REVIEWS

Click on a review title in the alphabetical list below,
or scroll down to browse the most recent reviews.

 
Alex’s Adventures in Numberland
A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics
Bright Boys
Carl Friedrich Gauß: Biographie und Documente
Circles Disturbed: The Interplay of Mathematics and Narrative
Cows in the Maze and other mathematical explorations
Decoding Reality: The Universe as Quantum Information
Driven to Innovate: A Century of Jewish Mathematicians and Physicists
Duel at Dawn
Euclid’s Elements
Femininity, Mathematics and Science, 1880–1914
Galileo’s Muse
Carl Friedrich Gauß: Biographie und Documente
Hidden Harmonies, The Life and Times of the Pythagorean Theorem
How to Fold It: The Mathematics of Linkages, Origami and Polyhedra
In Pursuit of the Travelling Salesman: Mathematics at the Limits of Computation
Is God A Mathematician?
In Pursuit of the Travelling Salesman: Mathematics at the Limits of Computation
Mathematics: A Beautiful Elsewhere
Mathematics and Music
Mathematics in Nature
Mathematics in Victorian Britain
Mathematics of Life: Unlocking the Secrets of Existence
Mathematics of the Heart
Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice
Nets, Puzzles and Postmen: An Exploration of Mathematical Connections
Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction
Numbers Rule
Secrets of Creation: The Mystery of Prime Numbers
Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-First Century
Tenet
The Best Writing on Mathematics 2010
The Big Questions – Mathematics
The Genius in my Basement: the Biography of a Happy Man
The Great Mathematicians
The Mystery of Prime Numbers
The Num8er My5teries
The Psychology of Learning Mathematics
The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions and Commentary
Trefethen's Index Cards: Forty Years of Notes About People, Words and Mathematics
What’s Luck Got to Do with It?
Why Beliefs Matter: Reflections on the Nature of Science
You Can Count on Monsters

Earlier Reviews can be found in the Published Issues.




  Book cover of 'In Pursuit of the Travelling Salesman'

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation by William J. Cook,  Princeton University Press, 2012, 272 pp, £19.95, $27.95 ISBN 978-0-6911-5270-7.

(Review published online 07 August 2012)

The Travelling Salesman Problem (TSP) – finding a lowest weight Hamilton circuit in an edge-weighted complete graph) is the poster child of an NP-complete problem - eminently relevant in practice, easy to explain to your neighbor, and incredibly hard in practice. Its tentacles are all over mathematics - from the theoretical question of P versus NP over algorithms and heuristics to computer code and heroic feats of large examples.

Much of this is discussed in this book, written by William Cook (not to be mistaken with Stephen Cook of NP fame), a researcher in combinatorial optimization and co-author of one of the standard TSP software solvers.

As the reader will likely surmise from the title (and pricing), this book is primarily aimed at the popular science market. They will find plenty of background material and diversions to questions of intelligence and art, while not having to face more advanced mathematics than a basic linear inequalities that are used to explain linear programming. In that, the book will be accessible to the interested layperson. It would make for an excellent gift for the A-level student who needs a nudge to realize that mathematics is a far sexier discipline than software engineering.

But for the self-proclaimed professional (such as this reviewer) this book is well worth the read. There is much to learn about the historical background of topics that are not found in the typical discrete mathematics textbook (such as the commercial failure of Hamilton’s dodecahedron game or the practices of 19th century travellers), about the history of solving TSP concretely on the computer, and about the methods used in solving. (While the book touches on P versus NP, it is more interested in solving problems than establishing that they are hard.)

The book’s meaty core is in Chapters 5 to 7 which describe the linear programming approach towards solving TSP. Besides the reduction to a widely studied and in general well behaving problem, this also provides a lower bound that can be used to establish optimality of a solution without the need to enumerate all possibilities. The initial price to pay is that initially a solution of a linear program, being rational and not 0-1, does not necessarily yield a Hamilton circuit. Cook describes nicely how to introduce further inequalities to eliminate such forbidden artifacts and how to use geometric constraints of the problem (such as clusters of towns away from the rest) for further improvements.

The book is generously illustrated, not only with diagrams describing the strategies, but also with many photographs (alas often quite small) of the protagonists and of applications. It will be a rich resource for anyone teaching a course on optimization or algorithms.

The book’s author maintains an attractive companion web page which among others offers an iPhone application that implements his “Concorde” TSP solver with a nice graphical interface that lets the reader investigate the methods described in further examples.

Alexander Hulpke
Colorado State University

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  Book cover of 'Sources in the Development of Mathematics'

Circles Disturbed: The Interplay of Mathematics and Narrative edited by Apostolos Doxiadis & Barry Mazur, 2012, 552 pp, £34.95, ISBN 978-0-69-114904-2.

(Review published online 16 July 2012)


‘Essays on the interplay of mathematics and narrative.' What could this mean? Is it a blow against the timeless, Platonic view of mathematics? Or about finding mathematical structure in literary plots, as in Proust? Or stories containing mathematical objects, such as those of Borges? Or just finding analogies between the two disciplines? (One of the editors, Barry Mazur, said, in an interview in Nature, that the common thread in mathematics and literature is urgency.)

Actually, it is all of the above. As a result, it is a long book, with much food for thought, but in places heavy going. I found the essays about mathematics the most satisfying, but I am not at home in the discourse of literary criticism, and people on the other side (including at least one contributor) find the mathematics hard.

