The Mathematical Universe

William Dunham


Many children begin reading with an alphabet book. Comfortably seated on a warm lap, youngsters listen as the alphabet unwinds from "A is for alligator" to "Z is for zebra." Such books may not be great literature, but they provide an effective introduction to letters, words, and language.

Echoing those alphabet readers of childhood, this volume surveys the discipline of mathematics in a series of essays running from A to Z. The content is considerably more sophisticated-D now stands for differential calculus rather than doggie-and a warm lap is no longer mandatory. But the basic idea of an alphabetical journey remains.

Such a format imposes severe restrictions upon a book meant to be read from cover to cover. Mathematical topics, after all, do not align themselves in a logical progression so as to mirror the Latin alphabet. Consequently, the transition between chapters is sometimes abrupt. Moreover, although certain letters of the alphabet are rich in possible subjects, others are quite barren, a situation reminiscent of children's alphabet books in which "C is for cat" but "X is for xenurus." As the reader will readily note, some of the following topics have been shoehorned into position, much as a size 16 foot must be squeezed into a size 8 boot. Devising a topical flow to coincide with the alphabetical flow has presented a true logistical challenge.

The book begins with the (apparently) simple subject of arithmetic. Later chapters have recurring and often interlaced themes. Sometimes consecutive chapters fit together, as with Chapters G, H, and I on geometry or the back-to-back Chapters K and L on seventeenth-century rivals Isaac Newton and Gottfried Wilhelm Leibniz. Some chapters focus on single mathematicians: We meet Euler in Chapter E, Fermat in Chapter F, and Bertrand Russell in Chapter R. Some describe specific results, such as the isoperimetric problem or Archimedes' determination of the surface area of a sphere. Some address such broader issues as the mathematical personality or the presence of women in the discipline. And whatever the subject, each chapter provides a strong dose of history.

Along the way, the major branches of mathematics-from algebra, to geometry, to probability, to calculus-make at least a brief appearance. Those sections designed to explain key mathematical ideas have the flavor of an informal textbook, and actual proofs (or at least "prooflets") appear here and there. Chapters D and L, for instance, introduce differential and integral calculus, and thus carry a bit more mathematical baggage.

In most chapters, however, there has been a deliberate attempt to avoid an overtly technical development. Virtually all of the mathematics is elementary-that is, pitched to those with some high school algebra and geometry under their belts. Professional mathematicians will find few surprises on these pages. The book is aimed at those whose interest in mathematics is at least as extensive as their training.

A few themes return again and again: that mathematics is an ancient yet vital subject, that it treats matters of everyday importance as well as matters of no utility whatever, and that it is a discipline whose remarkable breadth is matched only by its equally remarkable depth. To convey something of this in a sequence of alphabetically ordered chapters is the book's objective.

I would be remiss not to mention John Allen Paulos's book Beyond Numeracy(Knopf, New York, 1991), which he described as "in part a dictionary, in part a collection of short mathematical essays, and in part the ruminations of a numbers man." Paulos's lively volume charted a mathematical course from A to Z-in his case, from algebra to Zeno. By allowing multiple entries for some letters, he achieved greater breadth of coverage; by writing fewer but longer essays, I opted for greater depth. It is my hope that our two books can coexist peacefully as variations of the same alphabetical format.

Of course, there is no way an author can address every key point, introduce every important figure, or consider every pressing mathematical issue. Choices must be made at each turn, choices constrained by the demands of internal consistency, by the complexity of the subject, by the author's interest and expertise, and by the utterly artificial arrangement of the alphabet. A project of this type will omit a thousand times more than it can include, and scores of potential topics are bound to end up on the cutting room word processor.

In the end, this book is the response of a single individual to the immense mathematical universe. It represents one of countless journeys that could have been undertaken by countless authors, and I make no claim to having followed the comprehensive or definitive route from A to Z.

Such qualifications aside, I hope these chapters provide at least a glimpse into a subject of endless fascination. As the nineteenth-century mathematician Sofia Kovalevskaia observed, "Many who have never had the occasion to discover more about mathematics confuse it with arithmetic and consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination." And perhaps the book can serve to underscore Proclus' lofty sentiment from fifth-century Greece: "mathematics alone can revive and awaken the soul... to the vision of being, can turn her from images to realities and from darkness to the light of the intellect."


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