Customer Reviews

Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries)

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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.

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. In her illustration, for example, she says: "Suppose that GN(wffsub1) = 195589 and GN(wffsub2) = 317" (p. 175). But under Goldstein's own numbering system, 195589 and 317 would correspond, respectively, to ~'00)' and x~(, neither of which is a wff! By oversimplifying, Goldstein has made a mess.

The second main problem with the book is Goldstein's fascination with Wittgenstein and her comparison of him with Goedel. Any comparison between the two thinkers feels strained to begin with, and Goldstein's book does nothing to allay that feeling. It is a bit like writing a book about Vladimir Horowitz and then devoting a third of the book to comparing him with Liberace. Moreover, apart from whether any comparison is useful, Goldstein refuses to take Goedel at his word when he says that Wittgenstein had no influence on his work (p. 115-116). In fact, Goldstein takes Goedel's emphatic denial, coupled with what she sees as Goedel's resentment of Wittgenstein (p. 89), as evidence that Wittgenstein must have had an influence on Goedel, or "incentive" or "significant, if ambiguous, role," as Goldstein puts it (pp. 89, 116). This is just weird. If Goedel had written that he hated rock candy and didn't even like the looks of it, would Goldstein conclude either that Goedel really did like rock candy or that he ate filet mignon as a substitute? The ink that she spills on Wittgenstein could have been put to better use on Turing or von Neumann, both of whom get too few words.

The book contains strange repetitions. For example, twice Goedel's work is compared to Alice in Wonderland (p. 170, 252) and twice Goldstein tells us that her New York apartment has only one bathroom (p. 140, 184). Her catty remarks about Goedel's wife, his diet, and their home decorating are rude and irrelevant (pp. 208-209, 223). How poorly Goedel dealt with faculty politics is dull and irrelevant (pp. 234-245). Some extra proofreading wouldn't have hurt, either: "GN(p)" should be "GN(psub1)" (p. 174); "tilda" should be "tilde" (p. 174); "swiped" should be "swapped" (p. 210); and Waismann's name is misspelled twice (p. 105).

The book has its good points. The stories of Goedel's quirks and his friendship with Einstein are entertaining, the sketch of the Vienna Circle is okay, and Goldstein is right to point out that Einstein and Goedel should not be lumped together with Bohr, Heisenberg, and others as "destroyers of objectivity" (pp. 38-39). But that's about as good as the book gets.

For those that enjoy reading mathematics the best introduction to Godel's proof is the short, popular book Godel's Proof by Ernest Nagel and James R. Newman. But for readers more interested in Kurt Godel himself and in the philosophical implications of his remarkable theorems, there is no better starting point than Rebecca Goldstein's delightful book, Incompleteness - The Proof and Paradox of Kurt Godel.

This is a book to be relished, one that many readers will read more than once. Goldstein's engaging mix of biography, philosophy, and metamathematics operates successfully on two levels:

1) The reader new to Godel will be intrigued with Goldstein's biography of "the most famous mathematician that you have most likely never heard of". The section on Godel's innovative proof may remain a bit out of reach, but overall Goldstein's book is quite accessible.

2) Readers already familiar with Godel's proof, logical empiricism, predicate logic, and Hilbert's program will be surprised at how smoothly Goldstein weaves these topics together. Rebecca Goldstein is to be commended for her care and accuracy.

Goldstein begins with Godel in his later years as a close companion to Einstein at the Institute for Advanced Study at Princeton. She cogently argues that both Einstein and Godel, despite their fundamental contributions to twentieth century thought, were intellectual exiles, isolated by their firm rejection of subjectivism and positivism. Ironically, their own contributions - relativity and incompleteness - were used by others to foster intellectual viewpoints with which Einstein and Godel fundamentally disagreed.

Goldstein also offers a fascinating look at the development of logical positivism (or logical empiricism) that would dominate much of twentieth century philosophy. Godel, with his conviction that mathematics is a means of unveiling features of an objective mathematical reality (a Platonist position), was fundamentally at odds with the Vienna Circle, and especially with Wittgenstein's philosophy. But this most reserved individual remained a silent dissenter, waiting for that time when he could conclusively demonstrate mathematically what he wanted to say.

I found Goldstein's high level description of Godel's proof less satisfying than the more detailed explication found in Nagel's and Newman's book. However, her discussion of the philosophical implications of Godel's theorems, particularly regarding Hilbert's program and Wittgenstein's philosophy, is quite good. Also, I enjoyed her irreverent characterization of predicate calculus (i.e., first order logic) as limpid logic.

Incompleteness - The Proof and Paradox of Kurt Godel is an enjoyable book, one that warrants five stars.

(Note: I'm reposting this review, since apparently it was deleted due to a software glitch.)