The driving force behind the project was Apostolos Doxiadis, and his long essay was (for me) the best. Where does mathematical proof come from? Doxiadis traces its roots in the narrative poetry of the Homeric epics; the most important station on the streetcar line leading from epic narration to mathematical proof is rhetoric, in particular forensic rhetoric (the style of argument used in law courts). The forensic rhetorician has to show that a certain course of events probably happened, maybe because the alternatives are ruled out by the evidence; this recalls the mathematician's proof by contradiction. There are several specifics in which mathematics resembles narration or rhetoric: notably, technical features known as chiasmus and ring-composition are ubiquitous at all levels in epic verse, rhetoric and Euclid's proofs. (Readers who enjoyed the partisan account of the struggle between rhetoric and dialectic in Robert M. Pirsig's Zen and the Art of Motorcycle Maintenance will welcome this analysis.)

Among the best articles are those that tell the story behind the production of some piece of mathematics. The primary title of the book refers to the fatal encounter between Archimedes and the Roman soldier to whom he said, ‘Do not disturb my circles!’

Colin McLarty tackles Gordan's (possibly mythical) remark on reading Hilbert's proof of the finite basis theorem: ‘This is not mathematics, it is theology!’. This remark could be interpreted in many different ways, not all of them meaning a rejection of Hilbert's proof by Gordan. (In fact, Gordan refereed Hilbert's paper, and, while recognising the importance of the result, was critical of imprecision in the proof, saying memorably that ‘It is not enough that the author make the matter clear to himself. One demands that he build a proof following secure rules.’

Another remarkable story is told by Michael Harris, concerning Robert Thomason's paper on which his dead friend Tom Trobaugh is a co-author. Trobaugh's ghost appeared to Thomason in a dream and gave him an instruction which Thomason, on waking, knew could not work; yet it led to the key result of the paper. Harris bases a discussion of artificial intelligence in mathematical proofs on this incident. An android mathematician could not cooperate with us unless it could understand the message from Trobaugh's simulacrum in Thomason's dream.

From the other side, Jan Christoph Meister speculates on a story-telling algorithm which could pass the Turing test: ‘Tell me a story, and I will tell you whether you are human or a machine.’ Sometimes I wonder whether some blockbusters are already produced by such an algorithm…

Peter Cameron
Queen Mary, University of London

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  Book cover of 'Sources in the Development of Mathematics'

Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-First Century by by Ranjan Roy, Cambridge University Press, 2011, 974 pp, US$99, £65.00, ISBN 978-0-521-11470-7.

(Review published online 16 July 2012)


Ranjan Roy’s view, stated in the preface to this book, is that mathematicians should make a habit of studying the great mathematical works of the past. By doing this, he argues, they are sure to enrich their own thinking in the same way that writers of fiction can draw inspiration from classic literature. In particular, he encourages the use of historical texts in mathematics teaching: for example, he suggests that they can provide a rich resource for projects or seminar topics.

With this manifesto in mind, he has taken on the rather daunting task of providing a survey of sources on infinite series, products, and continued fractions from fifteenth century Kerala to twentieth century Europe. He presents a vast treasure trove of examples, several of which are only tangentially related to series and products: for example, the inclusion of a chapter on finite fields seemed to me rather surprising. Such chapters might easily have been cut during editing; however, the breadth of examples fits in fairly well with Roy’s aim of encouraging original perspectives on modern mathematics through the study of its development, and it highlights the unexpected ways in which series and products can be used.

Contrary to what is suggested by the title Sources in the Development of Mathematics, Roy does not provide extracts directly from the primary sources themselves. Rather, he presents a collection of brief commentaries on the original texts; these are accompanied by some background information, sets of exercises for the reader, and suggestions for further reading. The examples are grouped by topic into 41 chapters, arranged in approximately chronological order. The close connections between the topics mean that the work would have benefitted from systematic cross-referencing between related chapters; however, the book is easy to dip in and out of, and for those looking for a particular topic, the table of contents is detailed enough to serve as a useful navigational aid.

Sadly, this book would not be a suitable text for a course in the history of mathematics. Though the author does seem to have some appreciation of the sensitivities of presenting historical mathematics for modern audiences, he is not always successful in avoiding anachronism, and there are a number of howlers. For example, Chapter 7, on 'Geometric Calculus’, describes supposed attempts by mid-seventeenth century mathematicians such as Blaise Pascal, Isaac Barrow and James Gregory to calculate integrals, though the notion of an integral had not yet been articulated. Furthermore, Roy’s commentaries in the same chapter imply that Pascal, Barrow, and Gregory were working with functions; this, too, is a concept that simply had not yet been invented. Such modernisations severely obscure the meaning of the source texts and the intentions of their authors, and it is not always made clear where Roy’s notation or wording differs from the original.

Due to these shortcomings, I cannot recommend this book as a reliable authority on the history of series and products. However, the wide range of examples it contains make it a good resource for mathematicians and mathematics students to dip into to light the touchpaper of inspiration.

Rosanna Cretney
The Open University

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  Book cover of Trefethen's Index Cards

Trefethen’s Index Cards: Forty Years of Notes About People, Words and Mathematics by L.N. Trefethen, World Scientific Publishing, 2011, 368 pp, £13, US$19.95, ISBN 978-9-814-360692.

