Science popularization books generally fall into two categories. The first is chiefly concerned with the technical achievements in a particular field, often with an emphasis on the author's own theories and conjectures. These books serve to give professionals an idea of where their work is headed, but are often unintelligible to the general public. Other authors, sensing that readers will be less intimidated when the human element is included, concern themselves with the historical drama of science and the personal lives of scientific innovators. Unfortunately, these often read like: "Albert Einstein did poorly in grade school, escaped Nazi Germany, developed a charming wit, and then suddenly---e = mc2!"
Nowhere is this problem more acute than in the realm of mathematics. Not only is it difficult to make the search for the next prime number seem as socially relevant or dramatic as the search for cold fusion, but there is a sense that truly important mathematics and endless technical details go hand in hand. In The Mathematical Universe, author William Dunham surmounts these problems to create an enjoyable and provoking book for all levels of expertise.
In his previous book, Journey Through Genius, Dunham takes the reader on a historical ride through some of the great theorems from antiquity to the present. While it is a fantastic book, one of Dunham's main aims was to have the reader understand, as well as appreciate, the proofs to the theorems discussed. The reader who is put off by a page half-full of symbols may have trouble getting past page 14. In The Mathematical Universe, a more light-handed approach is taken, and the continuity of the essays is not seriously affected if the reader's eyes tend to become unfocused when looking at square roots and triangles. Further, the book consists of essays (labelled A to Z) that are more or less independent, allowing daunted readers to avoid painful memories of high-school algebra.
Together, the essays are intended as a survey of the mathematical world: its flavour, people, achievements, controversies, methods and history. While such a short book could not hope to be a very comprehensive survey, Dunham succeeds in selecting topics that span a great deal of material and history. Each essay is delivered with wit and insight, appealing to a broad spectrum of reader without being simplistic or condescending.
What separates Dunham's book from other math popularization books such as James Gleick's popular Chaos, is that it is an explanation and defence of mathematics by a mathematician. While it may seem like an indulgence in trivialities (and elitism) to distinguish Dunham from "mere" journalists such as Gleick, it is important because math is a hard sell to those who are not already buying. The relevance or appeal of their work is difficult to grasp since, unlike engineers and biologists, mathematicians are generally concerned with objects whose relation to the real world is tenuous at best. Applications for their work in, say, building better bridges or faster computers often come (if ever) to the complete surprise of mathematicians whose only concern was the beauty hidden within their diagrams and formulae.
It makes sense, then, that someone who seeks this beauty out as part of their job should try to explain this beauty. What compels people to stand entranced in front of a blackboard while their bodies and social lives waste away? Keeping his theories to himself, Dunham opts to show, rather than tell, the reader what the fuss is all about. He does this by presenting a handful of simple and elegant theorems, and by setting the stage for several other interesting and historically significant problems.
Dunham also sets out to convey the flavour and character of mathematics as an endeavour. Math is not a collection of equations which appeared out of nowhere; the directions it has taken throughout history reflect and affect the societies that have embraced it, and often tell us about some of the smartest people the world has ever known.
Concerning the current state of mathematics, little is said. This must be in part a wise move for an introductory book, as twentieth century math is marked by extreme abstraction and formalism. However, a passage is dedicated to how computers are changing the way math is done, and the "W" essay is devoted to the changing role of women in math.Together, the essays combine to form a highly readable, enjoyable book that does justice to an undervalued field, written by a person from a notoriously reclusive and uncommunicative discipline.
Day Kirby is a staff member of Duthie Books on Robson, a student of mathematics and computer science, and a director of Mathmania, a non-profit society for mathematics popularization and education.