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Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) Paperback – February 17, 2006

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Editorial Reviews Review

Kurt Gödel is often held up as an intellectual revolutionary whose incompleteness theorem helped tear down the notion that there was anything certain about the universe. Philosophy professor, novelist, and MacArthur Fellow Rebecca Goldstein reinterprets the evidence and restores to Gödel's famous idea the meaning he claimed he intended: that there is a mathematical truth--an objective certainty--underlying everything and existing independently of human thought. Gödel, Goldstein maintains, was an intellectual heir to Plato whose sense of alienation from the positivists and postmodernists of the 1940s was only ameliorated by his friendship with another intellectual giant, Albert Einstein. As Goldstein writes, "That his work, like Einstein's, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is ... more than a little ironic."

This and other paradoxes of Gödel's life are woven throughout Incompleteness, with biographical details taking something of a back seat to the philosophical and mathematical underpinnings of his theories. As an introduction to one of the three most profound scientific insights of the 20th century (the other two being Einstein's relativity and Heisenberg's uncertainty principle), Incompleteness is accessible, yet intellectually rigorous. Goldstein succeeds admirably in retiring inaccurate interpretations of Gödel's ideas. --Therese Littleton --This text refers to an out of print or unavailable edition of this title.

From Publishers Weekly

Gödel's Incompleteness Theorem, which proved that no formal mathematical system can demonstrate every mathematical truth, is a landmark of modern thought. It's a simple but profound statement, but the technicalities of Gödel's proof are forbidding. If MacArthur Fellow and Whiting–winning novelist and philosopher Goldstein (The Mind-Body Problem) doesn't quite succeed in explaining the proof's mechanics to lay readers, she does a magnificent job of exploring its rich philosophical implications. Postmodernists have appropriated it to undermine science's claims of certainty, objectivity and rationality, but Gödel insisted, to the contrary, that the theorem buttresses a Platonist conception of a transcendent mathematical reality that exists independent of human logic. Goldstein is an excellent choice for this installment of Norton's Great Discoveries series, which seeks to explain the ways of science to humanists. Her philosophical background makes her a sure guide to the underlying ideas, and she brings a novelistic depth of character and atmosphere to her account of the positivist intellectual milieu surrounding Gödel (including a caustic portrait of philosopher Ludwig Wittgenstein) and to her sympathetic depiction of the logician's tortured psyche, as his relentless search for logical patterns behind life's contingencies gradually darkened into paranoia. The result is a stimulating exploration of both the power and the limitations of the human intellect. Photos.
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. --This text refers to an out of print or unavailable edition of this title.

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Product Details

  • Series: Great Discoveries
  • Paperback: 224 pages
  • Publisher: W. W. Norton & Company; Reprint edition (February 17, 2006)
  • Language: English
  • ISBN-10: 0393327604
  • ISBN-13: 978-0393327601
  • Product Dimensions: 0.5 x 0.1 x 0.8 inches
  • Shipping Weight: 8.8 ounces (View shipping rates and policies)
  • Average Customer Review: 3.9 out of 5 stars  See all reviews (84 customer reviews)
  • Amazon Best Sellers Rank: #407,072 in Books (See Top 100 in Books)

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More About the Author

Rebecca Goldstein is a MacArthur Fellow, a professor of philosophy, and the author of five novels and a collection of short stories. She lives in Cambridge, Massachusetts.

Customer Reviews

Most Helpful Customer Reviews

101 of 111 people found the following review helpful By bensmomma on June 6, 2005
Format: Hardcover
Although I'll bet that readers more versed in the history of mathematics and philosophy will wish for more than Goldstein offers, I found "Incompleteness" to be a fascinating and well-written introduction to both Godel and the philosophy behind his incompleteness theorem (which proves, mathematically, that in any formal system, such as arithmetic, there will be propositions that are unprovable even though true).

Goldstein is such a clear writer that I finished the book feeling I actually understood this logic. More than simple clarity, though, she conveys a genuine affection for the subject (both Godel and his proofs). You can feel why she gets all worked up about its philosophical implications. It doesn't feel obscure in the least. How much writing about philosophy can say as much?

If you are looking for a complete description of ALL Godel's life work, you won't be happy (she deals almost exclusively with the incompleteness results, not his other work). Nor will you find this to be a standard-issue narrative biography (birth, education, marriage, death); although you can extract the basic facts from Goldstein's scant 260 pages, Godel's wife Adela doesn't appear until page 223; Godels' difficulties with his mental health are treated as non-issues rather than as defining or formative events.

