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Book Review, © Copyright 1997, Jim Loy
On the cover, the book Gödel, Escher, Bach (sometimes called "GEB") says, "A metaphorical fugue on minds and machines in the spirit of Lewis Carroll." This book was a winner of the Pulitzer Prize. And, it is one of the most amazing books of all time. It will stretch your mind.
This book is about Johann Sebastian Bach, and his amazing fugues. It is about M. C. Escher and his amazing drawings. And it is mainly about Kurt Gödel, and his amazing proof that "All consistent axiomatic formulations of number theory include undecidable propositions." The book builds up slowly and easily, so that you don't really notice that you are sometimes asked to consider fairly deep subjects.
Every chapter is introduced by a dialog between Achilles and the Tortoise, and sometimes other characters like the Crab or the Anteater. For example, the Crab has a perfect record player, which can reproduce any sound, with perfect clarity. And the Tortoise gives him a recording of the one sound that will destroy that particular record player (a song entitled, "I Cannot Be Played on Record Player 1"). Later, we encounter a song entitled, "I Cannot Be Played on Record Player 2."
The book wanders through many concepts in mathematics, logic (such as self reference), computer science (such as recursion), music (Bach fugues), art (Escher), philosophy (such as "What is 'thought'?"). But it mainly explores the ins and outs of an idea in logic called the Propositional Calculus, a fascinating way of manipulating symbols.
In the end, we follow the footsteps of Kurt Gödel, as he represents a classic paradox (a statement which can neither be proved nor disproved) in symbols, and then shows that these symbols can be used to transform this paradox into any non-trivial logical system, including algebra. In other words, algebra includes undecidable propositions.
Gödel, Escher, Bach is a fairly big book. But, it is a delight to read.
Another delightful book, on the subject of Gödel's proof, is Raymond Smullyan's Forever Undecided. He has written several books of logic puzzles and paradoxes, such as This Book Needs No Title, and What Is the Name of This Book?
Both authors spoon feed Gödel's proof to you, the reader. But while Hofstadter wanders through many related topics, Smullyan dives right in to manipulating symbols, using puzzles. Smullyan dedicates his book "to all consistent reasoners who can never know that they are consistent." Gödel's proof actually shows that they cannot prove that they are consistent.
I recommend both books.
To order Gödel, Escher, Bach Amazon.com (goes directly to this book). Forever Undecided is out of print. To attempt to order it anyway, click Amazon.com.