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The Princeton Companion to Mathematics

Timothy Gowers, editor
Princeton University Press
Publication Date: 
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
David P. Roberts
, on

Mathematics is rapidly growing as old subfields are deepened and new subfields are created. It is simultaneously integrating, as direct connections are found between subfields previously regarded as distant. It is dramatically increasing its role in other disciplines, both in science and beyond. These are exciting times for mathematics, and it is more important than ever that we mathematicians have a clear global view of our discipline.

However we are still rather stuck in our old ways. Our habits do not help us cultivate a broad view of mathematics, in either our students or ourselves. The demands of the traditional curriculum focus our teaching on narrow topics, such as how to integrate rational functions or how to diagonalize matrices. Our research efforts are much more likely to result in publications if we stay focused within our narrow areas of expertise.

The Princeton Companion to Mathematics aims to improve this situation. It is a monumental work aimed at readers ranging from undergraduate math majors to established researchers. Its goal is to assist these readers in cultivating a global view of mathematics. It thus strives to help individuals grow in a way that parallels the enormous growth of mathematics. The editor and driving force behind PCM is Fields medalist Timothy Gowers. He is assisted by almost 200 experts. Together they succeed completely.

Organization. PCM is divided into eight parts:

 Part  Pages  Articles  Average




I: Introduction 76 4 19.0   Logical
II: The Origins of Modern Mathematics 80 7 11.4   Chronological
III: Mathematical Concepts 158 99 1.6   Alphabetical
IV: Branches of Mathematics 366 26 14.1   Thematic
V: Theorems and Problems 52 35 1.5   Alphabetical
VI: Mathematicians 94 96 1.0   Chronological
VII: The Influence of Mathematics 128 14 9.1   Thematic
VIII: Final Perspectives 60 7 8.6   Random

As indicated by the table, the parts are quite different from one another.

The nature of the book is described in detail in the preface. The central focus is on modern pure mathematics. Indeed the focus of Parts I, III, IV, and V is very modern and pure. However Parts II and VI form a large historical component and Part VII represents applied mathematics. It is convincingly argued that these parts are necessary for balance: besides being subjects in their own right, mathematical history and applied mathematics provide important perspectives on modern pure mathematics.

A fundamental priority of the book is accessibility. The goal is to discuss a given mathematical idea at the "lowest level that is practical." To obtain this lowest level, examples and intuition are emphasized, and exposition is kept informal. The priority of maximum accessibility required "interventionist editing" throughout the six-year process of creating the book. Maximum accessibility is indeed evident in the final product. One of its consequences is that different sections are written at different levels.

Another priority is that the book should be much more than a collection of separate articles. One way this is achieved is by judicious use of cross-references, around five to a page. Another way is by the careful overall organization. Part I, for example, consists of material that is "part of the necessary background of all mathematicians rather than belonging to one specific area." Similarly, "the reflections of Part VIII are a sort of epilogue, and therefore an appropriate way for the book to sign off."

Parts I and VIII. Parts I and VIII are the most accessible parts. Part I is written entirely by Timothy Gowers and can best be described as an expert's overview of a solid undergraduate curriculum. The seriousness of the undertaking and the comprehensiveness of the coverage is clear from some of the subsection titles:

Sets; Functions; Relations; Binary Operations; Logical Connectives; Quantifiers; Negation; Free and Bound Variables; The Natural Numbers; The Integers; The Rational Numbers; The Real Numbers; The Complex Numbers; Groups; Fields; Vector Spaces, Rings; Substructures; Products; Quotients; Homomorphisms, Isomorphisms, and Automorphisms; Linear Maps and Matrices; Eigenvalues and Eigenvectors; Limits; Continuity; Differentiation; Partial Differential Equations; Integration; Holomorphic Functions; Geometry and Symmetry Groups; Euclidean Geometry; Affine Geometry; Topology; Spherical Geometry; Hyperbolic Geometry; Projective Geometry; Lorentz Geometry; Manifolds and Differential Geometry; Riemannian Metrics.

