Book Review

A Certain Ambiguity Reviewed by Danny Calegari

mathematical and historical reality is interspersed A Certain Ambiguity (A Mathematical Novel) Gaurav Suri and Hartosh Singh Bal with (fabricated) “diary entries” by such historical Hardcover, Princeton University Press figures as Pythagoras, Bolyai, and Cantor. US$27.95, 281 pages I find the premise of this book fascinating ISBN-13: 9780691127095 and am excited by the idea of a simultaneous examination of these issues from historical, sci- entific, and psychological viewpoints. I therefore In “A negative review of negative reviews” ([6]), regret (at the risk of arousing Zeilberger’s spleen) Doron Zeilberger contends that “(a)nyone who that, despite finding some things to like here, I wastes his time writing a review of a book that he or she dislikes, is a frustrated , who cannot recommend this book on literary, philo- has an axe to grind, and just enjoys being mean.” sophical, or mathematical grounds (however, read Writing a novel, especially a novel with se- on). The result is more of a novelty than a novel, rious philosophical aims, is not easy, and Suri and the authors’ ambitions seem more grandiose and Bal have made a serious attempt. They have than grand. Structurally, the book is sound, and chosen an ambitious and significant theme—the even innovative. Numerous historical and fictional story of the gradual discovery and development strands are played out, and the literary device of nonstandard axioms for (hyperbol- of switching back and forth between narrative, ic geometry) and for set theory (undecidability newspaper excerpt, transcripted dialogue, diary of the Continuum Hypothesis), and the implica- entry, and so forth does a good job of organizing tions for epistemology. This theme reveals itself and integrating these strands. through the story of a young man called Ravi, The main disappointment is the disparity be- who by chance takes a liberal arts tween the potential sophistication and depth of undergraduate course at Stanford. The topic of the subject matter, and the unremarkable depths the course is infinity, and Ravi’s intuitions and to which Suri and Bal plumb it. Easiest to criticize naive notions are sharpened by an exposure to (and least serious, in my opinion) is the curso- ideas such as cardinality, power sets, and so forth. ry investigation of the mathematics. We are led At the same time, Ravi stumbles on evidence of a through the usual pedestrian examples of sets that surprising episode inthe life ofhis deceasedgrand- are in bijection to proper subsets (the real line father Vijay and, by hunting through newspaper and an open interval, the positive integers and the clippings and court transcripts, comes to learn squares), Cantor’s diagonal argument that for any about a parallel formative experience his grand- set S the power set P(S) has greater cardinality father underwent at a young age. Ravi’s gradual than S, andsoon. Irritatingly, therearesomeminor discovery of the subtle and indirect nature of goofs even in the exposition of such basic material. Danny Calegari is professor of mathematics at the Cal- In order to disprove the existence of a bijection ifornia Institute of Technology. His email address is of a particular set A = {c, ∗,a, ?, #,q,t,... } with [email protected]. P(A), the authors instead start to give a partial