(Review published online 16 July 2012)


This is an unusual book. Since the age of 14, Nick Trefethen, FRS, Professor of Numerical Analysis at Oxford and current president of SIAM, has been collecting his thoughts on index cards. The result is presented for us here, and is definitely not a diary: it’s clear that these thoughts are meant for public consumption. Indeed, in a card dated from 1982, he fantasises that he will one day “find on my desk a book of Trefethen’s Duplicated Index Card Notes”. Instead, to compare this book to modern means of communication, each card is somewhere between a tweet and a blog post: a typical card contains around 900 characters, or 120 words.

The collection runs the gamut from religion, fatherhood, relationships, nuclear proliferation, mocking jokes about Oxford (Trefethen calculates that an undergraduate Oxford degree contains 1.37 bits of information), to, of course, mathematics. In addition, the benefits of touch typing play a surprisingly large role, mentioned no less than six separate times: “what a waste, to hunt and peck for a lifetime at 30 words per minute!” The end result is mixed, full of questions but with few answers.

Some of the most entertaining cards result when Trefethen considers the practical world. In this category, we have a prediction from 1987 that “dog toilets will appear on the market before too long” and a prediction from 1990 that we will soon see a “touch-of-a-button, cup-at-a-time coffee machine”. Achieving a 50% success rate on such predictions is impressive – and perhaps dog toilets lie in our future. We also find a theory of why sleeping babies are so hard to wake, a silly but entertaining argument in favour of not wearing bike helmets, and a timely metaphor for corporate lawyers (bankers too?): “I think of money as a system of rivers and tributaries flowing through the population [...] even a mediocre corporate lawyer can expect the money to pour in by the tens of thousands of dollars, camped as he is by the Amazon”.

Trefethen’s thoughts on mathematics will probably be the highlight for most readers. One of my favourite cards considers the similarity between mathematical precision and literary ambiguity. I also, even as a pure mathematician, appreciated his analogy between classical/rock music and pure/applied maths: “classical music and pure mathematics get trapped in their own history, their own self-awareness, their own high standards”, concluding “it is the rockers and rollers in both fields, looser and sloppier but with their heads properly attached, who do much of what’s really important”.

You will enjoy this book on your coffee table, so long as you understand beforehand what you are reading. Do not read it as an autobiography. Trefethen, at least as he presents himself, follows a geodesic from determined and high-performing adolescent (at the age of 14 he writes “I cannot bear the idea of going through life without being the most effective person who ever lived”) to determined and high-performing academic. Also do not read this book intending to agree at every turn. Trefethen himself admits “I’ve never had a conversation with Nick Trefethen, but I’m guessing I might not like him so much”. Whether you like him or not, most of his Index Cards will leave you with something to think about.

Vince Vatter
University of Florida

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Poster for the play 'Tenet'  

TENET

(Review published online 16 July 2012)

by Lorne Campbell and Sandy Grierson
directed by Lorne Campbell;
Gate Theatre, Notting Hill, London, May 2012

The playwright Lorne Campbell had wanted to write a play about Évariste Galois ever since reading about him ten years ago, but “whenever I tried to turn the story into a show it would either veer into mawkish sentimentality or sound completely overwrought with meaning in the works possible way.” But in collaboration with Sandy Grierson – astonishingly, introducing Julian Assange was the key – and the result was Tenet, A True Story About the Revolutionary Politics of Telling the Truth as Edited by Someone Who is Not Julian Assange in Any Literal Sense, presented by Greyscale Theatre at the Gate Theatre in London during May.

This turned out to be an astonishing night at the theatre, with sensational performances by the two actors, Lucy Ellinson as Assange and Jon Foster as Galois. There was a lot of mathematics in the play, and it was presented intelligently and accurately, but there was much more to the play than that.

Galois’s concern is pure, abstract, mathematical truth: if humans had never lived, then Galois’s mathematical structures would still exist (or so many of us like to believe). Assange is concerned with revealing the truth about the very human world around us. But the personal circumstances of both are extremely murky. We don’t know why Galois fought the duel in which he died (indeed, it has apparently been suggested that there never was such a duel) and what we do or do not know about Assange is equally complicated.

So the parallels are fascinating: two revolutionaries, fighting for truth where truth is very slippery. Having gone to the theatre out of curiosity, expecting (to be frank) at best an interesting failure, I found this to be an exceptionally rich experience, leaving the audience with much to think about regarding politics, mathematics and truth. I was fortunate that on the evening I went, there was a post-performance discussion with Peter Neumann, whose insights and good humour added to the occasion.
It is rare for theatre to explore mathematics so provocatively. I very much hope that there will be further opportunities to see Tenet: I certainly want to see it again!

Tony Mann
University of Greenwich

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The Big Questions – Mathematics by Tony Crilly, Quercus Publishing, 2011, 208 pp, £12.99, ISBN 978-1-8491624-0-1.

(Review published online 20 June 2012)

This book is the fourth in the The Big Questions series published by Quercus and edited by Simon Blackburn. The book under review is a nifty-looking little blue-cloth hard cover with an elasticised black-band page marker attached to the back. As one of my students put it, the book just feels ’cool’. All of the texts of the series make excellent travelling books and are very entertaining reads for the layperson, with the prior books in the series generally receiving very positive reader reviews on Amazon.com. To date, the titles in the series include Philosophy, Physics, The Universe and the book under review, Mathematics.

