In Search of Infinity 1995th Edition by N.Ya. Vilenkin (PDF)

Description

The concept of infinity is one of the most important, and at the same time, one of the most mysterious concepts of science. Already in antiquity many philosophers and mathematicians pondered over its contradictory nature. In mathematics, the contradictions connected with infinity intensified after the creation, at the end of the 19th century, of the theory of infinite sets and the subsequent discovery, soon after, of paradoxes in this theory. At the time, many scientists ignored the paradoxes and used set theory extensively in their work, while others subjected set-theoretic methods in mathematics to harsh criticism. The debate intensified when a group of French mathematicians, who wrote under the pseudonym of Nicolas Bourbaki, tried to erect the whole edifice of mathematics on the single notion of a set. Some mathematicians greeted this attempt enthusiastically while others regarded it as an unnecessary formalization, an attempt to tear mathematics away from life-giving practical applications that sustain it. These differences notwithstanding, Bourbaki has had a significant influence on the evolution of mathematics in the twentieth century. In this book we try to tell the reader how the idea of the infinite arose and developed in physics and in mathematics, how the theory of infinite sets was constructed, what paradoxes it has led to, what significant efforts have been made to eliminate the resulting contradictions, and what routes scientists are trying to find that would provide a way out of the many difficulties.

User’s Reviews

Editorial Reviews: Review “In fewer than 150 pages Vilenkin has packed a whole course on infinity: its history, philosophy, and mathematical theory. Abe Shenitzer’s translation is very smooth and natural… When fractals have gone out of style, and chaos is no longer the flavor of the month, some kind of ‘geometric set theory’ or ‘set-theoretic geometry’ may very well remain, and In Search of Infinity may still be the best introduction to it.”–American Mathematical Monthly From the Back Cover The concept of infinity has been for hundreds of years one of the most fascinating and elusive ideas to tantalize the minds of scholars and lay people alike. The theory of infinite sets lies at the heart of much of mathematics, yet is has produced a series of paradoxes that have led many scholars to doubt the soundness of it foundations. The author of this book presents a popular-level account of the roads followed by human thought in attempts to understand the idea of the infinite in mathematics and physics. In doing so, he brings to the general reader a deep insight into the nature of the problem and its importance to an understanding of our world. “When I read the first edition of the book, about 20 years ago, I was carried away by Vilenkin’s storytelling and his ability to bring subtle mathematical ideas down to earth. He stretches our imagination and educates our intuition…The second edition improves on the first by omission of some routine material about finite properties of sets, and increased attention to infinity. There is a wealth of new material on the position of infinity in human thought, from philosophy to physics, and also on its role in the history of mathematics. In particular, there are now biographical notes on over 100 mathematicians.Abe Shenitzer’s elegant translation makes this a rare work of literature—a serious mathematical book that will be read from over to cover.” —John Stillwell, Monash University, Australia

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As I pored over this captivating exploration into one of the deepest topics of mathematics and philosophy, I thought often of how Vilenkin’s smart and breezy book would make the perfect companion to Rudy Rucker’s magisterial

⭐. Rucker is bullish on infinity and takes more of a set-theoretic approach to the subject (unsurprising, since set theory is where most mathematical work pertaining to infinity is done). Vilenkin is much more ready to show how pernicious infinity can be when unleashed on the field of mathematical analysis (which informs most of his work) – infinite procedures have wreaked havoc on formerly-uncontroversial notions of continuity, differentiability, and dimension. But on the flip side, these stresses have led analysts to better understandings of these notions, the process culminating (in this account) in the idea of Lebesgue measure. This is a sort of analogue to the account in Stilwell’s

⭐, which details how recurring questions about the nature of numbers and arithmetic led to a widening of the horizons of mathematics and to more intricate, abstract, and far-reaching theory.After some preliminary fulminating, the meat of the book begins with V’s own version of the “Hilbert’s Hotel” story, told as a science-fiction tale from the viewpoint of a flamboyant space pilot who packs the (countably) infinitely-many rooms of the fantastic hotel with ever more late arrivals. This is a standard parable that math wonks employ to illustrate the hunt to find one-to-one correspondences between the integers and other infinite sets. Eventually, a wave of guests arrive that cannot be accommodated, demonstrating the existence of uncountably-infinite sets. From there, infinitely many succesively-smaller holes are cut from a carpet so that it seems that there can be no carpet left (but there is) and kinks are introduced into curves until the “curve” seems to consist solely of “kinks”.If the exuberant prose of this book is basis for judgement, I’d be proud to study under this man.If you read and enjoyed Rucker’s book, you’ll probably appreciate how this book supplements and furthers the intellectual adventure. If you’ve read neither, but enjoy witty and well-informed tours of big ideas, get both and clear your calendar.

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