February 2008 Notices of the AMS 235 bijection of A with P(P(A)) preaches a kind of relativism in which truth is provisional and relative to a set of axioms that are c ←→{?,a}, {q} accepted on some unknown basis. Further, there ∗ ←→{c,a}, {#, ?, ∗},... is an important distinction between the consis- a ←→{} tency of a theory and the question of whether that theory accurately models some phenomenon and so on. Actually, I assume this is what they are under discussion. Fine. A page and a half is de- doing; the notation makes it unclear exactly what voted to a token comparison of several schools set the range is supposed to be. However, in the of thought on the interpretation of mathematical next paragraph, the confusion is compounded by knowledge, e.g., Platonism, formalism, construc- the assertion that “the element c is mapped to tivism, quasi-empiricism, but the real target is the subsets {?,a}, {q}, and so elsewhere. The subplot revolving around Vijay is a c does not occur in the set it sustained effort to “equate ... faith in God with a is mapped into” (p. 171). Set or mathematician’s belief in absolute mathematics” sets? If the authors cannot get (p. 258). In this subplot, opposite (but apparently basic things right, at least they symmetric) points of view are taken up by Vijay can eulogize them: “(f)rom my and by Judge John Taylor. The independence of current perch in retrospective the parallel postulate is taken to undermine Vi- adulthood I consider Cantor’s jay’s initial position that the Christian belief in proof about power sets to be the authority of the Bible is illogical and has no one of the flagship creations place in America, the “land of ‘rationality and of the human race. We have objectivity’” (p. 51). Leaving aside the question thought nothing more elegant of whether a published professional number the- or powerful, only different” (p. orist working in 1919 could be unaware of the 170). If “perch” doesn’t make existence of a consistent theory of non-Euclidean you wince, the contrast between geometry, there is no man in this argument who this encomium and the sub- is not a straw man. I was irritated on behalf of sequent muddle will. There is and theists alike by their banal inconsistent notation (the power set of A is P[A] and predictable interchanges. Ultimately, Vijay’s at one point, and [A] later on, and then finally position is undermined when he discovers that the P(A)) and confusing exposition of elementary the- “self-evident” parallel postulate may not correctly orems in plane geometry. And for the reader who describe nature after all. Judge Taylor undergoes already has some familiarity with mathematics, a similar examination of his faith. “That night I let there is boredom. Mathematically speaking, there myself see the world as an atheist must: a desolate is nothing here that is not already in every other planet occupied by people who had abandoned popular or informal account of non-Euclidean ge- themselves to amoral meaninglessness” he writes ometry or the Continuum Hypothesis. All the high (pp. 248-249). As for the axiomatic method, we points are hit, like a tour bus doing Europe in seven get the following platitude: “(a)s long as (a man) days. Tangible examples of negative curvature in is true to some core beliefs, he can’t go too far nature (lettuce leaves, seaweed) or sewing (pleats) wrong. Which starting point is true is not some- are missing here; instead we read a newspaper thing we humans can make much progress on” account of an expedition to measure the effect of (p. 255). Bleagh. The best one can say about this gravity on light rays. At one point, Nico the course bit of homespun wisdom is that it has the virtue instructor declares “I’ve read some things about of being hopelessly naive about something impor- this but I don’t understand it fully, so I’m not going tant. From the Habermas-Lyotard debate (see [1] to talk about it” (p. 222). One gets the impression for an introduction) to the Sokal hoax ([4]), to re- that the authors have taken a different position. cent atheist manifestos on the bestseller lists (e.g., There is a lot of waffle about the implications of [2]) the question of foundations for intellectual the work of Gödel and Cohen, but no discussion thought and especially for intellectual debate has of what they actually did. Mathematics in this never been more critical or urgent. Never mind. book is a strut on which a range of philosophical Turning to the literary dimensions of the work, opinions are propped up. Perhaps in reaction to I am compelled to say that the prose is usually this, I was provoked into reading something more workmanlike, and the main characters’ attitudes substantial about the mathematics of the Contin- are generally sophomoric. But at least in this do- uum Problem. To the nonspecialist like myself, main there are parts of the novel worth enjoying. I can recommend Set Theory and the Continuum The description of Ravi’s emotional state as he Problem by Smullyan and Fitting ([3]), or Woodin’s lights his grandfather’s funeral pyre is delicate pair of articles which appeared in the Notices ([5]). and moving. And the story of his ambivalent quest The philosophical ambitions of the novel are to join the firm of Goldman-Sachs is compelling- more off-putting. In a broad sense, the novel ly factual (and, given the background of one of

236 Notices of the AMS Volume 55, Number 2 the authors, one suspects factually correct) and concrete. The character of Nico is sympathetic and admirable, and mainly by a process of tactful omission (“from the lower pitch of his voice I got that he didn’t really want to debate his beliefs with me” (p. 157)) is made to seem wise and generous. The narrative is well-plotted, dynamic, and compelling, and rarely drags. There is even a latent love interest. I found it easy to start reading this book, and to keep on reading. While some of the fictitious “diary entries” are somewhat uncon- ventional (e.g., Cantor’s journal entry in which he records his wife saying “You are a huggable bear, Georg” (p. 105)), one can read these in the playful spirit in which they are offered without taking offense. The book plausibly succeeds somewhere at the interface of entertainment and journalism. Readers with no prior interest in mathematics or philosophy may find these aspects of the book at their level and be drawn in by the entertainment the book provides to a fascinating subject whose rewards and subtleties the book points to even if it does not illustrate them.

References [1] M. Bérubé, What’s Liberal about the Liberal Arts? Classroom Politics and “Bias” in Higher Education, W. W. Norton, 2006. [2] S. Harris, Letter to a Christian Nation, Knopf, 2006. [3] R. Smullyan and M. Fitting, Set Theory and the Continuum Problem, Oxford Logic Guides, 34, Ox- ford Science Publications, The Clarendon Press, , 1996. [4] A. Sokal and P. Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, 1999. [5] H. Woodin, The continuum hypothesis, Parts I and II, Notices AMS 48, no. 6, pp. 567–576 and no. 7, pp. 681–690. [6] D. Zeilberger, A negative review of negative reviews, http://www.math.rutgers.edu/ ∼zeilberg/Opinion63.html.

February 2008 Notices of the AMS 237