The tone throughout is remarkably gentle, given the rapidly changing material. The last article, The General Goals of Mathematical Research, would be of particular interest to aspiring mathematicians. It, like all of Part I, is especially well-balanced. For example, it includes a long discussion about the relative place of rigorous and nonrigorous reasoning in mathematics. This discussion is not in the least a call to devalue rigor, but it is highly respectful of nonrigorous reasoning. It concludes, "The best way to describe the situation is perhaps to say that the two styles of argument have profoundly benefited each other and will undoubtedly continue to do so."

Part VIII's articles are very different from one another. Michael Harris's "Why Mathematics?" You Might Ask is very philosophical. Hilbert S. Wilf's Mathematics: An Experimental Science is a succinct and elegant argument for the value of computers in pure mathematics. VIII.6 is Advice to a Young Mathematician, with separate sections written by Atiyah, Bollobás, Connes, McDuff, and Sarnak. Adrian Rice's A Chronology of Mathematical Events, concludes the book by a five-page summary of the history of mathematics. Part VIII, in its liveliness and subjectivity, illustrates one of the points made strongly in the preface: PCM is a companion, not an encyclopedia.

Parts II and VI. Parts II and VI consist of historical material. The first six articles of Part II are historical surveys on broad topics: numbers, geometry, algebra, algorithms, rigor in analysis, and proof. The last article focuses on a shorter period, the "crisis in the foundations of mathematics" in the first third of the twentieth century.

The ninety-six short articles of Part VI are each about a single mathematician, except for VI.18 on the Bernoullis and VI.96 on Bourbaki. The focus here is on contributions prior to 1950, and the delicate choice of which mathematicians to include seems impeccable. For example, the sixteen mathematicians included who were born before 1650 are Pythagoras, Euclid, Archimedes, Apollonius, Al-Khwarizmi, Fibonacci, Cardano, Bombelli, Viète, Stevin, Descartes, Fermat, Pascal, Newton, and Leibniz. Similarly, eight mathematicians are given special prominence by the inclusion of a portrait: Descartes, Newton, Leibniz, Euler, Gauss, Riemann, Poincaré, and Hilbert.

Parts III and V. Part III consists of short articles on concepts. The emphasis on intuition and examples is clear everywhere. Terence Tao's article on compactness and compactification is illustrative of this style. It begins by carefully discussing how finite sets and infinite sets are different. It goes on to discuss, with reference to the unit interval [0,1], how some topological spaces behave very much like finite sets. It is these spaces that one would like to call compact. Only after all this preparation does the formal definition appear, that a space is compact exactly when all its open covers have finite subcovers. In just a few paragraphs, the reader is given a rather refined appreciation for this definition and some of its various near-equivalents. Similarly, the part of the article on compactification goes rather far, but is gently guided by considering the real line and its two most familiar compactifications: the extended line \([-\infty,\infty]\) and the projective line \(\mathbb{R}\cup\{\infty\}\).

Part V is similar to Part III except the focus is shifted to theorems and problems. Most of the articles are again on topics that play an extremely important role in mathematics: the central limit theorem is central to our understanding of data; the uniformization theorem is central to our understanding of Riemann surfaces; the resolvability of singularities is central to our understanding of algebraic varieties. Given the all-star nature of the list, readers will be particularly enticed by the articles on unfamiliar topics. For professional mathematicians, the level is generally non-technical and welcoming.

Part IV and VII. Part IV, on branches of mathematics, is described in the preface as "the heart of the book." The branches are a well-chosen sampling, including a healthy dose of mathematical physics of various sorts. Probabilistic Models of Critical Phenomena by Gordon Slade is representative of Part IV. It is as gentle as possible on the reader but goes deeply into its topic. An early example in this article, easier than the main examples later, involves branching processes. Suppose individuals in a certain population have zero, one, or two children with respective probabilities \((1-p)^2\), \(2p(1 – p)\), and \(p^2\). The average number of children is \(2p\). Accordingly, one can expect that for \(p < .5\) a given individual's descendants will eventually die out whereas for \(p > .5\) there is positive probability that the descendants will never die out. In fact, for \(p\) slightly smaller than \(.5\), the expected number of total descendants is approximately \(.5(.5 - p)^\gamma\) with \(\gamma=1\).. For \(p\) slightly larger than \(.5\), the chance that an individual will have infinitely many descendants is approximately \(8(p - .5)^\beta\) with \(\beta=1\). The "critical exponents" \(\gamma=1\) and \(\beta=1\) are remarkably stable: one can modify the branching process in a great many ways and the final formulas still have the same exponents. The idea of critical exponents has an amazing universality. For example, one has analogous quantities for percolation of fluids through porous materials and for ferromagnetism. In dimension two, a recent result is that \((\gamma,\beta)=(43/18,5/36)\) for percolation and \((\gamma,\beta)=(7/4, 1/8)\) for ferromagnetism. In dimensions greater than two, there are predictions from experiment but rarely rigorous confirmation.

