The art of the infinite : the pleasures of mathematics

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. 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 the 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 intuitionist 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 "This is mathematics with a plot and characters, as well as diagrams and formulas. In the process of discussing numbers, natural and rational, real and complex, the Kaplans introduce readers to the historical figures who were challenged by their mysteries. The authors explore math in ways that will be new to students whose education has been confined to the classroom. Readers learn not only that a number can be squared, but also that it can be "triangled," and that the sum of two adjacent triangular numbers always makes a square one. The book shows how all the concepts of different types of numbers lead to the notion of infinity, and how one can prove things through geometry that would normally appear to have nothing to do with shapes and lines. Most of the math discussed can be followed by anyone with a smattering of algebra and geometry, and always it is accompanied by stories of how people first discovered the mathematical principles, with illustrations of the protagonists. 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 Acknowledgements / ix An Invitation / 1 Chapter One / Time and the mind / 3 Chapter Two / How do we hold these truths? / 29 Chapter Three / Designs on a locked chest / 56 Interlude / The Infinite and the definite / 75 Chapter Four / Skipping stones / 77 Chapter Five / Euclid Alone / 100 Interlude / Longing and the Infinite / 131 Chapter Six / The Eagle of the Algebra / 133 Chapter Seven / Into the Highlands / 167 Interlude / The Infinite and the Unknown / 200 Chapter Eight / Back of Beyond / 202 Chapter Nine / The Abyss / 228 Appendix / 263 Bibliography / 315 Index 317