Godel's Theorem: An Incomplete Guide to Its Use and Abuse (Paperback)

by Torkel Franzen (Author) "Few theorems, of pure mathematics have attracted much attention outside the field of mathematics itself..." (more)
Key Phrases: Incompleteness Everywhere, Ayn Rand, David Hilbert (more...)

Editorial Reviews

Review
"Among the many expositions of Godel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzen gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstanding and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature" - John W. Dawson, Jr., author of Logical Dilemmas: The Life and Work of Kurl Godel Godel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself. - An Incomplete Education, by Judy Jones and William Wilson"

About the Author
A philosopher by training (PhD, University of Stockholm), Torkel Franzén has for the past twenty years been active working in computer science (at the Swedish Institute of Computer Science) and teaching programming (at Luleå University of Technology). He is the author of a number of books, among them Inexhaustibility: A Non-Exhaustive Treatment.

Product Details

• Paperback: 172 pages
• Publisher: A K Peters, Ltd. (May 25, 2005)
• Language: English
• ISBN-10: 1568812388
• ISBN-13: 978-1568812380
• Product Dimensions: 8.8 x 5.9 x 0.5 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars See all reviews (16 customer reviews)
• Amazon.com Sales Rank: #221,274 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

85 of 88 people found the following review helpful:

4.0 out of 5 stars Knowing when to invoke Gödel's Incompleteness theorems, August 6, 2005

By	T. Gwinn (USA) - See all my reviews (REAL NAME)

As evidenced from the title, the primary focus of the book is to identify the specific nature of these theorems, where they apply directly, and where they do not apply directly, and where they are interpreted entirely erroneously.

Although the book is aimed at non-mathematicians and those with no knowledge of formal logic, I can't really imagine someone with no understanding of logic and some fair amount of math comprehension benefitting alot from this book. I mean, by p. 10 he talking about Diophantine equations and Goldbach-like conjectures, and soon after, "PA" and "ZFC" are tossed about as if they were practically everyday acronyms for most people. The book is however, largely free of formulas and proofs, for those who are dissuaded by such. The overviews of the theorems themselves is not as lucid as I imagine they could be (which is why I rate it a 4 instead of a 5). The overviews will also seem a bit alien to someone expecting and Nagel & Newman kind of treatment; instead, this is discussed from a more abstract perspective of the characteristics and properties of formal systems, which avoids getting into the gritty details (even Gödel-numbering is not explained in detail!) but may be hard to grasp for someone not used to thinking at this level of abstraction about mathematical systems.

With that said, I still think it is quite worthwhile reading, and at a slim 170ish pages, it is a fairly quick read. After the overviews, he takes on various applications/misapplications of the theorems by topic. So, there are discussion of the theorems' relevance or applicability to things such as TOE (Theory of Everything), Turing machines, skepticism, minds, inexhaustibility, computability and so on. He does so typically by first giving several quotes that appear either in the literature or commonly on the internet, and then proceeds to either correct or clarify those quotes. Such notables as Roger Penrose, Freeman Dyson, and Stephen Hawking are among the quoted who are scrutinized.

I think the primary goal of the book is accomplished in its debunking of outright misuses of the theorems, and by way of correction and clarification of other uses, it accomplishes its pedagogical goal. I know it will cause me to strive to be more precise in any future invocations of these theorems.

29 of 30 people found the following review helpful:

5.0 out of 5 stars Just about the best book on Godel's Theorem I have read, April 5, 2006

By	sigfpe "sigfpe" (Oakland, CA United States) - See all my reviews

As a mathematician I thought I had a good understanding of Godel's Theorem but Franzen highlighted a bunch of misconceptions that I had. It went on to answer a bunch of questions I had wondered about but had never had a chance to ask a logician. So in a sense this book is precisely what I wanted from a book on Godel's theorem and I can't help but give it 5 stars.

But that's not all. Over the years I have read countless papers, articles and books by author who invoke Godel's Theorem in the most inapposite paces without understanding it. I've found this to be pretty annoying and in many cases there is no rebuttal. Franzen tackles many of these misuses whether they are comments on USENET or published arguments by Lucas, Chaitin and Penrose. It's great to see someone put pen to paper and reply to these abuses in one place. That would bump my rating up to 6 stars if I were able.

But be warned, this book is challenging. I'd suggest that as a prerequisite you need to be a mathematician, philosopher or computer scientist with at least some familiarity with Godel's Theorem.

15 of 15 people found the following review helpful:

5.0 out of 5 stars Valuable Counterbalance to Widespread Misconceptions and Nonsense in Print and on the Internet, August 13, 2006

By	Michael Wischmeyer (Houston, Texas) - See all my reviews (TOP 500 REVIEWER)    (REAL NAME)

Torkel Franzen has created an immensely valuable, deeply fascinating examination of misunderstandings, misconceptions, and outright abuse of Godel's theorems frequently found on the Internet (and occasionally in print). He does so in a cogent, non-confrontational style that makes enjoyable reading. Godel's Theorem - An Incomplete Guide to Its Use and Abuse warrants five stars.

A word of caution is appropriate, however. Chapters 2 and 3 will be heavy going for readers not familiar with formal logic. Although Franzen avoids the details of Godel numbers in his explication of Godel's proof, he does delve into topics like self-referential arithmetical statements, Tarski's theorem, Rosser sentences, weaker variants of the first incompleteness theorem, computably decidable sets, Turing's proof of the undecidable theorem, and the MRDP theorem.

Furthermore, the appendix offers both a formal definition of the concept of a Goldbach-like arithmetical statement and comments on the significance of Rosser's strengthening of Godel's first incompleteness theorem. (Any reader that stays the course with the early chapters will be able to handle the appendix discussions. The short chapter 7 is also more technical as it discusses the completeness of first order logic.)

A word of encouragement is equally appropriate. Chapters 2 and 3 can be browsed, even skipped outright. The later chapters are much more accessible and don't require that the earlier chapters have been mastered; instead, they focus on examples of the misuse of Godel's theorems - from the merely technically inaccurate to the humorously nonsensical. It is these later chapters that makes this book special.

Although words like consistent, inconsistent, complete, incomplete, and system have been carefully defined within the context of formal logic, in normal discourse these words have varied meanings, often leading to vagueness and confusion in discussions of Godel's theorems. Furthermore, Godel's theorems often serve in an inspirational fashion, that is being used as analogies and metaphors in which the essential condition that a system must be capable of formalizing a certain amount of arithmetic is largely ignored.

Invocations of Godel's incompleteness theorems in theology, in physics (like the theory of everything), and in the philosophy of the mind (the Lucas-Penrose arguments) are found in chapters 4, 5, and 6. Chapter 8 addresses the widely publicized philosophical claims of Geoffrey Chaitin on the relationship between incompleteness and complexity, randomness, and infinity.

Godel's Theorem - An Incomplete Guide to Its Use and Abuse may be too much too soon. A reader new to Godel's work might consider starting with Godel's Proof (by Ernest Nagel and James R. Newman) or Incompleteness - The Proof and Paradox of Kurt Godel (by Rebecca Goldstein).