The Princeton Companion to Mathematics

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:

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.