The Random Walks of George Pólya

George Pólya (1887–1985) was an eminently enigmatic personality. This is brought to the fore in this very readable, carefully researched and objective tribute to Pólya's great intellect and his varied scientific achievements. It is not easy to do justice to a person with so much depth and breadth, and whose contributions have been so useful in influencing mathematics and mathematicians, but Alexanderson has accomplished just that.

The book under review begins by painting a large canvas of the social and intellectual climate in Budapest at the end of the 19th century, touching upon the status of Jewish intellectuals. This intellectual tapestry is set up by creating a backdrop of the emergence of Budapest from its peasant past, into a modern, industrial society. The history of the family is carefully documented, with some subtle, yet vivid details of the nuances of being Jewish. The author then describes the early mathematical influences, the various positions held and the personalities Pólya encountered, in great detail.

Pólya's love and enjoyment of mathematics is brought across very clearly. His cheerfulness and glee at discovering new results, and then polishing them into something simple and elegant is illustrated throughout the book. Pólya comes across as a story-teller, an animated person with a subtle and refined sense of humor.

The title of the book is as much a literal reference as it is allegorical. Literal since he actually coined the term, and allegorical since that is essentially what his intellectual pursuits and early career resembled.

Chapters 1-4 present Pólya's childhood and formative years. Chapter 5 provides a detailed description of the Pólya-Szegö two-volume analysis, problem book, Aufgaben und Lehrsätze, and its influence on subsequent mathematical research. Chapters 5-9 concentrate on the events before, during and after his move to the United States. Chapters 10-12 present the latter part of his career, when he began to teach in summer institutes. After the youthful and energetic years, one is left with a distinctly sad feeling as Alexanderson describes Pólya's "Later Years", a feeling akin to that of reading the latter parts of Hardy's A Mathematician's Apology. The book itself touches on Pólya's various research interests and his general philosophy of teaching and doing mathematics. Including letters from many eminent mathematicians, and a generous sprinkling of anecdotes, the book provides a glimpse of Pólya through his interactions with the best mathematicians, and his honest perspectives on them. These include Lipót Fejér, G. H. Hardy, J. E. Littlewood, Gabör Szegö, Richard Courant, Donald Knuth, Karl Loewner, Felix Klein, David Hilbert, Adolf Hurwitz, Ludwig Bieberbach, Oswald Veblen and John von Neumann among others.

The list of references suggests further leads and Pólya's bibliography is enumerated in the order of the publication year, providing yet another view on the liveliness and vast scope of his simultaneous researches. To non-mathematicians (and maybe to many mathematicians too), he is perhaps best known for his books How to Solve It and Mathematics and Plausible Reasoning, but the bibliography makes it clear that his work was wide-ranging and influential.

Of the 14 Appendices, the first seven are by well-known mathematicians who provide an objective and non-technical review of particular and distinct aspects of Pólya's work. Appendix 8 is particularly fascinating: it lists the phrases (appearing in contemporary mathematics) that bear Pólya's name. Appendix 9 is a list of awards, honors and lectures dedicated to him. Appendices 10-13 are written by Pólya himself, appropriately having the last word. The last Appendix gives a list of all the personalities who encountered Pólya, along with a short description of each person.

Alexanderson is successful in depicting the many facets of Pólya's life and thought, masterfully weaving together anecdotes, articles, letters, photographs, book reviews and research problems. Without any direct reference (except toward the end), the author is also able to communicate Pólya's wife Stella's quiet support and strength of character.

Missing from the book is a portrait of Pólya's personal life and habits, but one could argue that this is not the author's objective anyway. The "Cast of Characters" might be more useful if the individuals' interactions with Pólya were mentioned briefly. But this information can still be obtained through the index.

This book is easily accessible and requires no prior knowledge or appreciation of mathematics. At the same time, it is a rich resource for serious students of mathematics, students of East-European intellectual history prior to the Second World War, historians of science and serious mathematicians alike. Although the research articles are widely available, this book provides an insight into the man behind these ideas, and thereby offers greater depth and perspective on the theorems themselves. This is a magnificent testament to the scientific legacy and versatility of George Pólya.

Manav Das (manav@louisville.edu) is Assistant Professor of Mathematics at the University of Louisville.