In the end, it's all about the math, and I enjoyed it.
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170 of 192 people found the following review helpful By C. Bracken on March 11, 2005
Format: Hardcover
I was not as enthusiastic about this book as most other reviewers seem to be. This is a book with some important high points, but also some serious flaws.

I was disappointed mainly by the biographical parts of the book. This is a very dry retelling of what is known of Godel's life. Other biographies of seemingly boring mathematicians have been engrossing (read the excellent "The Man Who Knew Infinity" or "A Beautiful Mind"), but this book misses the mark in terms of giving us a picture of who Godel really was.

Godel was part of the Vienna Circle, so we get a lot of history about the Vienna Circle in general. He was later at the Institute for Advanced Studies, so we get a lot of IAS history. But we seem to get little about the man himself, and more about the groups around him.

The meat of the book focuses on the Vienna Circle, and the author's main point: that Godel was a Platonist among the Positivists, and that his incompleteness theorems have been hijacked and misinterpreted by positivists over the years. This part is important and interesting, but I would have liked to have heard more about Von Neumann (who gets only a brief mention) and less about Wittgenstein. But this is in keeping with the book's bias toward the philosophical side of the story.

The explanation of Godel's main proof seemed a bit unclear to me, but I give Goldstein a lot of credit for not simply glossing over the details like most over-simplified explanations one reads. My suspicion is that most who read this book's description of the proofs will laud its clarity while quietly admitting to themselves that they didn't quite understand it all.

Bottom line: if this book will be on your bookshelf next to books of philosophy and logic, it will make a welcome addition.
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53 of 58 people found the following review helpful By Jesse Steven Hargrave on June 2, 2005
Format: Hardcover
It seems to me that, with increasing frequency, two books on the same or closely related subjects come out from different publishers almost simultaneously. I suspect an epidemic of corporate espionage. In 2003/4, did we really need two books with the identical title "Lincoln at Copper Union" about a pre-campaign speech in New York by the eventual president? Why was "The Empire of Tea" published within 6 months of "Tea: Addiction, Exploitation, and Empire"? (Perhaps they were tied to an epic mini-series that I missed.)

Kurt Gödel and his work have been largely ignored of late, yet now we suddenly have two books attempting to resurrect interest. Palle Yourgrau's "A World Without Time: The Forgotten Legacy of Gödel And Einstein" was published in January 2005, and "Incompleteness: The Proof and Paradox of Kurt Gödel" by Rebecca Goldstein just one month later.

Both are small-format books, and thus both attempt to squeeze already dense subject matter into unreasonably constricted space. Both use Gödel's personal and intellectual friendship with Einstein as a systematizing motif. Each author dedicates considerable time to rehearsing the history of The Vienna Circle, where Gödel spent formative years, and the Institute for Advanced Studies in Princeton, where Gödel and Einstein completed their careers. And both Goldstein (a mathematician and novelist) and Yourgrau (a professor of philosophy) attempt to give a summary of Gödel's important theorems that would make them accessible to the non-specialist.

However, the two books differ in important respects.

Goldstein, when dealing with Gödel's professional work, focuses almost exclusively on that concerned most directly with mathematical logic: his Incompleteness Theorems.
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37 of 41 people found the following review helpful By Edward T. Brading on July 9, 2006
Format: Paperback
The book has two main problems. The first is Goldstein's explication of the incompleteness theorem. The theorem is the reason for reading a book about Goedel. For the most part, the worth of a book about him for a general reader is measured by the clarity of an explication of the theorem. Goldstein's audience comprises readers who are not logicians or mathematicians, and so a lack of rigor is expected (p. 172). But Goldstein simplifies too much. Her explication is somewhat less clear than both the longer explication in Goedel's Proof by Nagel and Newman and the more technical introduction by Braithwaite in the Dover Publications reprint of Goedel's original paper.

Goldstein's numbering system (p. 172-175) is an example of oversimplification. Goedel's numbering system "used the exponential products of prime numbers and relied on the prime factorization theorem which states that every number can be uniquely factored into the products of primes" (p. 172). In this way, the "metasyntactic relation of provability will become an arithmetical relationship" (p. 176). Under Goldstein's simplified system of numbering, however, it is not at all clear that provability relationships among propositions would translate into arithmetical relationships among numbers, as they do under Goedel's numbering system. Why does Goldstein offer an alternative numbering system for illustrative purposes? I can't tell. She says that her system, if it were made rigorous, would be just as complicated as Goedel's own system (p. 172). So rather than invent her own, why doesn't she just set out a non-rigorous version of Goedel's own system? (That is what Nagel and Newman do.) Not only does Goldstein not improve matters, but also she loses clarity.
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