This book, written by Tony Crilly, continues the strong tradition of the series; it is very entertaining and puts twenty questions of and about mathematics on the table, addressing each with a compact essay, and with interconnections between topics well cross-referenced. The author’s expertise in the history of mathematics shows in his ability to place the discussion of his selected Big Questions into context. Laypersons reading the text will be entertained and enlightened as to the ’why’s and ’how’s of the appearances of mathematics in everyday life, and if mathematics students at university read this text early on in their coursework, then likely they would be much improved; perhaps even understand why we recommend the studies we do, and how it is we claim that mathematics is the lively, organic and dramatically growing field that it is today. In short, I must give this little blue book a strong positive review.

The author applies high art in order to discuss the development of real mathematics within entertainingly titled chapters like ’What is mathematics for?’, ’How big is infinity?’, or `Can a butterfly’s wings really cause a hurricane?’. For the reviewer, favourite chapters included ’Are statistics lies?’, ’Is mathematics true?’ and ’Is there anything left to solve?’. Ironically, two of these last chapters also played host to the few minor issues I had with the text.

Lest I sound too positive, let me now mention those few minor quibbles. Firstly, I felt that some mention should have been made, within the discussion on the debates between intuitionists and formalists in the early 1900s, of Poincaré’s wonderful contributions to those discussions. Secondly, perhaps the discussion of the Navier–Stokes equations was a bit light in the context of the rest of the discussions of the Clay Prize questions in the chapter ’Is there anything left to solve?’. And, finally, after doing an excellent job covering the difficult Gödel’s Incompleteness Theorem, the text ends the discussion of ’Is mathematics true?’ with a bit of a sad poetic bailout into academic relativism ending with the phrase ’… it could be said that mathematicians today pursue truth as they see it.’ Well, let me say that this, perhaps, is not how I interpret the meaning of that amazing theorem.

All in all, a nifty and fun book that does more than this reviewer expected.

Collin Bleak
University of St Andrews


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Mathematics in Victorian Britain edited by Raymond Flood, Adrian Rice and RobinWilson, Oxford University Press, 2011, 466 pp, £29.99, ISBN 978-0-19-960139-4.

(Review published online 20 June 2012)

According to its editors, Mathematics in Victorian Britain is the first general survey of Victorian mathematics. The editors are right. Other major English-language surveys of mathematics include Eleanor Robson’s and Jacqueline Stedall’s The Oxford Handbook of the History of Mathematics, which offers short chapters covering varying cultures in mathematics from antiquity to modernity, as well as Ivor Grattan-Guinness’s Landmark Writings in Western Mathematics: 1640–1940, which offers snippets of key texts published in the history of mathematics over the past 300 years. But neither of those previous publications focuses specifically on British mathematics.

With an A-list of contributors, Mathematics in Victorian Britain meets high expectations. The intention of the compendium is to provide an overview of the breadth, scope and nature of mathematics throughout the Victorian period.

It was a time during which Britain was transformed politically, culturally, industrially, and mathematically. As is argued throughout the publication, the legacy of Victorian mathematics resides in mathematical physics.Names such as WilliamThomson, George Gabriel Stokes, Peter Guthrie Tait and James Clerk Maxwell serve as thematic descriptors of the period in general, in particular in theories of heat, thermodynamics and electro-magnetism. Iron, steel and steam engine industries propelled applied mathematics into an era of great scientific innovation, including the mathematisation of ’energy’ and the formalisation of vectors.

In pure mathematics, on the other hand, British mathematicians performed weakly. Arthur Cayley, James Joseph Sylvester and their Oxford colleague Henry Smith did help to transform the notion of matrices into a full-blown theory of linear algebra. Cayley and Sylvester further worked to develop invariant theory. Yet, despite Cayley’s and Sylvester’s prolific and productive engagements, pure mathematics in Britain failed to rival its European counterparts.

In the opening chapters, Tony Crilly, Keith Hannabuss, Adrian Rice, Raymond Flood and June Barrow-Green cover the development of Cambridge’s Tripos mathematics system, Oxford’s lack of mathematical culture, and attitudes towards mathematics in universities in Scotland, Ireland and throughout the Commonwealth. Sloan Evans Despeaux then explores the publications through which Victorian mathematics was disseminated.

A.D.D. Craik, Allan Chapman, Doron Swade and M. Eileen Magnello describe areas of applied mathematics that were especially strong throughout this period, including astronomy, calculating engines, as well as vital and mathematical statistics. Magnello’s chapter, ’Vital Statistics: The measurement of public health’, explores Florence Nightingale’s use of statistics in tracking illness and hygienic standards. It stands out as a colourful account of nursing-as-statistics in the Victorian period.

The final chapters provide readers with a survey of pure mathematics in Britain from the development of rigorous calculus, geometry, and algebra to the rise of logic and combinatorics. Two chapters stand out as especially technical, namely Karen Hunger Parshall’s ’Victorian Algebra: The freedom to create new mathematical entities’, and I. Grattan-Guinness’s ’Victorian Logic: from Whately to Russell’. Yet both contributions still appeal to the general reader, as they contextualise algebra and logic by placing them within the fluctuating social conditions of Britain at a time when citizens were questioning absolutist norms of truth and veracity in politics, religion and, most especially, science.

The compendium ends on a contrarian note, however, with Jeremy Gray’s ’Overstating their case? Reflections on British pure mathematics in the nineteenth century’. Gray argues that Victorian-era mathematicians in Britain are often over-rated. Vis-à-vis their European and continental colleagues, British mathematicians underperformed and under-innovated. The supposed ’Greats’, including Cayley, Sylvester and perhaps the young William Kingdon Clifford, whose work on quaternions and non-Euclidean geometry was cutting edge, did not meet European standards. "British pure mathematicians of the nineteenth century have been overrated, to the detriment of historical writing on the subject," Gray writes, adding "what is striking about so many British mathematicians is their belief that they were the equals of their Continental peers, when no such comparison can be entertained".

