Undergraduate Texts in Mathematics

Editorial BoardS. Axler

K.A. Ribet

For other titles published in this series, go tohttp://www.springer.com/series/666

John Stillwell

Mathematicsand Its History

Third Edition

123

John StillwellDepartment of MathematicsUniversity of San FranciscoSan Francisco, CA 94117-1080USAstillwell@usfca.edu

Editorial BoardS. AxlerMathematics DepartmentSan Francisco State UniversitySan Francisco, CA 94132USAaxler@sfsu.edu

K.A. RibetMathematics DepartmentUniversity of California at BerkeleyBerkeley, CA 94720-3840USAribet@math.berkeley.edu

ISSN 0172-6056ISBN 978-1-4419-6052-8 e-ISBN 978-1-4419-6053-5DOI 10.1007/978-1-4419-6053-5Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010931243

Mathematics Subject Classification (2010): 01-xx, 01Axx

c© Springer Science+Business Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer soft-ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

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To Elaine, Michael, and Robert

Preface to the Third Edition

The aim of this book, announced in the first edition, is to give a bird’s-eye view of undergraduate mathematics and a glimpse of wider horizons.The second edition aimed to broaden this view by including new chapterson number theory and algebra, and to engage readers better by includingmany more exercises. This third (and possibly last) edition aims to increasebreadth and depth, but also cohesion, by connecting topics that were previ-ously strangers to each other, such as projective geometry and finite groups,and analysis and combinatorics.

There are two new chapters, on simple groups and combinatorics, andseveral new sections in old chapters. The new sections fill gaps and updateareas where there has been recent progress, such as the Poincare conjec-ture. The simple groups chapter includes some material on Lie groups,thus redressing one of the omissions I regretted in the first edition of thisbook. The coverage of group theory has now grown from 17 pages and 10exercises in the first edition to 61 pages and 85 exercises in this one. As inthe second edition, exercises often amount to proofs of big theorems, bro-ken down into small steps. In this way we are able to cover some famoustheorems, such as the Brouwer fixed point theorem and the simplicity ofA5, that would otherwise consume too much space.

Each chapter now begins with a “Preview” intended to orient the readerwith motivation, an outline of its contents and, where relevant, connectionsto chapters that come before and after. I hope this will assist readers wholike to have an overview before plunging into the details, and also instruc-tors looking for a path through the book that is short enough for a one-semester course. Many different paths exist, at many different levels. Upto Chapter 10, the level should be comfortable for most junior or seniorundergraduates; after that, the topics become more challenging, but also ofgreater current interest.

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All the figures have now been converted to electronic form, which hasenabled me to reduce some that were excessively large, and hence mitigatethe bloating that tends to occur in new editions.

Some of the new material on mechanics in Section 13.2 originally ap-peared (in Italian) in a chapter I wrote for Volume II of La Matematica,edited by Claudio Bartocci and Piergiorgio Odifreddi (Einaudi, Torino,2008). Likewise, the new Section 8.6 contains material that appeared inmy book The Four Pillars of Geometry (Springer, 2005).

Finally, there are many improvements and corrections suggested to meby readers. Special thanks go to France Dacar, Didier Henrion, DavidKramer, Nat Kuhn, Tristan Needham, Peter Ross, John Snygg, Paul Stan-ford, Roland van der Veen, and Hung-Hsi Wu for these, and to my sonRobert and my wife, Elaine, for their tireless proofreading.

I also thank the University of San Francisco for giving me the opportu-nity to teach the courses on which much of this book is based, and MonashUniversity for the use of their facilities while revising it.

John StillwellMonash University and the University of San Francisco

March 2010

Preface to the Second Edition

This edition has been completely retyped in LATEX, and many of the figuresredone using the PSTricks package, to improve accuracy and make revisioneasier in the future. In the process, several substantial additions have beenmade.

• There are three new chapters, on Chinese and Indian number theory,on hypercomplex numbers, and on algebraic number theory. Thesefill some gaps in the first edition and give more insight into laterdevelopments.

