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Wednesday, June 29, 2005

Franzén on Use and Abuse of Gödel's Theorem

Don't you wish someone would write a book that catalogs all the various ways in which one can misstate, misunderstand, and misapply Gödel's theorems, and how to correct such misunderstandings? A book that you can send your students off to read when they say stuff like, "Gödel showed that there is no mathematical truth," or "The mind can go outside the system, but no formal system can because of incompleteness, so the mind is not a formal system." Well, it's here. Torkel Franzén has been tireless for at least 15 years in correcting misunderstandings relating to logic on sci.logic (which was a lot of fun in the pre-AOL days), on FOM, and elsewhere. He has a book out last year in the ASL's Lecture Notes in Logic series, which is an excellent technical treatment, and now a fine popular book: Gödel's Theorem. An Incomplete Guide to its Use and Abuse (A K Peters, 2005). A brief excerpt from a forthcoming review in History and Philosophy of Logic:
On the heels of Franzén's fine technical exposition of Gödel's incompleteness theorems and related topics comes this survey of the incompleteness theorems aimed at a general audience. Gödel's Theorem. An Incomplete Guide to its Use and Abuse is an extended and self-contained exposition of the incompleteness theorems and a discussion of what informal consequences can, and in particular cannot, be drawn from them. The book is divided into seven chapters. A brief introduction outlines the aims and contents of the book, a lengthy second chapter introduces the incompleteness theorems and outlines their proofs in non-technical terms, and chapter 3 discusses computability and its connections with the incompleteness theorems. Chapter 7 deals with the completeness theorem, and chapter 8 outlines and criticizes Chaitin's work on information-theoretic complexity and its relationship to incompleteness. An appendix fills in some of the technical details. The remaining three chapters (4−6) are devoted to dispelling confusions about incompleteness. Chapter 4, “Incompleteness Everywhere”, dispenses with some basic misconceptions, examples range from atrocious yet all-too-common claims made in Internet discussions (“Gödel's theorems show that the Bible is either inconsistent or incomplete.”) to published remarks by the likes of Freeman Dyson and Stephen Hawking. As one might expect, the corrections here are often basic (e.g., pointing out that the Bible is not a formal system of arithmetic), but just as often they are quite subtle. The (purported) implications of Gödel's theorems for the character of mathematical knowledge and for the nature of the mind (the anti-mechanist arguments of Lucas and Penrose) receive extended treatment in chapters 5 (“Skepticism and confidence”) and 6 (“Gödel, minds, and computers”), respectively.
It's out now from A K Peters; Powell's and Amazon don't ship it yet, but you can preorder from Amazon.

UPDATE: The book's shipping now, and is also available from Powells.

ANOTHER UPDATE: Sol Feferman kindly sent a link to his letter to the editor of the New Your Review of Books on Dyson's review in which he (Dyson) appealed to Gödel's theorem.

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