At 466 pages, the compendium is worth the read and certainly worth the price for both the mathematical dabbler and the historical debutant. For specialist researchers in the history of mathematics, the most valuable component is its brilliant ’Notes, References, and Further Reading’, which provides thorough literature reviews of each topic discussed on a chapter-by-chapter basis. Similar compendia exploring mathematics throughout the seventeenth, eighteenth and twentieth centuries would be welcome additions to the libraries of mathematical historians.

Josipa Petrunic
SSHRC Research Fellow, History of Mathematics
University of Alberta


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The Genius in my Basement: the Biography of a Happy Man by Alexander Masters, Fourth Estate, 2011, 352 pp, £16.99, ISBN 978-0-00-724338-9.

(Review published online 15 May 2012)

The genius in my basement is, superficially at least, a biography of Simon Norton, probably best known as one of the authors of the Atlas of Finite Simple Groups. The biographer, Alexander Masters, was Simon’s tenant and friend for many years, and the book is as much a tale of their friendship as a description of its subject. One of its main themes is mathematical genius, why it occurs when it does and why some people produce amazing work at certain periods of their life and then stop. Along the way, it explains the very basics of group theory.

The style is extremely engaging: Masters writes fluently and entertainingly, and the text is interspersed with cartoons, drawn by the author. One unusual aspect is that the subject frequently interjects, even argues with, the text, making corrections and disputing phrasing. At first, this made me feel relaxed about the ways in which the book makes fun of Simon, as it felt as if he was very much in on the joke.

However, and this is a big however, it became increasingly apparent to me that the book has been far less collaborative than it first appears. The reason for this is nothing to do with its rude, but funny, descriptions of Simon, but is the fact that there are several mathematical howlers in the text. For example, the Monster is not "the largest finite group in the universe" (p. 296). I cannot believe that these would have passed unnoticed, if the book had really been read by its subject. This sense that the book might be exaggerating the extent of Simon’s editorial control meant that on a second read I was wincing, rather than laughing, at the book’s frequent grotesque flights of fancy.

This is not to say, however, that the author is just being snide. An awful lot of self-mockery takes place, and some of the funnier moments are where Simon interjects (entirely accurately) to correct Masters’ tendency to overstate and oversimplify. It’s written with a great deal of warmth, and this was one of the reasons why at first I found it such an amusing read. However, on further thought it feels as if the author has abused a long-term friendship for a few laughs, to make the book sell better. It could be argued that Masters is making a point about how biographies are inevitably distorting, since one person is necessarily being seen through the biased eyes of another, but this seems to me to be an insufficient justification for the intrusion.

A more minor complaint is that, once again, mathematicians are characterised as not really living in the same world as the rest of humanity. This undoubtedly makes for a more entertaining book, but is a little tiresome: it would be lovely to read a biography which celebrated a mathematician’s combination of creativity and precision, without making them seem in some way incapable of functioning in the real world.

This book is funny, entertaining, and often intriguing, but it has made me extremely uncomfortable, and I am not at all sure that I can recommend it.

Colva Roney-Dougal
University of St Andrews


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Secrets of Creation – Volume 1: The Mystery of the Prime Numbers by Matthew Watkins, The Inamorata Press, 2010, 362 pp, £15, ISBN 978-0-9564879-0-2.

(Review published online 2 March 2012)

As the title of this volume, and that of the proposed trilogy which it begins (Secrets of Creation), suggest, Matthew Watkins thinks someone or something has a secret. His opening observations concern the over-quantification of the world, the use of numbers where they are inappropriate, and the marginalisation of what cannot be quantified. These opening sentiments are typical of a volume where the expression of opinions is neither restrained nor limited to the topic at hand. He complains about how few, even in the mathematical community, know the prime number theorem, to say nothing of the improvements that go back to Riemann. What he proposes to do, in this volume and those to follow, is to make the material accessible to those with limited mathematical background. He seeks to avoid algebra except in the appendices, and the result is an unhurried survey of the interaction between addition and multiplication.

One of the notions the author employs crucially is that of a spiral. First, he wants to give readers still wary of logarithms from school the chance to appreciate what a logarithm is, by counting windings of a spiral. Then he turns to spirals as a way of generating the kind of waves into which he decomposes the function that represents the deviations between what appears in the prime number theorem and the count of something he calls ’primeness’. These so-called ’spiral waves’ appear in the title of the second volume of the series, and his development of them here gives the reader a sense of how they are a generalization of sine waves. 

Ramanujan is said to have known numbers as individuals. In a similar way, Watkins argues that primes can be studied by a sort of ’social statistics’ rather than like pebbles. He decries the value of ’prime hunting’, the quest for yet larger prime numbers, by comparison with understanding the distribution of primes. His plentiful use of analogies should aid the reader for whom the underlying algebraic ’reality’ would be a formidable challenge.

The author displays a touching confidence in logic, as he repeatedly observes that its conclusions generate absolute confidence.  He argues that the prime numbers are as close to fundamental as anything studied by human beings, and speculates about how efficient they might be for communication with other civilisations. There is an unevenness in his historical observations, as he is careful to point out the lack of evidence that Euclid discovered the fundamental theorem of arithmetic, but then unreservedly gives him credit for the proof of the infinitude of primes. He has striking images, like the notion of a function as a spell cast on the number line. When he finds a couple of authors referring to the primes as ’weeds’, he takes the occasion to criticize the notion of ’weed’ as a byproduct of capitalism run amok.