I didn't expect to learn anything from this book, but (as she did with several of her novels) Rebecca Goldstein surprised me. The book is basically a prolonged attempt to get inside Godel's head -- which, as a previous reviewer noted, means that it talks a lot about philosophy. That was fine with me, since I already knew the math and the basic facts of Godel's biography. Goldstein tells what I think is a new but largely persuasive story: (1) that Godel saw his incompleteness results as affirming the Platonic reality of the integers, and their irreducibility to Wittgensteinian language games; (2) that this was part of his motivation for proving the theorems (especially after he saw firsthand the Vienna Circle's unjustified Wittgenstein-worship); (3) that most people interpreted the theorems as showing the exact opposite of what Godel intended; and (4) that this lack of comprehension was one reason for Godel's paranoia and isolation at IAS, particularly after his fellow realist Einstein died.

My one criticism is that, in explaining the incompleteness results themselves, the book follows a needlessly cumbersome "old-school" approach. The only reason Godel had to futz around with prime numbers for 30 pages is that the concept of a computer had not yet been invented! Once you have Turing machines, the proof of the first incompleteness theorem is maybe one sentence ("If arithmetic was complete, then we could solve the halting problem, but we can't").

As a side comment, the book says little about Godel's work on the continuum hypothesis, and nothing at all about his remarkable letter to von Neumann, which first posed the "P versus NP" question. I consider both of these contributions to be on a par philosophically with the incompleteness theorems. But perhaps they're a subject for a different book.

This is a good companion and counterpoint to "Wittgenstein's Poker" that starts off well but ends as a book with no payoff. Other than her well-argued characterization of Godel as a Platonist among positivists whose ideas were misunderstood or ignored, Goldstein presents neither coherent biography, nor any explanation of the development and significant influence of Godel's work after 1931 (a subject that seems beyond Goldstein's capabilities). Despite an occasional mention of an important date and a few details of political intrigue at the Institute for Advanced Study in between, the book has almost no content about Godel between 1931 to the 1970's, not long prior to Godel's starving himself to death.

Goldstein does present a decent overview of the first two incompleteness theorems and the goals of the formalists who preceded Godel, though the view she presents is very limited because it ignores the issues of pervasive errors in mathematical reasoning about the infinite in analysis and other fields that led to the formalist point of view in the first place. She is somewhat fuzzy, though, on the relationsip between completeness and incompleteness. There are some obvious errors in terms of the surrounding explanations from small details - Hilbert presented 23 major problems, not 10, at the 1900 Math Congress - to a misleading implication that the theory of arithmetic underlies all of mathematics and its incompleteness implies that the formal inference no value for complex mathematical domains (ignoring efficiently decidable theories like that of real-closed fields, for example).

As other reviewers have noted, Goldstein has almost nothing to say about Godel's relationship with von Neumann who has his biggest champion. Despite her mention of Turing's work, other than saying that Godel was pleased by it she seems to describe it as little more than changing the terminology of Godel's work. Finally, she repeats the Lucas argument and Godel's lack of sanction for it, but she does not seriously discuss the refutations of the Lucas argument that have appeared. The focus of her text seems to imply that, in his heart of hearts, Godel would have liked to believe it.

I didn't expect to learn anything from this book, but (as she did with several of her novels) Rebecca Goldstein surprised me. The book is basically a prolonged attempt to get inside Godel's head -- which, as a previous reviewer noted, means that it talks a lot about philosophy. That was fine with me, since I already knew the math and the basic facts of Godel's biography. Goldstein tells what I think is a new but largely persuasive story: (1) that Godel saw his incompleteness results as affirming the Platonic reality of the integers, and their irreducibility to Wittgensteinian language games; (2) that this was part of his motivation for proving the theorems (especially after he saw firsthand the Vienna Circle's unjustified Wittgenstein-worship); (3) that most people interpreted the theorems as showing the exact opposite of what Godel intended; and (4) that this lack of comprehension was one reason for Godel's paranoia and isolation at IAS, particularly after his fellow realist Einstein died.

My one criticism is that, in explaining the incompleteness results themselves, the book follows a needlessly cumbersome "old-school" approach. The only reason Godel had to futz around with prime numbers for 30 pages is that the concept of a computer had not yet been invented! Once you have Turing machines, the proof of the first incompleteness theorem is maybe one sentence ("If arithmetic was complete, then we could solve the halting problem, but we can't").

As a side comment, the book says little about Godel's work on the continuum hypothesis, and nothing at all about his remarkable letter to von Neumann, which first posed the "P versus NP" question. I consider both of these contributions to be on a par philosophically with the incompleteness theorems. But perhaps they're a subject for a different book.

You don't have to be crazy to write a biography of Kurt Goedel--
but IT HELPS! Rebecca Goldstein's stirring INCOMPLETENESS : The
Proof and Paradox of Kurt Goedel is a case in point. Goldstein is an
unabashed admirer of the logician; being in love with her hero, she
does her best to make us fall in love with him too.