Part VII illustrates how mathematics influences other fields. Chemistry, biology, engineering, computer science, economics, statistics, medicine, philosophy, music, and art are all represented by at least one article. A short and particularly intriguing article is Mathematical Statistics by Persi Diaconis. It explains how common uses of averages and least-squares estimators are sometimes inappropriate, and more sophisticated concepts need to be used instead. Many readers are likely to find these replacements quite counterintuitive. In general, the articles in Parts IV and VII present the most challenging reading in PCM.

Conclusion. Page 1 of PCM begins, "It is notoriously hard to give a satisfactory answer to the question, What is Mathematics?" Indeed, it is not reasonable to try to capture what mathematics is in a short paragraph in the style of a mathematical definition. On the other hand, "What is Mathematics?" is surely a fundamental question all of us must answer in our own way. PCM gives us very valuable support in trying to come up with our own answers. It is unprecedented in its signature combination of depth and accessibility. It takes us beyond our own experience in teaching and research, and lets us share in the experience of many experts. It gives us a balanced and broad overview of mathematics in one single volume. For readers of MAA Online, there is no better way to invest $75 than to buy the Princeton Companion to Mathematics.

David Roberts is a professor of mathematics at the University of Minnesota, Morris.

Preface ix
Contributors xvii

Part I Introduction
I.1 What Is Mathematics About? 1
I.2 The Language and Grammar of Mathematics 8
I.3 Some Fundamental Mathematical Definitions 16
I.4 The General Goals of Mathematical Research 48

Part II The Origins of Modern Mathematics
II.1 From Numbers to Number Systems 77
II.2 Geometry 83
II.3 The Development of Abstract Algebra 95
II.4 Algorithms 106
II.5 The Development of Rigor in Mathematical Analysis 117
II.6 The Development of the Idea of Proof 129
II.7 The Crisis in the Foundations of Mathematics 142