Even those for whom all the mathematics in the volume is familiar are bound to find Watkins’ presentation original. It would have benefited from some editing to remove repetitions of text and especially of quotations.  The illustrations are entertaining without always helping to illuminate the argument further. It is perhaps unfair to judge the volume by itself when it is only the first of a trilogy. There is, however, the assurance that readers are likely to be eager to follow the story of spiral waves as the basis for the prime numbers in subsequent volumes.

Thomas Drucker
University of Wisconsin-Whitewater

Note: Volume 2 of the trilogy was published in December 2011: The Enigma of the Spiral Waves, 249 pp, £12.50, ISBN 978-0-9564879-1-9.


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MATHEMATICS OF THE HEART

(Review published online 28 February 2012)
7 February – 3 March 2012
Theatre 503, London SW11

Paul MacMillan is a mathematician in mid career. He is neither successful nor especially unsuccessful, and he is still in the shadow of his late father, a domineering man who was widely respected in the world of mathematics. He is involved in the modelling of storm patterns, and his own work is concerned with North Utsire, a bleak area which we are not surprised to learn he has never visited. Not for him a trip to the beaches of Hawaii at some Research Council's expense.

Paul has all the negative characteristics that many people associate with mathematicians and, as if that weren't enough, a younger brother who is as outgoing as he is introverted, and who has moved in with him for an indefinite stay. Paul is also in a long-term relationship that is not going well.

As the play opens, a new graduate student, Zainab, arrives in Paul's shabby north London flat. She is keen about mathematics and about research, highlighting Paul's own diminishing enthusiasm. And in the space of a few minutes she inadvertently makes matters worse by saying how much she admires the work of Paul's father, by creating even more tension between Paul and his girlfriend, and by falling under the spell of his brother.

That's pretty bleak for Paul, and things are considerably worse by the end. It's not a bleak play, though, at least not most of the time. A lot of it is very funny and there are also some surrealist touches, some of which look like the results of the director, Donnacadh O'Briain, turning to advantage the limitations of the venue, currently the upstairs room of a pub in Battersea.

In Paul's first conversation with Zainab they are both obviously alluding to chaos (well, obviously if you're a mathematician). Neither of them actually mentions the word, though, and I didn't feel that the author, Kefi Chadwick, had attempted to work chaos into the plot, in the way that, say, Tom Stoppard brought concepts of quantum mechanics into Hapgood. I think she was right: the play is about how difficult someone like Paul can find it to cope with ordinary life, and it makes sense for events to flow more or less directly from his weaknesses rather than owing too much to chance and tiny perturbations.

I went to see Mathematics of the Heart with three friends, none of whom are mathematicians, and on the basis of that blatantly non-random sample, I'm happy to recommend it to mathematicians and non-mathematicians alike. Though it might not be such a good choice if you're with someone you're trying to reassure that we mathematicians are perfectly normal people.

[ See POSTER AND PROMOTIONAL OFFER ]

Peter Saunders
King's College London

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Number-crunching: Taming unruly computational problems from mathematical physics to science fiction by Paul J. Nahin, Princeton University Press, 2011, 400 pp, £20.95, $29.95, ISBN 978-0-6911 4425-2.

(Review published online 1 February 2012)

If this book were a three-course meal in a posh restaurant, I’d know exactly what to write: dodgy starter, yummy main, a most peculiar dessert, and a waiter who drove me slightly bonkers but gave me a lovely cuddle as I left. Now to translate all that into a book review....

Number-Crunching is a book for people with a pretty high level of mathematical sophistication: Nahin expects the reader to manipulate equations, change variables in an integral, apply various rules of differentiation, and so on. It’s not a relaxing read: there is a good level of rigour to the mathematics and one is expected to follow fairly subtle lines of argument carefully.

Which makes it all the more unfortunate that he spends half of the introduction discussing ’TUPA’, an imaginary ’Universal Photo Album’ containing all the images ever occurring in the universe. The punch line is that TUPA cannot exist because the number of such images is many orders of magnitude greater than any feasible storage device. It’s a poor start: if you’re writing a book for people who can integrate by parts, you shouldn’t expect them to be impressed by the fact that 10100 is much bigger than 1010.

So much for the dodgy starter. The main course, though, I enjoyed: Nahin discusses several important problems in mathematical physics; he uses sophisticated mathematical techniques to analyse them and then, when the mathematics hits its limits, he shows how computers can be used to uncover unexpected behaviour.

The first big example is the hotplate problem (solving the heat equation on a unit square with prescribed boundary conditions). I hadn’t thought about such things since undergraduate days but Nahin drew me in and reminded me that the mathematics in this area is a delight. He solves different cases of the problem using different analytic methods, and then uses MATLAB to implement two different algorithms (one iterative, one probabilistic) which exhibit interesting behaviour.

This is followed up by an historically important piece of mathematical physics: the study of mixing in a one-dimensional crystal. The crystal is modeled as a string of particles linked by springs, and a complete analysis is given for the case where particle interaction is linear. The discussion is beautiful and the result entirely satisfactory. This is followed up by a numerical analysis (again using MATLAB) of the non-linear situation; this analysis is now known as the ’Fermi–Pasta–Ulam Computer Experiment’ after the three physicists who first performed this analysis on MANIAC I. The behaviour that MATLAB displays is unexpected and gives the reader a good sense of the excitement that Fermi and company must have felt. It is a splendid example of the power, and importance, of computers in the study of mathematical physics.

