129 of 132 people found the following review helpful:
5.0 out of 5 stars
An Abstruse Mathematical Proof Made Fascinating, December 24, 2001
This is a remarkable book. It examines in considerable detail Godel's proof, a mathematical demonstration noted for its difficulty in its novel logical arguments. The chapter topics - the systematic codification of formal logic, an example of a successful absolute proof of consistency, the arithmetization of meta-mathematics - appear almost unapproachable. And yet, Ernest Nagel and James R. Newman have created a delightful exposition of Godel's proof. I actually read this book in one sitting that took me late into the night. I simply didn't want to stop; it is really a good little book.
Godel's proof is not easy to follow, nor easy to grasp the full implications of its conclusions. Many mathematical texts, overviews, and historical summaries avoid directly discussing Godel's proof as these quotes indicate: "Godel's proof is even more abstruse than the beliefs it calls into question." "The details of Godel's proofs in his epoch-making paper are too difficult to follow without considerable mathematical training. "These theorems of Godel are too difficult to consider in their technical details here." Such is the common reference to Kurt Godel's milestone work in logic and mathematics.
In their short book (118 pages) Nagel and Newman present the basic structure of Godel's proof and the core of his conclusions in a way that is intelligible to the persistent layman. This is not an easy book, but it is not overly difficult either. It does require concentration and a willingness to reread some sections, especially the second half.
"Godel's Proof" begins with an explanation of the consistency problem: how can we be assured that an axiomatic system is both complete and consistent? The next chapter reviews relevant mathematical topics, modern formal logic, and places Godel's work in a meaningful historical context. Following chapters explain Hilbert's approach to the consistency problem - the formalization of a deductive system, the meaning of model-based consistency versus absolute consistency, and gives an example of a successful absolute proof of consistency.
The plot now begins to twist and turn. We learn about the Richardian Paradox, an unusual mapping that proves to be logically flawed, but nonetheless provided Godel with a key to mapping meta-mathematics to an axiomatic deductive system. (I forgot to explain meta-mathematics; you will need to read the story.) And then we learn about Godel numbering, a mind boggling way to transform mathematical statements into arithmetic quantities. This novel approach leads to conclusions that shake the foundations of axiomatic logic!
The authors carefully explore and explain Godel's conclusions. For the first time I began to comprehend Godel's fundamental contribution to mathematics and logic. I am almost ready to turn to Godel's original work (in translation), his 1931 paper titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems". But first, I want to read this little book, this little gem, a few more times.
Help other customers find the most helpful reviews
Was this review helpful to you? Yes
No
57 of 59 people found the following review helpful:
5.0 out of 5 stars
Wish I'd read it first ..., May 1, 2003
This review is from: Godel's Proof (Hardcover)
I read Godel's paper in grad school. I wish I had read this first, because it lays out the structure of the argument clearly. N&N are particularly good on clarifying what Godel did and did not prove. This is important because of all the loose mystical obfuscation out there about this theorem.
N&N clearly explain what formal "games with marks" methods are, and why mathematicians resort to them. They then walk through what Godel proved, with a bit on how he proved it. The basic idea of his (blitheringly complex) mapping is explained quite well indeed.
Suitable for mathematicians, or philosophy students tired of mystical speculations. Also goo for anyone with an interest in computability theory or any formal logic. And read it before you read Godel's paper!
Help other customers find the most helpful reviews
Was this review helpful to you? Yes
No
48 of 50 people found the following review helpful:
5.0 out of 5 stars
An excellent guide to Gödel, June 26, 1999
By A Customer
Simply magnificent. This book meets and exceeds the description on its back cover -- offering "any educated person with a taste for logic and philosophy the chance to satisfy his intellectual curiosity about a previously inaccessible subject." This book gives anyone with the interest and the motivation a solid, if not complete, understanding of the ideas underlying the proof. While it's true that someone very unfamiliar with mathematics (or, more importantly, with logic and mathematical thinking) would not get as much out of the book, it does a very good job of walking the reader through Gödel's complex but breathtakingly elegant reasoning. I wholeheartedly recommend this book.
Help other customers find the most helpful reviews
Was this review helpful to you? Yes
No