• There are many more exercises. This, I hope, corrects a weakness ofthe first edition, which had too few exercises, and some that were toohard. Some of the monster exercises in the first edition, such as theone in Section 2.2 comparing volume and surface area of the icosa-hedron and dodecahedron, have now been broken into manageableparts. Nevertheless, there are still a few challenging questions forthose who want them.

• Commentary has been added to the exercises to explain how theyrelate to the preceding section, and also (when relevant) how theyforeshadow later topics.

• The index has been given extra structure to make searching easier.To find Euler’s work on Fermat’s last theorem, for example, one nolonger has to look at 41 different pages under “Euler.” Instead, onecan find the entry “Euler, and Fermat’s last theorem” in the index.

• The bibliography has been redone, giving more complete publica-tion data for many works previously listed with little or none. I havefound the online catalogue of the Burndy Library of the Dibner In-stitute at MIT helpful in finding this information, particularly for

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early printed works. For recent works I have made extensive use ofMathSciNet, the online version of Mathematical Reviews.

There are also many small changes, some prompted by recent mathe-matical events, such as the proof of Fermat’s last theorem. (Fortunately,this one did not force a major rewrite, because the background theory ofelliptic curves was covered in the first edition.)

I thank the many friends, colleagues, and reviewers who drew my at-tention to faults in the first edition, and helped me in the process of revision.Special thanks go to the following people.

• My sons, Michael and Robert, who did most of the typing, and mywife, Elaine, who did a great deal of the proofreading.

• My students in Math 310 at the University of San Francisco, whotried out many of the exercises, and to Tristan Needham, who invitedme to USF in the first place.

• Mark Aarons, David Cox, Duane DeTemple, Wes Hughes, ChristineMuldoon, Martin Muldoon, and Abe Shenitzer, for corrections andsuggestions.

John StillwellMonash UniversityVictoria, Australia

2001

Preface to the First Edition

One of the disappointments experienced by most mathematics students isthat they never get a course on mathematics. They get courses in calculus,algebra, topology, and so on, but the division of labor in teaching seems toprevent these different topics from being combined into a whole. In fact,some of the most important and natural questions are stifled because theyfall on the wrong side of topic boundary lines. Algebraists do not discussthe fundamental theorem of algebra because “that’s analysis” and analystsdo not discuss Riemann surfaces because “that’s topology,” for example.Thus if students are to feel they really know mathematics by the time theygraduate, there is a need to unify the subject.

This book aims to give a unified view of undergraduate mathematics byapproaching the subject through its history. Since readers should have hadsome mathematical experience, certain basics are assumed and the mathe-matics is not developed formally as in a standard text. On the other hand,the mathematics is pursued more thoroughly than in most general historiesof mathematics, because mathematics is our main goal and history onlythe means of approaching it. Readers are assumed to know basic calcu-lus, algebra, and geometry, to understand the language of set theory, and tohave met some more advanced topics such as group theory, topology, anddifferential equations. I have tried to pick out the dominant themes of thisbody of mathematics, and to weave them together as strongly as possibleby tracing their historical development.

In doing so, I have also tried to tie up some traditional loose ends. Forexample, undergraduates can solve quadratic equations. Why not cubics?They can integrate 1/

√1 − x2 but are told not to worry about 1/

√1 − x4.

Why? Pursuing the history of these questions turns out to be very fruitful,leading to a deeper understanding of complex analysis and algebraic ge-ometry, among other things. Thus I hope that the book will be not only a

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bird’s-eye view of undergraduate mathematics but also a glimpse of widerhorizons.

Some historians of mathematics may object to my anachronistic use ofmodern notation and (fairly) modern interpretations of classical mathemat-ics. This has certain risks, such as making the mathematics look simplerthan it really was in its time, but the risk of obscuring ideas by cumber-some, unfamiliar notation is greater, in my opinion. Indeed, it is practicallya truism that mathematical ideas generally arise before there is notation orlanguage to express them clearly, and that ideas are implicit before theybecome explicit. Thus the historian, who is presumably trying to be bothclear and explicit, often has no choice but to be anachronistic when tracingthe origins of ideas.

