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The Pea and the Sun: A Mathematical Paradox

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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.

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.

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.

I'm admittedly "overqualified" for this book, but I enjoy reading math books for non-mathematicians for inspiration and breadth. The first chapter on the lives and history of the people and theorems repeats the same teasers a number of times. It would be better to intermix the math and the history.

The theme of the book is nice, though, and it is the only one I've read that really addresses lots of seeming paradoxes about infinity in a way anyone could appreciate. By the time it gets to Banach-Tarski, there have been so many very similar theorems and so many teasers that it is actually quite a let-down. The bit on decomposition puzzles was quite fun, though, so this book is worth at least checking out of the library and skimming.

And on a mathematical note, the book's characterization of the Axiom of Choice as something you either accept or not is a total misrepresentation - there are numerous intermediate axioms (dependent choice, countable choice, etc) that allow lots of useful results, and you just need to indicate when you use one of them.

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.

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.

wapner made it clear and shift the argument from formal to non-mathematitians level.
i loved the review of set theory and cantor mathematics about arithmatic of transfinite,but the thing that was not clear in this book is "axiom of choice"
i highly advice this book for lay person who love the essence of math,especially counter-intutive one.

This is what I would call a comedy math book. In it the author has taken a solidly provable mathematical theorem that makes no sense at all when you say it and used it as an example to describe some pretty elegant mathematics.

The area he talks about is given in the title. Of course if you're in the know it's called the Banach-Tarski theorem. It says that you can take a solid object like for instance a pea. You then cut it into pieces and you can then reassemble the pieces and get something as big as the sun.

Obviously untrue, but you can prove it mathematically. Of course you can mathematically prove that a bumblebee can't fly. The bumblebee doesn't know math, so he just keeps on flying. There are a lot of mathematical things, E=MC-Squared for instance, that just can't possibly be true.

Written for the non-mathematician, this book is a discussion on the history of the B-T theorem. But it's at a level that my mother isn't going to understand it.

For the casual mathematican, a delightful read.