The Art of the Infinite: The Pleasures of Mathematics

by Robert Kaplan, Ellen Kaplan

Overview

Robert Kaplan's The Nothing That Is: A Natural History of Zero was an international best-seller, translated into eight languages. The Times called it "elegant, discursive, and littered with quotes and allusions from Aquinas via Gershwin to Woolf" and The Philadelphia Inquirer praised it as "absolutely scintillating."
In this delightful new book, Robert Kaplan, writing together with his wife Ellen Kaplan, once again takes us on a witty, literate, and accessible tour of the world of mathematics. Where The Nothing That Is looked at math through the lens of zero, The Art of the Infinite takes infinity, in its countless guises, as a touchstone for understanding mathematical thinking. Tracing a path from Pythagoras, whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret (the existence of irrational numbers); through Descartes and Leibniz; to the brilliant, haunted Georg Cantor, who proved that infinity can come in different sizes, the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics. The Kaplans guide us through the "Republic of Numbers," where we meet both its upstanding citizens and more shadowy dwellers; and we travel across the plane of geometry into the unlikely realm where parallel lines meet. Along the way, deft character studies of great mathematicians (and equally colorful lesser ones) illustrate the opposed yet intertwined modes of mathematical thinking: the intutionist notion that we discover mathematical truth as it exists, and the formalist belief that math is true because we invent consistent rules for it.
"Less than All," wrote William Blake, "cannot satisfy Man." The Art of the Infinite shows us some of the ways that Man has grappled with All, and reveals mathematics as one of the most exhilarating expressions of the human imagination.

Product Details

ISBN-13:	9780195176063
Publisher:	Oxford University Press
Publication date:	11/18/2004
Edition description:	New Edition
Pages:	336
Product dimensions:	8.90(w) x 5.50(h) x 0.90(d)

About the Author

Robert and Ellen Kaplan are the founders of The Math Circle, a school, open to anyone of any age, that teaches the enjoyment of mathematics. They have been invited to lecture on mathematics teaching and the Math Circle to organizations such as the American Mathematical Society and universities in Spain and Switzerland. They live in Cambridge, Massachusetts.

Table of Contents

Acknowledgments ix

An Invitation 1

Chapter 1 Time and the Mind 5

Chapter 2 How Do We Hold These Truths? 37

Chapter 3 Designs on a Locked Chest 71

Interlude: The Infinite and the Indefinite 93

Chapter 4 Skipping Stones 95

Chapter 5 Euclid Alone 123

Interlude: Longing and the Infinite 165

Chapter 6 The Eagle of Algebra 167

Chapter 7 Into the Highlands 209

Interlude: The Infinite and the Unknown 249

Chapter 8 Back of Beyond 251

Interlude: The Infinite There-But the Finite Here 283

Chapter 9 The Abyss 287

Appendix 329

Bibliography 391

Index 393

Editorial Reviews

Robert Kaplan can make even nullity interesting. His The Nothing That Is: A Natural History of Zero confounded the numerically challenged and became an international bestseller. In The Art of the Infinite, he and his wife, Ellen, extend that success to the nth degree, providing a elegant, often playful excursion into the mathematics of infinity. They explain that in the Republic of Numbers, the attempt to grasp the ungraspable is an essential activity.

In The Art of the Infinite, Robert and Ellen Kaplan take us on a grand tour, leading us from the terra firma of the simple counting numbers (one, two, three, four and so on) through the discovery of the rationals, the irrationals, the negatives and the complex numbers that combine the ordinary, or real, numbers and the imaginaries to generate a two dimensional number-space known as the complex plane. It is here that the famous Mandelbrot set lives, that enigmatic emblem of chaos theory. — Margaret Wertheim