Mathematicians may object to my choice of topics, since a book ofthis size is necessarily incomplete. My preference has been for topics withelementary roots and strong interconnections. The major themes are theconcepts of number and space: their initial separation in Greek mathemat-ics, their union in the geometry of Fermat and Descartes, and the fruitsof this union in calculus and analytic geometry. Certain important topicsof today, such as Lie groups and functional analysis, are omitted on thegrounds of their comparative remoteness from elementary roots. Others,such as probability theory, are mentioned only briefly, as most of their de-velopment seems to have occurred outside the mainstream. For any otheromissions or slights I can only plead personal taste and a desire to keep thebook within the bounds of a one- or two-semester course.

The book has grown from notes for a course given to senior undergrad-uates at Monash University over the past few years. The course was ofhalf-semester length and a little over half the book was covered (Chapters1–11 one year and Chapters 5–15 another year). Naturally I will be de-lighted if other universities decide to base a course on the book. There isplenty of scope for custom course design by varying the periods or topicsdiscussed. However, the book should serve equally well as general readingfor the student or professional mathematician.

Biographical notes have been inserted at the end of each chapter, partlyto add human interest but also to help trace the transmission of ideas fromone mathematician to another. These notes have been distilled mainly fromsecondary sources, the Dictionary of Scientific Biography (DSB) normallybeing used in addition to the sources cited explicitly. I have followed theDSB’s practice of describing the subject’s mother by her maiden name.

Preface to the First Edition xiii

References are cited in the name (year) form, for example, Newton (1687)refers to the Principia, and the references are collected at the end of thebook.

The manuscript has been read carefully and critically by John Crossley,Jeremy Gray, George Odifreddi, and Abe Shenitzer. Their comments haveresulted in innumerable improvements, and any flaws remaining may bedue to my failure to follow all their advice. To them, and to Anne-MarieVandenberg for her usual excellent typing, I offer my sincere thanks.

John StillwellMonash UniversityVictoria, Australia

1989

Contents

Preface to the Third Edition vii

Preface to the Second Edition ix

Preface to the First Edition xi

1 The Theorem of Pythagoras 11.1 Arithmetic and Geometry . . . . . . . . . . . . . . . . . . 21.2 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . 41.3 Rational Points on the Circle . . . . . . . . . . . . . . . . 61.4 Right-Angled Triangles . . . . . . . . . . . . . . . . . . . 91.5 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . 111.6 The Definition of Distance . . . . . . . . . . . . . . . . . 131.7 Biographical Notes: Pythagoras . . . . . . . . . . . . . . 15

2 Greek Geometry 172.1 The Deductive Method . . . . . . . . . . . . . . . . . . . 182.2 The Regular Polyhedra . . . . . . . . . . . . . . . . . . . 202.3 Ruler and Compass Constructions . . . . . . . . . . . . . 252.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . 282.5 Higher-Degree Curves . . . . . . . . . . . . . . . . . . . 312.6 Biographical Notes: Euclid . . . . . . . . . . . . . . . . . 35

3 Greek Number Theory 373.1 The Role of Number Theory . . . . . . . . . . . . . . . . 383.2 Polygonal, Prime, and Perfect Numbers . . . . . . . . . . 383.3 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . 413.4 Pell’s Equation . . . . . . . . . . . . . . . . . . . . . . . 443.5 The Chord and Tangent Methods . . . . . . . . . . . . . . 48

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3.6 Biographical Notes: Diophantus . . . . . . . . . . . . . . 50

4 Infinity in Greek Mathematics 534.1 Fear of Infinity . . . . . . . . . . . . . . . . . . . . . . . 544.2 Eudoxus’s Theory of Proportions . . . . . . . . . . . . . . 564.3 The Method of Exhaustion . . . . . . . . . . . . . . . . . 584.4 The Area of a Parabolic Segment . . . . . . . . . . . . . . 634.5 Biographical Notes: Archimedes . . . . . . . . . . . . . . 66

