Overview

How can the infinite, a subject so remote from our finite experience, be an everyday working tool for the working mathematician? Blending history, philosophy, mathematics and logic, Shaughan Lavine answers this question with clarity. An account of the origins of the modern mathematical theory of the infinite, his book is also a defense against the attacks and misconceptions that have dogged this theory since its introduction in the late 19th century. With his development of set theory in the 1880s, Georg Cantor introduced the infinite into mathematics. But his theory, both critics and supporters have charged, was subject to paradoxes proceeding from Cantor's naive intuitions , and this verdict has had an enormous impact on the philosophy of mathematics. Lavine effectively reverses this charge by showing that set theory is in fact an excellent example of the posititve and necessary role of intuition in mathematics. His history, moving from Greek geometry through the development of calculus to the evolution of set theory, ultimately leads to the crux of the issue: the source of our intuitions concerning the infinite. Along the way, he offers a careful and critical discussion of differing views across the philosophical spectrum. Making use of the mathematical work of Jan Mycielski, formerly accessible only to logicians, Lavine demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge. He shows that the source of our intuitions concerning Cantor's infinite, as a matter of historical and psychological fact, is extrapolation from ordinary experience of the indefinitely large.

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How, in a finite world, does one obtain any knowledge about the infinite? Lavine argues that intuitions about the infinite derive from facts about the finite mathematics of indefinitely large size...The issues are delicate, but the writing is crisp and exciting, the arguments original. This book should interest readers whether philosophically, historically, or mathematically inclined, and large parts are within the grasp of the general reader. Highly recommended.--D. V. Feldman Choice

Understanding the Infinite is a remarkable blend of mathematics, modern history, philosophy, and logic, laced with refreshing doses of common sense. It is a potted history of, and a philosophical commentary on, the modern notion of infinity as formalized in axiomatic set theory...An amazingly readable [book] given the difficult subject matter. Most of all, it is an eminently sensible book. Anyone who wants to explore the deep issues surrounding the concept of infinity...will get a great deal of pleasure from it. -- Ian Stewart New Scientist

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