Incompleteness: The Proof and Paradox of Kurt Gödel

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.

This engrossing book is more than just a biography of a brilliant but crazy mathematician - another "beautiful mind." It seeks to change our understanding of the significance of one of the most important discoveries in the history of mathematics, perhaps the history of thought. "Incompleteness," like "Relativity" and "Uncertainty," are pithy labels for counterintuitive discoveries from the early decades of the twentieth century that got appropriated by café intellectuals to mean that all knowledge is subjective. Goldstein (who is best known as a novelist, but is also a Princeton-trained philosopher who studied logic and teaches philosophy of science) shows that in the case of Gödel's Incompleteness theorems, this understanding is exactly backwards. The most defensible interpretation of his results imply that mathematical truths exist apart from any human activity to capture them in our formal systems. Moreover, Goldstein show that Gödel himself saw the significance of his own proofs in this light-that he was a mathematical Platonist-and may even have been motivated to pursue his stunning proof out of a desire to vindicate Platonism, in defiance of the prevailing wisdom (associated with Wittgenstein and the logical positivists) that mathematics is just a game of manipulating symbols according to rules. Goldstein is not unafraid of controversy. There are still some logicians who spin the Incompleteness theorems in a way that is consistent with the "formalist" (symbol-manipulating-game) view of mathematics. And some historians of mathematics suspect that Gödel only became a Platonist later in his life. But the case that Goldstein makes in this book is persuasive. Most mathematicians these days really are Platonists, in large part because of Gödels' results, and the ones I have spoken with all agree with Goldstein's interpretation. (A number of them wrote laudatory reviews on this very site, which mysteriously vanished.) As for Gödel's own motives, we'll never really know what went on his mind in those crucial years, but Goldstein presents the closest thing we have to a smoking gun-a letter she discovered in which Gödel tells a sociologist that he was a Platonist in the early 1920s, well before he began work on his theorems. This isn't the only popular book on Gödel, but it is the one I would recommend to anyone wanting an accessible but technically accurate explanation of what the fuss is all about. For that matter, it is the book I would also recommend to mathematicians and philosophers who want to learn more about the intellectual context of the Incompleteness theorems - who and what Gödel was reacting to, and how his results were understood and misunderstood. Hofstadter's "Gödel Escher Bach," as one reviewer of Goldstein pointed out, "labors under the handicap that Goedel, Escher, and Bach have no nontrivial connections with one another." And Yourgrau's recent "World Without Time" (which is about Gödel's views on cosmology, briefly but movi

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.

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.