Part III Mathematical Concepts
III.1 The Axiom of Choice 157
III.2 The Axiom of Determinacy 159
III.3 Bayesian Analysis 159
III.4 Braid Groups 160
III.5 Buildings 161
III.6 Calabi-Yau Manifolds 163
III.7 Cardinals 165
III.8 Categories 165
III.9 Compactness and Compactification 167
III.10 Computational Complexity Classes 169
III.11 Countable and Uncountable Sets 170
III.12 C*-Algebras 172
III.13 Curvature 172
III.14 Designs 172
III.15 Determinants 174
III.16 Differential Forms and Integration 175
III.17 Dimension 180
III.18 Distributions 184
III.19 Duality 187
III.20 Dynamical Systems and Chaos 190
III.21 Elliptic Curves 190
III.22 The Euclidean Algorithm and Continued Fractions 191
III.23 The Euler and Navier-Stokes Equations 193
III.24 Expanders 196
III.25 The Exponential and Logarithmic Functions 199
III.26 The Fast Fourier Transform 202
III.27 The Fourier Transform 204
III.28 Fuchsian Groups 208
III.29 Function Spaces 210
III.30 Galois Groups 213
III.31 The Gamma Function 213
III.32 Generating Functions 214
III.33 Genus 215
III.34 Graphs 215
III.35 Hamiltonians 215
III.36 The Heat Equation 216
III.37 Hilbert Spaces 219
III.38 Homology and Cohomology 221
III.39 Homotopy Groups 221
III.40 The Ideal Class Group 221
III.41 Irrational and Transcendental Numbers 222
III.42 The Ising Model 223
III.43 Jordan Normal Form 223
III.44 Knot Polynomials 225
III.45 K-Theory 227
III.46 The Leech Lattice 227
III.47 L-Functions 228
III.48 Lie Theory 229
III.49 Linear and Nonlinear Waves and Solitons 234
III.50 Linear Operators and Their Properties 239
III.51 Local and Global in Number Theory 241
III.52 The Mandelbrot Set 244
III.53 Manifolds 244
III.54 Matroids 244
III.55 Measures 246
III.56 Metric Spaces 247
III.57 Models of Set Theory 248
III.58 Modular Arithmetic 249
III.59 Modular Forms 250
III.60 Moduli Spaces 252
III.61 The Monster Group 252
III.62 Normed Spaces and Banach Spaces 252
III.63 Number Fields 254
III.64 Optimization and Lagrange Multipliers 255
III.65 Orbifolds 257
III.66 Ordinals 258
III.67 The Peano Axioms 258
III.68 Permutation Groups 259
III.69 Phase Transitions 261
III.70 p 261
III.71 Probability Distributions 263
III.72 Projective Space 267
III.73 Quadratic Forms 267
III.74 Quantum Computation 269
III.75 Quantum Groups 272
III.76 Quaternions, Octonions, and Normed Division Algebras 275
III.77 Representations 279
III.78 Ricci Flow 279
III.79 Riemann Surfaces 282
III.80 The Riemann Zeta Function 283
III.81 Rings, Ideals, and Modules 284
III.82 Schemes 285
III.83 The Schrödinger Equation 285
III.84 The Simplex Algorithm 288
III.85 Special Functions 290
III.86 The Spectrum 294
III.87 Spherical Harmonics 295
III.88 Symplectic Manifolds 297
III.89 Tensor Products 301
III.90 Topological Spaces 301
III.91 Transforms 303
III.92 Trigonometric Functions 307
III.93 Universal Covers 309
III.94 Variational Methods 310
III.95 Varieties 313
III.96 Vector Bundles 313
III.97 Von Neumann Algebras 313
III.98 Wavelets 313
III.99 The Zermelo-Fraenkel Axioms 314

Part IV Branches of Mathematics
IV.1 Algebraic Numbers 315
IV.2 Analytic Number Theory 332
IV.3 Computational Number Theory 348
IV.4 Algebraic Geometry 363
IV.5 Arithmetic Geometry 372
IV.6 Algebraic Topology 383
IV.7 Differential Topology 396
IV.8 Moduli Spaces 408
IV.9 Representation Theory 419
IV.10 Geometric and Combinatorial Group Theory 431
IV.11 Harmonic Analysis 448
IV.12 Partial Differential Equations 455
IV.13 General Relativity and the Einstein Equations 483
IV.14 Dynamics 493
IV.15 Operator Algebras 510
IV.16 Mirror Symmetry 523
IV.17 Vertex Operator Algebras 539
IV.18 Enumerative and Algebraic Combinatorics 550
IV.19 Extremal and Probabilistic Combinatorics 562
IV.20 Computational Complexity 575
IV.21 Numerical Analysis 604
IV.22 Set Theory 615
IV.23 Logic and Model Theory 635
IV.24 Stochastic Processes 647
IV.25 Probabilistic Models of Critical Phenomena 657
IV.26 High-Dimensional Geometry and Its Probabilistic Analogues 670

