The Princeton Companion to Mathematics

A valuable companion for professional mathematicians as well as students because it provides an astute and thoughtful perspective

Most people are familiar with the word “mathematics” and may use it frequently. However, if asked what mathematics is, they could have very different answers. To many, mathematics is the study of numbers, magnitudes, and the relationships between them. Those with a little training in formal mathematics may say that mathematics consists of rigorous deductions of theorems from axioms. To scientists and engineers, mathematics is a toolbox of techniques which can be used to analyze complex real-life problems. Then there are the aesthetes, to whom mathematics is like an art—a domain where beauty can be discovered in its purest and most abstract form. The truth, however, is that there is no simple, satisfactory answer to the question of what mathematics is. Even among mathematicians, disagreement over what constitutes good mathematics, and how it should be done, is frequent.

Thus, to address this question, it is necessary to try and understand the subject matter of mathematics. The trouble is that mathematics has become so vast that it is no longer easy for a person to get even a peripheral understanding of all its branches. The Princeton Companion to Mathematics (henceforth referred to as The Companion) may be viewed as a thousand-page long answer to this question.

The Companion is a snapshot of mathematics as it stands today, taken from the point of view of professional mathematicians. In a little more than one thousand pages, it undertakes to explain the background, the tools, the branches, the achievements, and the ethos of mathematics. It is a carefully edited compilation of the contributions of about 130 authors. The chief editor, Timothy Gowers (University of Cambridge) and the associate editors, June Barrow-Green (The Open University) and Imre Leader (University of Cambridge) have succeeded in the gargantuan task of assembling this coherent whole from the contributions of mathematicians who have very different styles and perspectives.

The Companion is divided into “Parts”, each of which is a collection of articles. The table of contents for Part III lists 99 major mathematical concepts, including, for example, one-and-a-half pages on the number $\pi$, and half-a-dozen pages on the Ricci flow. But at the heart of The Companion lies Part IV, on “Branches of Mathematics”, covering 26 branches of mathematics, each discussed in ten to twenty pages. Each branch is surveyed by a specialist in that area. Mathematicians who try to explain their research to non-specialized audiences know how hard it is to give an accessible account. Therefore, it is very impressive that The Companion succeeds in doing exactly this for every major active area in contemporary mathematics, without going much beyond mathematics at the M.Sc. level. Even without such a background, it is possible for a reader to get some understanding of each branch of mathematics, including the motivating problems, the important examples, the techniques used, the connections with other areas, and the important open problems.

The Companion is a snapshot of mathematics as it stands today, taken from the point of view of professional mathematicians

This is followed by a Part with 35 articles on theorems and problems, each a page or two in length. Each article addresses either one major theorem, conjecture, open problem, or a closely related collection. Then there is a Part on mathematicians—all deceased, with the exception of Nicolas Bourbaki, who is not a person but a collective pseudonym that a group of mathematicians, mainly French, used in their efforts to rebuild mathematics on more abstract, rigorous foundations. This list of 96 names is arranged in chronological order, starting with Pythagoras and ending with Bourbaki. (Surprisingly, there is only one Indian mathematician in this list—Srinivasa Ramanujan.)

Next appears a Part on “The Influence of Mathematics”. Besides the expected discussion of how mathematics influences chemistry, biology, and technology, this section also has articles on the influence of mathematics on music and art. Given that modern physical theories are invariably expressed in the language of mathematics, it is a surprise that there is no mention of physics in this section. Perhaps the association is so obvious that it does not need pointing out. After all, general relativity and the Einstein equations appear in Part IV among the branches of mathematics. On the other hand, how mathematics is used in the analysis of traffic in networks, which is one of the topics discussed in Part VII, is not something that everyone is aware of.