In my case, she succeeded-only in part against my better judgment.
Goedel is the man who, among other things, proved that

GT: In every sufficiently strong consistent formal system S there is
some statement G which is (1) true and (2) unprovable in S. (G is a
"Goedel sentence" for S.)

How can GT be? It can be, as the author deftly explains, because
G says (in code) that G is unprovable in S. Working out the code,
it must be confessed, took some doing for the hero. The author
supplies a simplified version, beginning from a rudimentary numeric
code that she and her schoolmates exchanged as children.

So what's the big deal? If little girls in New York can work out for
themselves the (rudiments of) the most famous proof procedure in
all of Logic, how did this book end up in a famous publisher's
Great Discoveries series? Granted, the author wasn't just any little
girl; she grew up to be a skillful philosophical logician and an even
more famous novelist.

Even more to the point, the biographical subject was, quite frankly,
a nut! Goldstein takes us through his life, from the boyhood in which
his nickname was Mr. Why, through his (silently defiant) conviction
of the Platonic reality of numbers and other mathematical objects
(against the logical positivist views of his interwar seniors in the
famed Vienna circle) to his great discoveries from 1930 on.

In truth, there were not so many of these discoveries. But each of
them solved an extremely important mathematical problem, stopping
some directions of foundational research in their tracks while giving
rise to whole new areas of Logic, Mathematics, Physics and Philosophy.
In consequence Goedel escaped the horrors of Nazified Vienna (not
that he thought of it as an escape) for the bucolic life of a full-time
researcher at the newly founded Institute for Advanced Study in
Princeton, New Jersey.

Meanwhile, he went even more mad. Shrinking from personal
confrontation, he preferred to let his proofs talk for him-and
then grew distraught when less insightful folks missed the point of
what he had shown! What is it about our heroes, anyway? As a
species, we human beings don't have all that much to brag about.
Even the brightest of us push our understandings on a little. The
price they too often pay-which Goedel paid-is to take leave of
the humdrum rationality on which we mix with our fellows and
help ourselves to three square meals a day. Horrible result-after
Einstein died Goedel was left without intimate friends, starving
himself to death for fear of being poisoned.

There are some AWFUL errors in this book-and I read through it
quickly, like a novel. The author knows who the famous logician
S. C. Kleene is-and ought accordingly to have caught his incorrect
identification as Kline on p. 195. David Hilbert listed 23 problems-not
10, as is asserted on p. 138 (and elsewhere)--in his famous "pep
talk" to mathematicians in 1900. (The automated reasoning
pioneer Dr. Larry Wos of Argonne National Laboratory claims that
Hilbert intended 24 problems, the 24th of which had to do with
finding truly efficient proofs.)

The second of these Hilbert problems, on which the author does
dwell, was indeed that of the consistency of arithmetic. But Hilbert
meant much more by arithmetic than it has come to mean, after
the work of Goedel. For Hilbert was concerned, at the very least,
with ninth grade arithmetic, when we begin to worry about the
properties of the (incredibly dense) real numbers. Goedel showed
that we are already in trouble in third grade arithmetic, when all
we have to worry about are adding and multiplying the natural
numbers 0, 1, ..., n, n+1,. .... That was a surprise!

Please pass on to Norton, accordingly, the modest suggestion that
they hire a graduate student (or, perhaps, a recent Ph. D.) to check
the facts. There are certainly enough unemployed and very highly
qualified smart people, these days, to be worth-say-ten dollars
an hour for a very useful service. And I note all too often in the
books I buy such egregious errors in the works of incredibly
famous authors. Please, publishers, do something about it!

Still, INCOMPLETENESS is a really good read. For all my
grousing, Rebecca Goldstein is to be congratulated for having
produced a wonderful book, which I recommend most highly
to your other customers.

After reading the rave reviews posted for this book I must admit my expectations concerning this book were set in advance. My impression was that this book would provide a lucid historical narrative complete with literary flare concerning the life and thought of Kurt Godel. While this seems to have been the intent of the author I must say I was quite disappointed by what can only be described as poor writing. Too many sentences are fragmentary, or run-ons, or unfocused. The chapters are long sometimes untidy, as if neglected by the editor. The author seems to have a hard time presenting her ideas in a concise, logical structured manner. She uses commas and hyphens so much that they often disrupt the flow of the narrative. She also employs sentences consisting of unusually large numbers of modifiers (words that modify the subject and objects of the sentence). Sometimes her sentences begin and end on completely different topics altogether. With that being said I found much of the narrative informative and valuable. The perspective is philosophical, not mathematical, so expect to read more about the philosophical implications more than the mathematical implications (although the two are closely tied.) Her exposition of the proof itself was good, but Nagel and Newman do a better job in their work "Godel's Proof". I would recommend this book to any interested reader with the reservation that the reader expect a somewhat bumpy ride.