5 Number Theory in Asia 695.1 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . 705.2 The Chinese Remainder Theorem . . . . . . . . . . . . . 715.3 Linear Diophantine Equations . . . . . . . . . . . . . . . 745.4 Pell’s Equation in Brahmagupta . . . . . . . . . . . . . . 755.5 Pell’s Equation in Bhaskara II . . . . . . . . . . . . . . . 785.6 Rational Triangles . . . . . . . . . . . . . . . . . . . . . . 815.7 Biographical Notes: Brahmagupta and Bhaskara . . . . . . 84

6 Polynomial Equations 876.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 Linear Equations and Elimination . . . . . . . . . . . . . 896.3 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . 926.4 Quadratic Irrationals . . . . . . . . . . . . . . . . . . . . 956.5 The Solution of the Cubic . . . . . . . . . . . . . . . . . . 976.6 Angle Division . . . . . . . . . . . . . . . . . . . . . . . 996.7 Higher-Degree Equations . . . . . . . . . . . . . . . . . . 1016.8 Biographical Notes: Tartaglia, Cardano, and Viete . . . . . 103

7 Analytic Geometry 1097.1 Steps Toward Analytic Geometry . . . . . . . . . . . . . . 1107.2 Fermat and Descartes . . . . . . . . . . . . . . . . . . . . 1117.3 Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . 1127.4 Newton’s Classification of Cubics . . . . . . . . . . . . . 1157.5 Construction of Equations, Bezout’s Theorem . . . . . . . 1187.6 The Arithmetization of Geometry . . . . . . . . . . . . . 1207.7 Biographical Notes: Descartes . . . . . . . . . . . . . . . 122

Contents xvii

8 Projective Geometry 1278.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 1288.2 Anamorphosis . . . . . . . . . . . . . . . . . . . . . . . . 1318.3 Desargues’s Projective Geometry . . . . . . . . . . . . . . 1328.4 The Projective View of Curves . . . . . . . . . . . . . . . 1368.5 The Projective Plane . . . . . . . . . . . . . . . . . . . . 1418.6 The Projective Line . . . . . . . . . . . . . . . . . . . . . 1448.7 Homogeneous Coordinates . . . . . . . . . . . . . . . . . 1478.8 Pascal’s Theorem . . . . . . . . . . . . . . . . . . . . . . 1508.9 Biographical Notes: Desargues and Pascal . . . . . . . . . 153

9 Calculus 1579.1 What Is Calculus? . . . . . . . . . . . . . . . . . . . . . . 1589.2 Early Results on Areas and Volumes . . . . . . . . . . . . 1599.3 Maxima, Minima, and Tangents . . . . . . . . . . . . . . 1629.4 The Arithmetica Infinitorum of Wallis . . . . . . . . . . . 1649.5 Newton’s Calculus of Series . . . . . . . . . . . . . . . . 1679.6 The Calculus of Leibniz . . . . . . . . . . . . . . . . . . . 1709.7 Biographical Notes: Wallis, Newton, and Leibniz . . . . . 172

10 Infinite Series 18110.1 Early Results . . . . . . . . . . . . . . . . . . . . . . . . 18210.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . 18510.3 An Interpolation on Interpolation . . . . . . . . . . . . . . 18810.4 Summation of Series . . . . . . . . . . . . . . . . . . . . 18910.5 Fractional Power Series . . . . . . . . . . . . . . . . . . . 19110.6 Generating Functions . . . . . . . . . . . . . . . . . . . . 19210.7 The Zeta Function . . . . . . . . . . . . . . . . . . . . . . 19510.8 Biographical Notes: Gregory and Euler . . . . . . . . . . 197

11 The Number Theory Revival 20311.1 Between Diophantus and Fermat . . . . . . . . . . . . . . 20411.2 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . 20711.3 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . 21011.4 Rational Right-Angled Triangles . . . . . . . . . . . . . . 21111.5 Rational Points on Cubics of Genus 0 . . . . . . . . . . . 21511.6 Rational Points on Cubics of Genus 1 . . . . . . . . . . . 21811.7 Biographical Notes: Fermat . . . . . . . . . . . . . . . . . 222

xviii Contents

12 Elliptic Functions 22512.1 Elliptic and Circular Functions . . . . . . . . . . . . . . . 22612.2 Parameterization of Cubic Curves . . . . . . . . . . . . . 22612.3 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . 22812.4 Doubling the Arc of the Lemniscate . . . . . . . . . . . . 23012.5 General Addition Theorems . . . . . . . . . . . . . . . . 23212.6 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . 23412.7 A Postscript on the Lemniscate . . . . . . . . . . . . . . . 23612.8 Biographical Notes: Abel and Jacobi . . . . . . . . . . . . 237