The Los Angeles Times

While Kaplan (The Nothing That Is: A Natural History of Zero) and his wife intend this volume to delight the numerophobic into seeing the beauty in math, the "art" they describe is hidden in a thicket of dry proofs. And yet they've written a lovely and erudite history of the subject in spite of that, one that will absorb anyone who already fancies numbers and all their possibilities. Hand-drawn diagrams accompany dense explanatory prose in this exploration of infinity, as the authors chart mathematical discoveries and great thinkers throughout history. Frequent references to luminaries from the humanities (Shakespeare, Baudelaire, Gaudi, Robert Graves) would earn this book comfortable shelving in a liberal arts library if the math weren't so devilishly hard to grasp. (A typical passage compares the way great changes happen in mathematics with the way important figures enter the action in Proust.) The authors acknowledge that even math basics can be tricky: that the product of two negatives is a positive, for instance, is a puzzle that the Kaplans say "put too many people off math forever, convinced that its dicta were arbitrary or spiteful." The authors write that "[m]athematics is permanent revolution," and indeed, some may find their heads spinning. Nevertheless, a patient reader who loves thinking about thinking will be rewarded by the book's end; by the final pages, he or she will have personally experienced, via these diagrams and problems, many of the great discoveries in mathematics. Graphs and illustrations throughout. (Apr.) Copyright 2003 Reed Business Information.

Publishers Weekly

The Kaplans are founders of the Math Circle, a school that teaches the enjoyment of mathematics, and Robert is the author of The Nothing That Is: A Natural History of Zero. In this new book, the authors cover some of the elements of such areas as plane geometry, algebra, and trigonometry. The contents are chosen and arranged so as to lead into a concluding discussion of Georg Cantor's remarkable discoveries/inventions concerning the nature of infinity. All of this is related in a cheerful conversational tone with frequent allusions to, and quotations from, many other fields of knowledge, including literature, history, and philosophy. At times, a meander through a different discipline distracts from the main argument, but overall the Kaplans' approach makes for very enjoyable reading. This volume should appeal to a broad spectrum of readers interested in learning more about the beauty of mathematics. Recommended for all public and academic libraries.-Jack W. Weigel, Ann Arbor, MI Copyright 2003 Reed Business Information.

Library Journal

Adult/High School-This exploration of mathematics and its history involves plot and characters as well as numbers; the stories of the thinkers who were challenged by its mysteries and discovered its principles range from tragic to laugh-out-loud funny. Copyright 2004 Reed Business Information.

School Library Journal

"Robert and Ellen Kaplan clearly relish the chance to expound the beauty of their subject, in prose that performs some glorious turns. They mix weighty but approachable maths with imagery and allusion, beginning with number and heading persuasively into the unknown. As the awesome presence of those infinite infinities finally takes hold, the mind reels and hairs stand on end. This is mathematics for the soul—just the way it should be."—New Scientist

"Anyone interested in a serious introduction to mathematics will delight in this volume. The Kaplans' background in languages and linguistics inclines them to a depth of literary allusion that few writers in any technical field can match. Robert Kaplan's prior book, 'The Nothing That Is: A Natural History of Zero,' remains, for my money, the best popular mathematics book ever written."—Margaret Wertheim, Los Angeles Times Book Review

"Guides the reader through some extremely difficult mathematical ideas in ways that are both imaginative and diverting. Mathematics is often said to be the science of the infinite; the Kaplans want us to appreciate mathematics as the art of the infinite, an art which involves invention, narrative and an inexhaustible pursuit of variations on themes."—London Review of Books

"Very enjoyable reading.... Related in a cheerful conversational tone with frequent allusions to, and quotations from, many other fields of knowledge, including literature, history, and philosophy.... This volume should appeal to a broad spectrum of readers interested in learning more about the beauty of mathematics."—Library Journal

"This is a mathematics with a plot and characters, as well as diagrams and formulas. These accounts vary from tragic to laugh-out-loud funny. Those who love math won't want to miss this one, and those who would like to love it but never have should give the book a try."—School Library Journal

From the Publisher