The dessert in Number-Crunching is a series of short science fiction stories probing the various possibilities for computers in modern life. The stories themselves are mildly enjoyable, although they represent a strange non-sequitur in the context of the book. They are also rather crude and, accompanied as they are by a discussion of the place of Sci-Fi in literature, I found them very limited in scope.

And, finally, that infuriating waiter. The book is littered with personal anecdote of the ’wudja believe it?!’ variety, includes frequent reference (including lengthy quotes) to the author’s other books, and periodically lapses into the tone of a fireside chat. Still, for all that, the self-effacing enthusiasm of the author carries the day: I might have been seriously annoyed by some of the hokum, but instead I ended up charmed by Nahin’s obvious sincerity. This is a man who really believes in the value of computers to the work of a mathematical physicist and, after reading the book, I can see his point.

Nick Gill
Open University

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The Great Mathematicians by Robin Wilson and Raymond Flood, Arcturus Publishing, 2011, 208 pp, £9.99, ISBN: 978-1-84837-902-2.

(Review published online 1 February 2012)

The Great Mathematicians is a largely chronological account of the development of mathematics from the Ancient Egyptians to the present day. Material is presented in a series of two-page spreads, most of which focus on the life and selected achievements of a single mathematician. In the words of the authors: ’This book aims to present mathematics with a human face.’

It divides the history of mathematics into five main areas. Ancient mathematics is focussed for the most part on Greek mathematics, but also includes material on Chinese, Indian, Mayan and Arabic mathematics. Early European mathematics stretches from Fibonacci to Desargues. Awakening and Enlightenment traces the development of mathematics in the seventeenth and eighteenth centuries. The Age of Revolutions begins with Gauss and ends with Klein. Finally, The Modern Age is a selection of mathematician from the twentieth and twenty-first centuries, concluding with a discussion of Perelman and the Poincaré conjecture.

The book is attractively bound, and at £9.99, provides a competitively priced nontechnical overview of the development of mathematics. I found the exposition of the first two chapters particularly appealing, but as the book progresses the biographical sketches become thinner and the authors struggle occasionally to present more technical material in an accessible fashion. Thus the coverage of figures such as Gerbert (later Pope Sylvester II) and Recorde (who introduced the × and = signs) appealed greatly to me. I was disappointed, however, by the coverage of Abel and Galois, who share a single spread with a description of the classical problems of doubling the cube, trisecting an angle and squaring the circle. Their stories are related in a telegraphic style, missing out on a golden opportunity to tell two of the more melodramatic stories from the history of mathematics.

While the book contains descriptions and illustrations of a great number and range of mathematical results, it contains relatively few formal proofs, with the notable exception of a presentation of Cantor’s diagonal argument on page 165. This is a pity: without proofs it can be difficult to appreciate the elegance of the material that is being presented. These are minor reservations however; the lack of depth in the treatment is counterbalanced by the comprehensive bibliography provided for further reading.

The authors have been successful in providing a comprehensive and non-technical introduction to the great mathematicians of history.

Padraig Ó Catháin
NUI, Galway

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(© Olivier Ouadah)
 
Hiroshi Sugimoto
Conceptual Form 011, 2008
 

MATHEMATICS: A BEAUTIFUL ELSEWHERE

(Review published online 10 January 2012)

Exhibition, 21 October 2011 – 18 March 2012, Fondation Cartier pour l’art contemporain, Paris

How can one communicate the ’beautiful elsewhere’ of abstract mathematics? If one is willing to accept Cédric Villani’s explanation, in the series of interviews created by filmmakers Raymond Depardon and Claudine Nougaret, the answer to this recurrent question for every working mathematician is right in front of us: there is nothing as effective as a blackboard! However, when trying to reach a general audience a more visual beauty often comes to the rescue. With this idea in mind, the Fondation Cartier pour l’Art Contemporain conceived the nearly impossible dream of establishing a fertile dialogue between many mathematicians and artists with whom it has worked closely in the past. The project came true thanks to the enthusiastic dedication of curators Jean-Pierre Bourguignon, Michel Cassé and Hervé Chandès, who relied on the patronage of the Institut des Hautes Études Scientifiques and UNESCO. The exhibition Mathematics: a Beautiful Elsewhere is on in Paris until 18 March 2012.

Japanese mathematicians of the Edo period already understood the preeminent role that art can play in the popularization of science, as they initiated the tradition of Sangaku: wooden tablets representing puzzles in Euclidian geometry that were placed as offerings at the entrances to temples. Using the latest technology in computer animation, BUF Compagnie has given life to them in a sequence of videos, which also show how Chinese people were able to perform quite complicated calculations with sticks. It is not the only homage to Sangaku: inspired by them, artist Beatriz Milhazes has composed a collage in which equations ruling phenomena such as irisation, morphogenesis or electromagnetism invite visitors to open the Book of Nature, where "numbers govern even fire". Joseph Fourier could hardly imagine that the quotation opening his monumental treatise on heat would be used two centuries later to design a fireplace next to Misha Gromov’s Library of Mysteries! The Russian mathematician proposed a selection of thirty major works in the history of mathematics and human thought, from the fragments of Heraclitus to Grothendieck’s Récoltes et Semailles. But how to actually exhibit them? To that end, David Lynch has created a structure in the shape of a zero, in which an audiovisual installation presents extracts from the books, in a journey from the smallest particles to the largest galaxies.

