Book Review Incompleteness: The Proof and Paradox of Kurt Gödel Reviewed by Juliette Kennedy Incompleteness: The Proof and Paradox of Kurt Incompleteness Theorems. This means that some Gödel of these areas are covered more comprehensively Rebecca Goldstein than others. W. W. Norton & Company In places the author succeeds creditably; for ex- February 2005 ample, her portrayals of behind-the-scenes acade- $22.95, 296 pages, ISBN 0393051692 mic life will likely be of interest to readers who enjoy such material—indeed, such portrayals seem Popular books on mathematics play an impor- to be her forte. tant role in the lay public’s education. But as is As is often the case with books about mathe- known to anyone who has given a popular mathe- matics written by nonmathematicians though, matics lecture or written about a famous theorem for an audience of nonmathematicians, doing jus- shortfalls of precision occurring here and there will tice to the mathematics in question is almost im- leave mathematicians unsatisfied; and the mis- possible in those circumstances. Rebecca Gold- statement of the Fixed Point Theorem on page 180, stein, the MacArthur Foundation fellow and author the heart of the matter technically, makes it, of The Mind-Body Problem (a novel which seems to unfortunately, quite impossible for anyone to be quite popular among mathematicians) attempts reconstruct the proof of the First Incompleteness an even more difficult task in her short new book Theorem from Goldstein’s account. Incompleteness: The Proof and Paradox of Kurt In this review I will comment on the three main Gödel, namely, to place a significant piece of math- aspects of Goldstein’s book, apart from her tech- ematics—Gödel’s Incompleteness Theorems—in nical account of the theorems: first, her portrayal the context of the wider intellectual currents of the of the life and personality of Gödel and the social twentieth century, both within the mathematical surroundings in which he worked; second, her logic and the philosophy of mathematics commu- main claim, which is that Gödel’s work has been nities, as well as within the intellectual culture at misunderstood and misused by postmodernists large. and other intellectuals; and third, her rather sub- The theorems are presented in the context of a stantial discussion of foundational issues, which somewhat detailed personal and intellectual biog- unfortunately is the weakest of these three aspects raphy of Gödel as well as in that of the various schools in foundations of mathematics in exis- of the book. tence at the time. A vast amount of material is The author’s novelistic skills are at their most covered: everything from the history of the Vienna conspicuous in the section of the book devoted to Circle, to Gödel’s philosophical differences with her colorful portrait of Gödel. From p. 59: Wittgenstein, to the Hilbert Program, to Gödel’s I think it is fair to say…that like so views on appointments at the Institute for Ad- many of us Gödel fell in love while an vanced Study, to many aspects of his personal bi- undergraduate. He underwent love’s ec- ography, in addition to the account of the two static transfiguration, its radical re- Juliette Kennedy is university lecturer in the Department ordering of priorities, giving life new of Mathematics and Statistics at the University of Helsinki. focus and meaning. One is not quite Her email address is [email protected]. the same person as before. 448 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Kurt Gödel fell in love with Platonism, Those familiar and he was not quite the same person with Gödel’s life may as he was before. find all this some- what reductive; al- Another example of this theme occurs on p. 110, though it is clear where Gödel is described as “a man whose soul had from the author’s been blasted by the Platonic vision of truth.” portrayal of him that Goldstein includes all of the standard anecdotes she very much sym- about Gödel, as reported by the occasional, usually pathizes with Gödel. confounded, eyewitness. Some of these are quite Some occasional amusing, for example, the story of Gödel’s citi- wrong notes include zenship hearings (pp. 232–233), in which a logical the discussion on pp. inconsistency Gödel discovered in the American 226–227, in which Constitution threatened to upset the proceedings. some key back- The psychological analysis the author sprinkles ground facts are here and there into the biographical material is omitted, as well as nothing if not enterprising. For example, Goldstein the discussion on p. 223, in which the author ven- has a theory about the source of Gödel’s psycho- tures to describe Gödel’s marriage to his wife Adele, logical difficulties (pp. 