13 Mechanics 24313.1 Mechanics Before Calculus . . . . . . . . . . . . . . . . . 24413.2 The Fundamental Theorem of Motion . . . . . . . . . . . 24613.3 Kepler’s Laws and the Inverse Square Law . . . . . . . . . 24913.4 Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . 25313.5 Mechanical Curves . . . . . . . . . . . . . . . . . . . . . 25513.6 The Vibrating String . . . . . . . . . . . . . . . . . . . . 26113.7 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 26513.8 Biographical Notes: The Bernoullis . . . . . . . . . . . . 267

14 Complex Numbers in Algebra 27514.1 Impossible Numbers . . . . . . . . . . . . . . . . . . . . 27614.2 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . 27614.3 Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . 27714.4 Wallis’s Attempt at Geometric Representation . . . . . . . 27914.5 Angle Division . . . . . . . . . . . . . . . . . . . . . . . 28114.6 The Fundamental Theorem of Algebra . . . . . . . . . . . 28514.7 The Proofs of d’Alembert and Gauss . . . . . . . . . . . . 28714.8 Biographical Notes: d’Alembert . . . . . . . . . . . . . . 291

15 Complex Numbers and Curves 29515.1 Roots and Intersections . . . . . . . . . . . . . . . . . . . 29615.2 The Complex Projective Line . . . . . . . . . . . . . . . . 29815.3 Branch Points . . . . . . . . . . . . . . . . . . . . . . . . 30115.4 Topology of Complex Projective Curves . . . . . . . . . . 30415.5 Biographical Notes: Riemann . . . . . . . . . . . . . . . . 308

Contents xix

16 Complex Numbers and Functions 31316.1 Complex Functions . . . . . . . . . . . . . . . . . . . . . 31416.2 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . 31816.3 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . 31916.4 Double Periodicity of Elliptic Functions . . . . . . . . . . 32216.5 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . 32516.6 Uniformization . . . . . . . . . . . . . . . . . . . . . . . 32916.7 Biographical Notes: Lagrange and Cauchy . . . . . . . . . 331

17 Differential Geometry 33517.1 Transcendental Curves . . . . . . . . . . . . . . . . . . . 33617.2 Curvature of Plane Curves . . . . . . . . . . . . . . . . . 34017.3 Curvature of Surfaces . . . . . . . . . . . . . . . . . . . . 34317.4 Surfaces of Constant Curvature . . . . . . . . . . . . . . . 34417.5 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 34617.6 The Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . 34817.7 Biographical Notes: Harriot and Gauss . . . . . . . . . . . 352

18 Non-Euclidean Geometry 35918.1 The Parallel Axiom . . . . . . . . . . . . . . . . . . . . . 36018.2 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . 36318.3 Geometry of Bolyai and Lobachevsky . . . . . . . . . . . 36518.4 Beltrami’s Projective Model . . . . . . . . . . . . . . . . 36618.5 Beltrami’s Conformal Models . . . . . . . . . . . . . . . 36918.6 The Complex Interpretations . . . . . . . . . . . . . . . . 37418.7 Biographical Notes: Bolyai and Lobachevsky . . . . . . . 378