Part V Theorems and Problems
V.1 The ABC Conjecture 681
V.2 The Atiyah-Singer Index Theorem 681
V.3 The Banach-Tarski Paradox 684
V.4 The Birch-Swinnerton-Dyer Conjecture 685
V.5 Carleson's Theorem 686
V.6 The Central Limit Theorem 687
V.7 The Classification of Finite Simple Groups 687
V.8 Dirichlet's Theorem 689
V.9 Ergodic Theorems 689
V.10 Fermat's Last Theorem 691
V.11 Fixed Point Theorems 693
V.12 The Four-Color Theorem 696
V.13 The Fundamental Theorem of Algebra 698
V.14 The Fundamental Theorem of Arithmetic 699
V.15 Gödel's Theorem 700
V.16 Gromov's Polynomial-Growth Theorem 702
V.17 Hilbert's Nullstellensatz 703
V.18 The Independence of the Continuum Hypothesis 703
V.19 Inequalities 703
V.20 The Insolubility of the Halting Problem 706
V.21 The Insolubility of the Quintic 708
V.22 Liouville's Theorem and Roth's Theorem 710
V.23 Mostow's Strong Rigidity Theorem 711
V.24 The P versus NP Problem 713
V.25 The Poincaré Conjecture 714
V.26 The Prime Number Theorem and the Riemann Hypothesis 714
V.27 Problems and Results in Additive Number Theory 715
V.28 From Quadratic Reciprocity to Class Field Theory 718
V.29 Rational Points on Curves and the Mordell Conjecture 720
V.30 The Resolution of Singularities 722
V.31 The Riemann-Roch Theorem 723
V.32 The Robertson-Seymour Theorem 725
V.33 The Three-Body Problem 726
V.34 The Uniformization Theorem 728
V.35 The Weil Conjectures 729

Part VI Mathematicians
VI.1 Pythagoras (ca. 569 B.C.E.-ca. 494 B.C.E.) 733
VI.2 Euclid (ca. 325 B.C.E.-ca. 265 B.C.E.) 734
VI.3 Archimedes (ca. 287 B.C.E.-212 B.C.E.) 734
VI.4 Apollonius (ca. 262 B.C.E.-ca. 190 B.C.E.) 735
VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800-847) 736
VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170-ca. 1250) 737
VI.7 Girolamo Cardano (1501-1576) 737
VI.8 Rafael Bombelli (1526-after 1572) 737
VI.9 François Viète (1540-1603) 737
VI.10 Simon Stevin (1548-1620) 738
VI.11 René Descartes (1596-1650) 739
VI.12 Pierre Fermat (160?-1665) 740
VI.13 Blaise Pascal (1623-1662) 741
VI.14 Isaac Newton (1642-1727) 742
VI.15 Gottfried Wilhelm Leibniz (1646-1716) 743
VI.16 Brook Taylor (1685-1731) 745
VI.17 Christian Goldbach (1690-1764) 745
VI.18 The Bernoullis (fl. 18th century) 745
VI.19 Leonhard Euler (1707-1783) 747
VI.20 Jean Le Rond d'Alembert (1717-1783) 749
VI.21 Edward Waring (ca. 1735-1798) 750
VI.22 Joseph Louis Lagrange (1736-1813) 751
VI.23 Pierre-Simon Laplace (1749-1827) 752
VI.24 Adrien-Marie Legendre (1752-1833) 754
VI.25 Jean-Baptiste Joseph Fourier (1768-1830) 755
VI.26 Carl Friedrich Gauss (1777-1855) 755
VI.27 Siméon-Denis Poisson (1781-1840) 757
VI.28 Bernard Bolzano (1781-1848) 757
VI.29 Augustin-Louis Cauchy (1789-1857) 758
VI.30 August Ferdinand Möbius (1790-1868) 759
VI.31 Nicolai Ivanovich Lobachevskii (1792-1856) 759
VI.32 George Green (1793-1841) 760
VI.33 Niels Henrik Abel (1802-1829) 760
VI.34 János Bolyai (1802-1860) 762
VI.35 Carl Gustav Jacob Jacobi (1804-1851) 762
VI.36 Peter Gustav Lejeune Dirichlet (1805-1859) 764
VI.37 William Rowan Hamilton (1805-1865) 765
VI.38 Augustus De Morgan (1806-1871) 765
VI.39 Joseph Liouville (1809-1882) 766
VI.40 Eduard Kummer (1810-1893) 767
VI.41 Évariste Galois (1811-1832) 767
VI.42 James Joseph Sylvester (1814-1897) 768
VI.43 George Boole (1815-1864) 769
VI.44 Karl Weierstrass (1815-1897) 770
VI.45 Pafnuty Chebyshev (1821-1894) 771
VI.46 Arthur Cayley (1821-1895) 772
VI.47 Charles Hermite (1822-1901) 773
VI.48 Leopold Kronecker (1823-1891) 773
VI.49 Georg Friedrich Bernhard Riemann (1826-1866) 774
VI.50 Julius Wilhelm Richard Dedekind (1831-1916) 776
VI.51 Émile Léonard Mathieu (1835-1890) 776
VI.52 Camille Jordan (1838-1922) 777
VI.53 Sophus Lie (1842-1899) 777
VI.54 Georg Cantor (1845-1918) 778
VI.55 William Kingdon Clifford (1845-1879) 780
VI.56 Gottlob Frege (1848-1925) 780
VI.57 Christian Felix Klein (1849-1925) 782
VI.58 Ferdinand Georg Frobenius (1849-1917) 783
VI.59 Sofya (Sonya) Kovalevskaya (1850-1891) 784
VI.60 William Burnside (1852-1927) 785
VI.61 Jules Henri Poincaré (1854-1912) 785 