48–49): a lively and witty woman who seems to have been as I hope will become ever clearer in the somewhat out of place in Princeton, as “weird”, “ac- cording to just about everyone”. One wonders chapters to come, the internal para- about this characterization of their marriage, when doxes in Gödel’s personality were at many of the Gödels’ friends seem to give a differ- least partially provoked by the world’s ent impression in their reports of it. In fact, on the paradoxical responses to his famous whole the portrait of Adele is a bit ungenerous, with work. the author making very heavy weather, for exam- See also p. 57: ple, about such things as Mrs. Gödel’s appalling— to Goldstein—taste in home decoration. …the precocious Gödel grasped the lim- Does the real person come through in this ac- its of parental omniscience at about the count? Gödel was an extremely private person who age of five. It would be comforting, in at the same time suffered, as many creative peo- the presence of such a shattering con- ple do, from disabling episodes of anxiety and de- clusion, especially when it’s reinforced pression. The episodes became worse with age. It by serious illness a few years later, to is not a very pretty story; but his productivity, derive the following additional conclu- given the circumstances, makes it a very moving sion…the grownups around me may be one—albeit one that in the end, at least in this a sorry lot, but luckily I don’t need to book, may remain to be told. depend on them. I can figure out every- That said, interviews the author has conducted thing for myself. The world is thor- with the principals, for example Armand Borel, oughly logical and so is my mind—a have yielded valuable new information about perfect fit. Gödel’s relationship with his colleagues at the In- Quite possibly the young Gödel had stitute for Advanced Study, answering the question why Gödel was so isolated from his colleagues dur- some such thoughts to quell the terror ing his later years there. Goldstein also draws upon of discovering at too young an age that her experience as a graduate student in philosophy he was far more intelligent than his par- at Princeton, which has put her in the position of ents. It would explain much about the being able to speak firsthand about the Princeton man he would become.1 academic culture of the time—even if her per- 1Some of the important memoirs about Gödel include spective is very much that of an awestruck student. those by his brother Rudolf Gödel, his classmate Olga The book is centered around the claim that, in Taussky-Todd [18], the obituary of Gödel for the Royal So- an ironic twist of events, the “intellectual commu- ciety by Georg Kreisel [11], as well as other writings of nity”, as Goldstein refers to it, used Gödel’s own Kreisel on Gödel, Hao Wang’s three books based on his con- incompleteness theorems to discredit his philo- versations with Gödel [14], [15], [16], as well as Stephen sophical Platonism; that the Incompleteness The- Kleene’s memoir [10], to name just a few. The reader is orems became “grist for the postmodern mill”, if also referred to the biography of Gödel by John Dawson not the main weapon in the contemporary “revolt entitled Logical Dilemmas [4], as well as to Palle Yourgrau’s against objectivity”; that consequently Gödel, to portrayal of Gödel in A World without Time [17], which focuses mostly on Gödel’s friendship with Einstein and the whom the notion of mathematical truth was an ab- scientific work which grew out of it. solute and objective one, had to battle to the end APRIL 2006 NOTICES OF THE AMS 449 of his days against postmodern misconceptions and view of Gödel’s time. For example, the Hilbert Pro- misconstruals of his theorems, which were misin- gram aimed to show that all that was required to terpreted to show that there is no such thing as formalize mathematics were finitary axioms stated truth. in a precise syntax together with finitary rules of Platonism in the context of foundations of math- proof (proving consistency was the second desider- ematics is essentially the view that mathematics is atum of it). In showing that mathematics can be con- a descriptive science, although, unlike the empir- strued as a “formal game of symbols”, a slogan ical sciences, the domain described is thought to which came into use at the time, mathematics consist of abstract objects. Another tenet of Pla- would be put on a firm foundation by eliminating tonism is that the concept of mathematical truth reference to infinite objects, as well as the use of is a meaningful one. Gödel held this view from unstated assumptions or proof procedures that about 1925 onwards (though he wavered a bit be- might lead to paradoxes.