The last Part of The Companion, titled “Final Perspectives”, is an eclectic collection of opinions and musings, which cannot be characterized in any simple terms. For example, there is an article by the late Herbert Wilf—who worked on combinatorics and graph theory at the University of Pennsylvania—expressing a point of view that is yet to catch on in a large part of the mathematical community. It is called “Mathematics: An Experimental Science”. Here, Wilf discusses the use of computers as an aid to mathematical experimentation. His proselytization is all the more convincing as it is backed up with concrete techniques, tools, and problems that have yielded to such attacks. He gives, for instance, a formula for $\pi$:

$$\pi = \sum_{i=0}^\infty \frac{1}{16^i}\left(\frac{4}{8i+1} - \frac{2}{8i+4} - \frac{1}{8i+5} - \frac{1}{8i+6}\right)$$

due to Bailey, Borwein, and Plouffe which was experimentally “discovered” using computer programs. Using this formula, it is possible to compute a single digit in the hexadecimal expansion of $\pi$ without computing all the previous digits. For example, the trillionth digit of $\pi$ can be computed in less time than would conceivably be needed to calculate the first trillion digits.

The collection of articles under the title “Advice to a Young Mathematician” is a rich source of wisdom, often expressed in lyrical language. For example, Béla Bollobas, of whom both Tim Gowers and Imre Leader were doctoral students, says that a career in mathematics is not something to be embarked upon lightly:

Do mathematics only if you are passionate about it, only if you would do it even if you had to find the time for it after a full day’s work in another job. Like poetry, mathematics is not an occupation but a vocation.

Alain Connes, a French mathematician who was a member of Bourbaki, writes about the unity of mathematics:

It might be tempting at first to regard mathematics as a collection of separate branches, such as geometry, algebra, analysis, number theory, etc., where the first is dominated by the attempt to understand the concept of “space”, the second by the art of manipulating symbols, the third by access to “infinity” and the “continuum” and so on.
This, however, does not do justice to one of the most important features of the mathematical world, namely, that it is impossible to isolate any of the above parts from the others without depriving them of their essence. In this way the corpus of mathematics resembles a biological entity, which can only survive as a whole and which would perish if separated into disjoint pieces.

Peter Sarnak, who works on analytic number theory, writes on dealing with the serious demands that one often faces in mathematical research:

Doing research in mathematics is frustrating and if being frustrated is something you cannot get used to, then mathematics may not be an ideal occupation for you. Most of the time one is stuck, and if this is not the case for you, then either you are exceptionally talented or you are tackling problems that you knew how to solve before you started. There is room for some work of the latter kind, and it can be of a high quality, but most of the big breakthroughs are earned the hard way, with many false steps and long periods of little progress, or even negative progress. There are ways to make this aspect of research less unpleasant. Many people these days work jointly, which, besides the obvious advantage of bringing different expertise to bear on a problem, allows one to share the frustration. For most people this is a big positive (and in mathematics the corresponding sharing of the joy and credit on making a breakthrough has not, so far at least, led to many big fights in the way that it has in some other areas of science).

The overall style of writing in The Companion is the dispassionate, precise, and careful style that characterizes most mathematical exposition these days. However, within this constraint the editors have allowed—perhaps even encouraged—the diverse personalities of some of the contributors to shine through. As a result, many articles do not just provide dry technical accounts of mathematical topics, but also share deep perspectives which their authors have developed after many years of immersion in their respective subjects.

In the age of Wikipedia and omniscient internet search engines, encyclopaedic information has little value. Definitions, theorems, histories are easily accessible to anyone with an internet connection. However, The Companion goes beyond providing just information, by also providing astute and thoughtful perspective. While even the different contributors to The Companion may not agree upon many mathematical issues, each of them has a point of view that comes from a deep understanding of the subject.

The editors have succeeded in the gargantuan task of assembling this coherent whole from the contributions of mathematicians with very different styles and perspectives

The Companion is a valuable resource for mathematicians who wish to understand mathematics outside their own area of specialization. It is also the perfect reference for beginning postgraduate students to get a broader perspective on an area of mathematics that they are learning about, or to get a quick overview of a concept, an area or a theorem in mathematics

Amritanshu Prasad is at the Institute of Mathematical Sciences, Chennai.