The Pea and the Sun: A Mathematical Paradox

by Leonard M. Wapner

Book Overview

Nehmen Sie einen Apfel und schneiden Sie ihn in f nf Teile. W rden Sie es f r m glich halten, dass Sie diese f nf Teile so zusammensetzen k nnen, dass Sie zwei pfel der gleichen Form und Gr e wie der... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 1568812132
• ISBN13: 9781568812137
• Release Date: April 2005
• Publisher: A K PETERS
• Length: 232 Pages
• Weight: 1.10 lbs.

Customer Reviews

Extraordinarily Successful and Accessible Look at Some Topics in Modern Mathematics

The central aim of the book is an understanding of the mathematics underpinning the Banach-Tarski Theorem (frequently mischaracterized as a "paradox"), a justifiably famous result that even many graduate students only hear about as a counterintuitive example without studying its details of proof. In accomplishing its objective, The Pea and the Sun successfully introduces the general, non-technically sophisticated reader to a variety of interrelated ideas and results in 20th century mathematics (set theory and measure theory) without oversimplifying the subject or patronizing the reader. Among the building blocks of the overall a treatment are an interesting discussion of genuine and spurious paradoxes that is interesting in its own right, a recapitulation of Cantor's methods of counting infinite sets, and a nice discussion of the Vitali non-measurable set, which in some ways is the kernel of the entire discussion. Remarkably, the discussion is self-contained, and the reader needs no special training in mathematics, only a willingness to follow the discussion and stretch his or her mind at each point. Moreover, it is well-written and manages to convey a flow of technical ideas lucidly without sacrificing a sense of their depth. This book is recommended for a very wide range of readers, all the way from bright high school students to graduate students in mathematics and teachers of technical subjects at all levels. A masterpiece of exposition.

It made my top ten list of best popular mathematics books

The Banach-Tarski paradox is a candidate for the most counter-intuitive mathematical result ever published. Basically, the conclusion of the theorem is that it is possible to decompose a small object into a finite number of pieces and then reconstruct the pieces a certain way to make two objects identical to the original. Phrased another way, it means that an object the size of a pea can be deconstructed into a finite set of objects that can be reconstructed in a manner to make an object the size of the sun. As bizarre as this sounds, the proof has stood up against all manner of assaults until there is no doubt that it is in fact true. Wapner does an excellent job in setting the mathematical, historical and philosophical groundwork for explaining the theorem. The book starts with a brief explanation of the lives of Alfred Tarski and Stefan Banach as well as others such as Georg Canto and Kurt Godel who helped create the mathematical framework. This is followed by a lengthy and thorough discussion of the mathematical background needed to understand the theorem and the proof. It begins at the very basic level, so very little mathematical knowledge is needed before you begin. The next step is the proof of the theorem, which by this time is very easy to understand. It is done step-by-step with not even the slightest "leap of faith." The final chapters deal with the consequences of the theorem. I found these chapters to be the most interesting in the book. In "Resolution", Wapner discussions the possible reactions to the theorem. They are: *) Declare the result fallacious. *) Accept the theorem at face value. *) Reinterpret the result. The first is not realistic as there is no longer any doubt that the theorem is true and the second is self-evident. Performing the mental gyrations necessary to accept the third option is the most interesting. Wapner resolves it by saying, "Yes, the theorem is true, but the actions needed to do something like duplicating a gold bar are not possible." Chapter 7, called "Real world" mentions some of the principles of quantum mechanics and how they can be related to the Banach-Tarski paradox. This book is one that will fascinate you, it proves in the mathematical sense what you "know" cannot be true in the real sense. It also demonstrates a fundamental problem of philosophy, which is to consider to what extent a mathematical result can be applied in the real world. I loved this book, it made my top ten list of best popular mathematics books. Published in Journal of Recreational Mathematics, reprinted with permission.

Actually proves the theorem

The book is wonderful because it actually proves the theorem in a way that a non-expert in mathematical foundations can actually understand. I wish all popular mathematics books were written at this level where the goal is to educate and entertain. Now as I suggested to the author all we need is a book like this one that will explain Godel's and Cohen's results on the independence of the continuum hypothesis.

A fascinating introduction to the Banach-Tarski Paradox

The Pea And The Sun: A Mathematical Paradox is a fascinating introduction to the Banach-Tarski Paradox, a mathematical riddle that asserts it could be possible to create something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again. Written to be accessible to lay readers and non-mathematicians, The Pea And The Sun outlines the history of the paradox, introduces readers to the basics of such matters as set theory, isometrics, scissors congruence and equidecomposability, and walks the reader through the theorem and proof that object duplication is indeed mathematically possible. But just because it is mathematically possible, is it physically possible? The highly counterintuitive nature of the mathematical theorem demands a critical response. A final chapter dwells on speculation as to how the Banach-Tarski Paradox may apply to the modern and future world. Written in a fresh, captivating, friendly style, The Pea And The Sun is remarkably engaging and will appeal to any reader with a discerning, inquisitive mind into the nature of the so-called impossible, regardless of their particular mathematical background.

Perfect blend of math, humor, and information for the layman as well as the math professional

This book gives an incisive look at a fascinating area of science. It is technical enough to hold the attention of the math whiz, while "gentle" enough to carry a complete layman along. I personally learned a great deal about this amazing paradox, and also about the world of higher mathamatics in general. Fascinating, but light hearted reading. Highly recommended for anyone with any interest in this type of field.