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REVIEWS Contents The Pea and the Sun: a Mathematical Paradox REVIEWS
The Pea and the Sun: a Mathematical Paradox by Leonard M. Wapner, A K Peters, 232 pp, 2005 hardback £21, ISBN 1-56881-213-2, 2007 paperback £11.00, ISBN 978-1-56881-327-1. The Banach-Tarski paradox is one of the strangest statements in mathematics, and it is one of my personal favorites. One formulation of the statement of the paradox is that a ball of radius R in Euclidean 3-space can be decomposed into finitely many disjoint pieces, which can then be moved by Euclidean motions to be reassembled into a ball of radius 2R. The Banach-Tarski paradox is equivalent to the Axiom of Choice, and the heart of the paradox lies in the existence of non-measurable sets. The book under review is written to explain the Banach-Tarski paradox to a general audience. It is a noble and difficult ambition, and it is one that the author succeeds at. He begins slowly, by introducing the main characters (Stefan Banach and Alfred Tarski, of course, but also Georg Cantor, Kurt Gödel and Paul Cohen), giving short biographies and discussing their work in Set Theory in basic terms. He then goes on to discuss the notion of paradox and then presents and discusses several jigsaw fallacies from the past century and a bit. The discussion of jigsaw fallacies is particularly good, I think, as it gets the reader in the right frame of mind for what is to come next. In Chapter 3, he starts getting to the heart of the matter, discussing scissors congruences and equidecomposibility, and the difference between them. In Chapter 4, he brings in what he refers to as 'baby BTs', or baby versions of the Banach-Tarski paradox. These two chapters are the most difficult, and the most interesting, because they form the bridge between the general discussion in the first two chapters, and the Banach-Tarski paradox itself which awaits in Chapter 5. These third and fourth chapters are written well and do a good job of conveying the interested reader along. Necessary topics such as matrix multiplication and Lebesgue measure are defined and discussed briefly, but in such a way that the general reader should not get bogged down. Largely because of the care that the author has taken in setting the stage, the Banach-Tarski paradox itself comes almost as an anti-climax. But only almost. The proof is given in Chapter 5 and is broken down into pieces that are individually relatively straightforward. There is an air of magic to the proof, but I find it a pleasing air of magic and it is an air that many good proofs have. The book closes with a discussion of the possible relevance of the Banach-Tarski to the world we live in, and in particular I found the discussion of possible connections between the Banach-Tarski paradox and physics to be fascinating. I would find it remarkable if the Banach-Tarski did indeed arise there. The Banach-Tarski paradox, particularly in its details, is hard to get a hold of. This is especially true for a non-mathematician, who isn't used to the basic axiomatic approach that mathematics takes to its universe. As is often the case of a book on a hard mathematical topic written for a general audience, there are places where the mathematician might quibble with details of the presentation, but I do not view these as serious. Rather, they are necessary if the flavour of the idea is to be presented without losing it in a forest of detail. For the general reader, this book is an excellent introduction to a fascinating mathematical idea, and it is a book that I am tempted to buy a few people for Christmas. Jim Anderson
The Triumph of Numbers: How Counting Shaped Modern Life by I Bernard Cohen, Norton & Co Ltd, 2006, Paperback, 209pp, £9.99, ISBN 0-393-32870-8 This is a modest, attractive, and readable introduction to the history of political statistics: it is, as the cover promises, 'brief, lively, and highly entertaining'. The author, I Bernard Cohen, was one of the greatest of twentieth-century historians of science, and this is the final book on which he worked before his death in 2003. Aiming well away from exhaustiveness, the book uses a series of extremely well-chosen mini-case-studies to illuminate 'how quantitative considerations have entered the conduct of government' (pp. 48–50) from about the middle of the seventeenth century to the beginning of the twentieth, from the work on mortality statistics by John Graunt and William Petty in Restoration England, through Jefferson's and Franklin's uses of numbers in politics, the introductions of national censuses around 1800, and Quetelet's work in nineteenth-century Belgium, up to Florence Nightingale's statistics-driven reforms of hospital practice in the later nineteenth century. The Triumph of Numbers also seems to be the incomplete relic of a rather different book, one which would have dealt with uses of numbers in general throughout (modern) history. This would have been an extraordinarily ambitious project: even in the book as we have it 'numbers' include both discrete and continuous quantities in cultures from ancient Egypt to nineteenth-century France. They may be natural numbers, integers, rationals, reals, or even ratios; they may be abstract concepts, the physical objects or collections which those concepts count or measure, or the written symbols used to record those counts and measures. I am not sure that even Cohen could have drawn coherence from such a protean concept. This book's two examples of 'numbers' from ancient Egypt and the Old Testament serve only to highlight how very different number concepts can be in different cultures, and what very different work they can do; and a similar point could be made about the short chapter on early modern numerology. As it is, the opening two chapters, and certain passages elsewhere, which extend the scope of the book beyond a history of political statistics from 1650 to 1900, strike me as something of a distraction. Even the very brief consideration of the role of numbers in the Scientific Revolution feels as though it has been fitted to the 'numbers' theme only with difficulty. For example, it is not obvious that 'an exact statement that leads to prediction and test' (p.36) need be numerical, nor why the law of reflection of light (p.37) should be considered a 'numerical law'. Any good book, especially one so brief, inevitably raises more questions than it answers: why did the rise of political statistics happen when and how it did? Who desired it, and who benefited from it? What were its effects, in general, on the practice of government? I could also have wished for the consideration of its critics to be more fully integrated into the narrative. A foreword notes that the subtitle is not one of which Cohen would have approved, but tells us nothing else about the state which the text had reached at his death, or about the level of editorial intervention which it has received. The epilogue and the chapter on the critics of statistics both end so abruptly as to feel, to me, unfinished. I have the impression that what I believe to be problems in the book are probably not the result of Cohen's intentions: it is therefore uncomfortable and perhaps unfair to draw attention to them. The Triumph of Numbers is a welcome and immensely able account of its subject. It is neither the vast survey implied by the title nor, perhaps, the monument to Cohen's gifts that one might wish it to be. Benjamin Wardhaugh