19 Group Theory 38319.1 The Group Concept . . . . . . . . . . . . . . . . . . . . . 38419.2 Subgroups and Quotients . . . . . . . . . . . . . . . . . . 38719.3 Permutations and Theory of Equations . . . . . . . . . . . 38919.4 Permutation Groups . . . . . . . . . . . . . . . . . . . . . 39319.5 Polyhedral Groups . . . . . . . . . . . . . . . . . . . . . 39519.6 Groups and Geometries . . . . . . . . . . . . . . . . . . . 39819.7 Combinatorial Group Theory . . . . . . . . . . . . . . . . 40119.8 Finite Simple Groups . . . . . . . . . . . . . . . . . . . . 40419.9 Biographical Notes: Galois . . . . . . . . . . . . . . . . . 409

xx Contents

20 Hypercomplex Numbers 41520.1 Complex Numbers in Hindsight . . . . . . . . . . . . . . 41620.2 The Arithmetic of Pairs . . . . . . . . . . . . . . . . . . . 41720.3 Properties of + and × . . . . . . . . . . . . . . . . . . . . 41920.4 Arithmetic of Triples and Quadruples . . . . . . . . . . . 42120.5 Quaternions, Geometry, and Physics . . . . . . . . . . . . 42420.6 Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . 42820.7 Why C, H, and O Are Special . . . . . . . . . . . . . . . . 43020.8 Biographical Notes: Hamilton . . . . . . . . . . . . . . . 433

21 Algebraic Number Theory 43921.1 Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . 44021.2 Gaussian Integers . . . . . . . . . . . . . . . . . . . . . . 44221.3 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . 44521.4 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44821.5 Ideal Factorization . . . . . . . . . . . . . . . . . . . . . 45221.6 Sums of Squares Revisited . . . . . . . . . . . . . . . . . 45421.7 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . 45721.8 Biographical Notes: Dedekind, Hilbert, and Noether . . . 459

22 Topology 46722.1 Geometry and Topology . . . . . . . . . . . . . . . . . . 46822.2 Polyhedron Formulas of Descartes and Euler . . . . . . . 46922.3 The Classification of Surfaces . . . . . . . . . . . . . . . 47122.4 Descartes and Gauss–Bonnet . . . . . . . . . . . . . . . . 47422.5 Euler Characteristic and Curvature . . . . . . . . . . . . . 47722.6 Surfaces and Planes . . . . . . . . . . . . . . . . . . . . . 47922.7 The Fundamental Group . . . . . . . . . . . . . . . . . . 48422.8 The Poincare Conjecture . . . . . . . . . . . . . . . . . . 48622.9 Biographical Notes: Poincare . . . . . . . . . . . . . . . . 492

23 Simple Groups 49523.1 Finite Simple Groups and Finite Fields . . . . . . . . . . . 49623.2 The Mathieu Groups . . . . . . . . . . . . . . . . . . . . 49823.3 Continuous Groups . . . . . . . . . . . . . . . . . . . . . 50123.4 Simplicity of SO(3) . . . . . . . . . . . . . . . . . . . . . 50523.5 Simple Lie Groups and Lie Algebras . . . . . . . . . . . . 50923.6 Finite Simple Groups Revisited . . . . . . . . . . . . . . . 51323.7 The Monster . . . . . . . . . . . . . . . . . . . . . . . . . 515

Contents xxi

23.8 Biographical Notes: Lie, Killing, and Cartan . . . . . . . . 518

24 Sets, Logic, and Computation 52524.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52624.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 52824.3 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 53124.4 Axiom of Choice and Large Cardinals . . . . . . . . . . . 53424.5 The Diagonal Argument . . . . . . . . . . . . . . . . . . 53624.6 Computability . . . . . . . . . . . . . . . . . . . . . . . . 53824.7 Logic and Godel’s Theorem . . . . . . . . . . . . . . . . 54124.8 Provability and Truth . . . . . . . . . . . . . . . . . . . . 54624.9 Biographical Notes: Godel . . . . . . . . . . . . . . . . . 549

25 Combinatorics 55325.1 What Is Combinatorics? . . . . . . . . . . . . . . . . . . 55425.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . 55725.3 Analysis and Combinatorics . . . . . . . . . . . . . . . . 56025.4 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . 56325.5 Nonplanar Graphs . . . . . . . . . . . . . . . . . . . . . . 56725.6 The Konig Infinity Lemma . . . . . . . . . . . . . . . . . 57125.7 Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . . 57525.8 Hard Theorems of Combinatorics . . . . . . . . . . . . . 58025.9 Biographical Notes: Erdos . . . . . . . . . . . . . . . . . 584

Bibliography 589

Index 629