Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) [Englisch] [Taschenbuch]

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. If you are setting it next to "Men of Mathematics," "The Man Who Knew Infinity," or other such rich biographies of mathematicians, you may be disappointed by this book's philosophical and academic tone.

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.

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. That means Gödel's more cosmological exertions, which came after he joined the Institute, are left untreated. And Goldstein has a theorem or two of her own: that the implications of Gödel's work in mathematical logic and metaphysics were seriously misconstrued even in his own day, that such misunderstanding was a gnawing disturbance to the logician, and that it contributed greatly to his increasingly pathological alienation from his colleagues and the world at large.

Yourgrau is more interested in the validity and implications of Gödel's later philosophical (or cosmological) work on the nature of time. Yourgrau published an earlier monograph which the book jacket claims "sparked a resurgence of interest in Gödel's ideas about time and relativity." Yourgrau comes across as Gödel's self-appointed apologist, armed to defend the logician against claims that these later philosophical applications were amateurish and easily dismissed.

Both books, I felt, succeeded in gaining the reader's sympathy for their respective perspectives. But neither could be suitably comprehensive in the relatively few pages allotted them. For me, Goldstein did the slightly better job of explaining the Incompleteness Theorems.

(It would be beyond the skills of even the most accomplished popularizer to fit a truly satisfying explanation into these abbreviated books. The reader is subjected in both to sentences such as this one from Yourgrau: "The representation occurs via the arithmetization of the syntax of FA, so corresponding to a given syntactical truth Bew(x,y) of MFA, there is an arithmetical truth Bew(x,y) of IA that corresponds to a formula Bew(x,y) in FA that can be interpreted as saying that the sequence of formulas with Gödel number x is a proof of the formula with Gödel number y, and this formula, Bew(x,y), is a theorem of FA.")

You thus get from Goldstein a better grounding in what is considered Gödel's true legacy. But you have to look to Yourgrau to get even a basic sense of what Gödel later had to say about cosmology. In that sense, Yourgrau's book is the more thought-provoking.

Both authors are gifted writers, although Yourgrau seems to loose some control over his metaphors as he gets increasingly worked up about the lack of respect given to Gödel's cosmological contributions. As Yourgrau tells of a 1995 symposium on "Gödel's General Philosophical Significance", readers may feel they have stumbled into a metaphysical food-fight.

The fact that these two books were published at almost the same time shows that there must be a significant audience of non-specialist readers interested in an updated accounting of Gödel's life and work. It's unfortunate that such readers have to buy both these books and navigate through so much redundant material to get even the beginnings of a complete perspective.