Nonplussed! Mathematical Proof of Implausible Ideas by Julian Havil, Princeton University Press, 2007, $24.95, ISBN 0-691-12056-0 This book describes surprising, even paradoxical, results for readers who have a sound understanding of A level mathematics; a second volume is promised. Here fourteen chapters alternate between ideas from probability, and from the rest of mathematics, with a steady increase in depth and complexity. Some of the later manipulations may be rather daunting to the faint-hearted. The general style is well illustrated by the account of Buffon's Needle. Some historical background, a side-track to the fairground game of rolling a coin to land within a ruled square, consideration of needles of length l tossed at random onto a surface ruled with parallel lines distance d apart, with the cases l<d and l>d both considered, and a neat finish: if the needle is replaced by a convex polygon, all of whose sides are shorter than d, then the chance this lamina crosses a line is simply the ratio of the length of its perimeter to that of the perimeter of a circle of diameter d, irrespective of its shape! (But one vital condition has been overlooked – the diameter of the lamina must also be less than d.) The mecha-nically minded reader, who wishes to build a device where-by a double cone can roll uphill along a pair of inclined rails will find a full recipe here. Readers of Mark Haddon's The Curious Incident of the Dog in the Night-Time will recall the invocation of Conway's Soldiers: a lucid exposition of John Conway's proof that no soldier can advance more than four ranks is given. Edwin Abbott's Flatland is the inspiration for the chapter that looks at the calculation of volumes in spaces with a large number (not necessarily an integer) of dimensions, giving an excuse to introduce the Gamma, Digamma and Error functions, and to sum (tricky) infinite series. Parrondo's paradox is beautifully presented. The variety and general quality of the material covered are both admirable. I do have quibbles. Always placing figures and tables at the top of a page disrupts the flow of the text, and puts some material in the wrong section. There are a few minor slips and omissions. One is noted above, another is in the very first problem considered, where the number of matches to be played needs to be specified. Most of the time, an explanation as to why the apparent paradox should not really be a surprise is offered, but more opportunities to offer this enlightenment could have been taken. This is a splendid collection of articles, inspired by Martin Gardner's writings. Old conundrums are given new twists and applications, newer perplexing ideas are described with panache. The forthcoming companion book has a high standard to maintain. John Haigh John Dee: Interdisciplinary Studies in English Renaissance Thought edited by Stephen Clucas, International Archives of the History of Ideas, Vol. 193. Dordrecht: Springer Verlag, 2006, 366 pp, £111.00, €144, ISBN 978-1-4020-4245-4. As Stephen Clucas (the editor of the volume under review) notes, more than any other figure in the English Renaissance, John Dee has been 'fragmented and dispersed across numerous disciplines' (p.1), and attempts to present a coherent image of his multifaceted persona have relied on assertive, but ultimately misleading, exercises in moulding his career to the model of a particular world view. One of this volume's great strengths lies in its determination to present the 'problematic multiplicity' of Dee's activities in a non-synthetic manner. Thus, the volume is divided into six sections: Astronomy and Astrology, Dee and Maritime Affairs, Dee and the Occult Sciences, Dee's Conversations with Angels, Dee and Kelley, and Library Catalogue and Bibliography (this last section containing Julian Roberts' valuable additions and corrections to 'John Dee's Library Catalogue' and a bibliography of recent works on Dee by Clucas). This arrangement inevitably leads to conflicting (sometimes even contrary) presentations of Dee's outlook and significance, but this is a virtue in itself, for it lays bare the complex historiography with which scholarship on Dee must now contend. The divisive heritage of Dee studies is ably examined in Clucas' detailed and instructive introduction, in which the volume's editor navigates expertly though a historiographical minefield, charting as he travels Dee's fortuna within intellectual history. At the heart of the 'historiographical tension' evident in Dee studies is the (continuing) debate over the relationship between magic and science at the beginning of the so-called Scientific Revolution. Indeed, one of the most illuminating aspects of Clucas' Introduction is the extent to which Dee has been at the centre of several decades of passionate argument about the foundations of modern science, from E.G.R. Taylor's and Francis Johnson's attempts to revive the 'mathematical' Dee, through I.R.F. Calder's 'Neoplatonic' Dee and (most controversially) Frances Yates' and Peter French's 'Elizabethan Magus', to the vituperative rebuttal of the Yates thesis by Robert S. Westman and John Heilbron, leading (eventually) to a more nuanced account of the relationship between the 'occult' and the 'scientific', amply evident in this volume. It is to be hoped, despite the obvious (and very serious) deficiencies of the Yates thesis, that this volume might prompt a more balanced, reflective evaluation of Yates' overall contribution to intellectual history. It is impossible to do justice to the fourteen essays that constitute this volume, but historians of mathematics will be most interested in the lucid, erudite, and compelling contributions by Robert Goulding and Stephen Johnston on, respectively The Parallactic Treatises of John Dee and Thomas Digges and John Dee, Thomas Digges and the Identity of the Mathematician as well as by Robert Baldwin's study of Dee's Interest in the Application of Nautical Science, Mathematics and Law to English Naval Affairs. Despite its long gestation, this volume is a timely and welcome contribution not only to Dee studies, but to Renaissance intellectual history as a whole. There remains much work to be done on Dee and his milieu; this book offers an inviting springboard for future research. Alexander Marr
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