VI.62 Giuseppe Peano (1858-1932) 787
VI.63 David Hilbert (1862-1943) 788
VI.64 Hermann Minkowski (1864-1909) 789
VI.65 Jacques Hadamard (1865-1963) 790
VI.66 Ivar Fredholm (1866-1927) 791
VI.67 Charles-Jean de la Vallée Poussin (1866-1962) 792
VI.68 Felix Hausdorff (1868-1942) 792
VI.69 Élie Joseph Cartan (1869-1951) 794
VI.70 Emile Borel (1871-1956) 795
VI.71 Bertrand Arthur William Russell (1872-1970) 795
VI.72 Henri Lebesgue (1875-1941) 796
VI.73 Godfrey Harold Hardy (1877-1947) 797
VI.74 Frigyes (Frédéric) Riesz (1880-1956) 798
VI.75 Luitzen Egbertus Jan Brouwer (1881-1966) 799
VI.76 Emmy Noether (1882-1935) 800
VI.77 Wac?aw Sierpinski (1882-1969) 801
VI.78 George Birkhoff (1884-1944) 802
VI.79 John Edensor Littlewood (1885-1977) 803
VI.80 Hermann Weyl (1885-1955) 805
VI.81 Thoralf Skolem (1887-1963) 806
VI.82 Srinivasa Ramanujan (1887-1920) 807
VI.83 Richard Courant (1888-1972) 808
VI.84 Stefan Banach (1892-1945) 809
VI.85 Norbert Wiener (1894-1964) 811
VI.86 Emil Artin (1898-1962) 812
VI.87 Alfred Tarski (1901-1983) 813
VI.88 Andrei Nikolaevich Kolmogorov (1903-1987) 814
VI.89 Alonzo Church (1903-1995) 816
VI.90 William Vallance Douglas Hodge (1903-1975) 816
VI.91 John von Neumann (1903-1957) 817
VI.92 Kurt Gödel (1906-1978) 819
VI.93 André Weil (1906-1998) 819
VI.94 Alan Turing (1912-1954) 821
VI.95 Abraham Robinson (1918-1974) 822
VI.96 Nicolas Bourbaki (1935-) 823

Part VII The Influence of Mathematics
VII.1 Mathematics and Chemistry 827
VII.2 Mathematical Biology 837
VII.3 Wavelets and Applications 848
VII.4 The Mathematics of Traffic in Networks 862
VII.5 The Mathematics of Algorithm Design 871
VII.6 Reliable Transmission of Information 878
VII.7 Mathematics and Cryptography 887
VII.8 Mathematics and Economic Reasoning 895
VII.9 The Mathematics of Money 910
VII.10 Mathematical Statistics 916
VII.11 Mathematics and Medical Statistics 921
VII.12 Analysis, Mathematical and Philosophical 928
VII.13 Mathematics and Music 935
VII.14 Mathematics and Art 944

Part VIII Final Perspectives
VIII.1 The Art of Problem Solving 955
VIII.2 "Why Mathematics?" You Might Ask 966
VIII.3 The Ubiquity of Mathematics 977
VIII.4 Numeracy 983
VIII.5 Mathematics: An Experimental Science 991
VIII.6 Advice to a Young Mathematician 1000
VIII.7 A Chronology of Mathematical Events 1010

Index 1015