.
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Figures ix
Preface xiii
Prolog xxi
ONE Introduction 1
TWO Convention: How Means and Variances are Entrenched as Statistics 27
THREE Subject: The Fields of Topology 54
Appendix: The Two-Dimensional Ising Model of a Ferromagnet 108
FOUR Calculation: Strategy, Structure, and Tactics in Applying Classical
Analysis 120
FIVE Analogy: A Syzygy Between a Research Program in Mathematics and a
Research Program in Physics 211
SIX In Concreto: The City of Mathematics 256
Epilog 296
APPENDICES
A: C.N. Yang, “The Spontaneous Magnetization of a Two-Dimensional
Ising Model” (1952) 298
B: C. Fefferman and L.A. Seco, “On the Dirac and Schwinger Corrections
to the Ground-State Energy of an Atom” (1994, pp. 1–14) 307
vii
CONTENTS
C: J. Leray, “Sur la forme des espaces topologiques et sur les points fixes
des représentations” (1945, pp. 96–99) 322

D: A. Weil, “Une lettre à Simone Weil” (1940) 325

Notes 339

Bibliography 431

Thematic Index 463
viii
Figures
P.1 A Riemann surface for z½ ii

P.2 Topical Table of Contents xvi

1.1 Brouwer’s indecomposable plane (1910) 8

1.2 Isles of Wada, from Yoneyama (1917) 8

1.3 Proof diagram from Brouwer (1909) 9

1.4 Three points in syzygy 15

3.1 Diagrammatic representation of commutativity of continuity and structure
58

3.2 An exact sequence 59

3.3 Riemann’s slices in domains (1857) 63

3.4 Relations between main classes of topological spaces 67

3.5 Proof diagram from Brouwer (1911) 69

3.6 Diagrams typical of algebraic topology 69

3.7 Possible streamings or flows in an annulus (Klein, 1893) 81

3.8 The two-dimensional Ising lattice 84

3.9 Parallel exact sequences mirroring each other 98

3.10 Commuting diagram for a natural transformation 101

3.11 Kunen-Miller chart 103

ix