Everybody agrees that infinity is one of the hardest concepts to comprehend. What about trying to learn it by looking at Hiroshi Sugimoto’s three metre high Surface of revolution with constant negative curvature, whose tip is only two millimetres in diameter? Henri Poincaré would have been fascinated by this modern representation of the pseudosphere because, as painter Jean-Michel Alberola shows, hyperbolic geometry had a central place in his mathematical world. What about the mathematical worlds of today’s researchers? Eight mathematicians coming from different fields have given eight beautifully framed answers. While Michael Atiyah thinks of mathematics as a "reflection of what people dream", for Don Zagier they form an open world in which the only way to get surprising results is to let oneself go by ideas. That is probably the best metaphor of the whole exhibition. As one visitor put it, "if I had known mathematics was like that…" 

Javier Fresán
Université Paris 13
 
(© Olivier Ouadah)
The Room of the Four Mysteries

 
 
 
 
Exhibition posterExhibition catalogue cover

 

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Hidden Harmonies, The Life and Times of the Pythagorean Theorem by Robert and Ellen Kaplan, 2011, Bloomsbury Press, 304 pp, $25.00, ISBN 978-1-596-915220.

(Review published online 10 January 2012)

I really enjoyed this book I had no idea that there was so much to say about Pythagoras’ theorem. I have been fascinated, and have probably bored my colleagues to death. Still, everyone wants to borrow my copy now...

The book starts with some history. The proof of the theorem is of course attributed to the Pythagoreans in the middle of the sixth century BC. But were they really the first to notice it? Was it already known to the Egyptians, or even the Babylonians?

Chapter 1 discusses the Egyptians. There’s evidence in a Middle Kingdom (2500–1800 BC) papyrus that they knew about 3,4,5 triangles, then there are references by Democritus in the fifth century BC to the ’rope stretchers of Egypt’, who were apparently constructing right angles two millennia earlier using ropes of lengths 3, 4, 5 units, and there is possibly (well, maybe) some evidence from megaliths.

But then Chapter 2 looks further back. Maybe the theorem actually originated even earlier in Mesopotamia. There are records of many gnomonic Pythagorean triples (those with two numbers consecutive) in records of old Babylonian mathematics. There’s the evidence of a broken tablet (the ’Babylonian Shard’) from around 1760 BC, and then there’s the palm-sized ’Yale tablet’, showing a calculation of the digits of root 2 (computed in the Babylonian base of 60) alongside a sketch of a square with its diagonals. Surely it was knowledge of the theorem that motivated that calculation?

But certainly the Pythagoreans knew the theorem. In Chapter 3 we learn about their society and style of proof, and the proof of irrationality of root 2 is attributed to the Pythagorean Hippasus.

Chapter 4 continues with the Greeks, and especially Euclid. His Proposition 1.47 proves our theorem, with three squares and a tangle of five additional lines constructed around the right-angled triangle. This proof has been variously called the ’mouse trap’, ’Dulcarnon’ (meaning ’two-horned’ in Persian), and ’the bride’s chair’; there’s a beautiful reproduction of a front cover from La Vie Parisienne, showing a soldier carrying his bride and all her chattels on his back, which really does seem to have been constructed on Euclid’s diagram.

There have been a few published collections of proofs of the theorem, the largest apparently being Loomey’s collection of 367 in Harvard library; Chapter 5 looks at a few. I’ve long liked the proof that places the hypotenuses of four a,b,c triangles against the inside edges of a square of edge c, revealing a square of length a b in the middle; although somehow I feel it is cheating, since I need to do some algebra to see that the combined area of those four figures is really a2+b2. But I found a new favourite on p. 131: since the a,b,c triangle (with hypotenuse c) is the union of similar triangles with hypotenuses a,b, its area is the sum of theirs, but of course their areas are in the same ratio as the squares of corresponding lengths. And I am amazed that Alvin Knoer could still find a new proof in 1924!

I thought that Chapter 5 would be the climax of the book. But it carries on. Chapter 6 is about consequences; we discuss Ptolemy, Pappus, Apollonius, Dijkstra, and more. Chapter 7 classifies all relatively prime Pythagorean triples. Chapter 8 moves beyond 2 dimensions, ultimately to infinite dimensions and Fourier series. Then in Chapter 9, we see the ’Deep Point of Dream’, that Pythagoras’ theorem is equivalent to Euclid’s parallel postulate.

This book is about far more than Pythagoras’ theorem. It’s accessible, and fun. I recommend it!

Sarah Rees
University of Newcastle

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Galileo’s Muse by Mark A. Peterson, Harvard University Press, 2011, 352 pp, £21.95, ISBN 978-0-674-05972-6.

(Review published online 10 January 2012)

In this book, structured around Galileo but constituting a much wider discussion on Renaissance thought, Peterson argues that Renaissance art was a key motivating force behind the rediscovery of hellenistic science and the development of new scientific thought. With the science of the classical period having suffered a reversal over time (mathematics in particular having become an ossified logical system in an obscure branch of philosophy), Peterson argues that the arts provided the stimulus for new, ’living’ mathematics and physics. He shows how problems arising in the arts led to the need for new mathematical techniques (e.g. perspective in painting inspired work in optics and gave geometry a new connection with describing space), while ways of thinking from the arts provided a fertile mental landscape for new scientific approaches (eg. considering multiple viewpoints, or the imprecision of observational data).

Peterson illustrates this cross-pollination between the arts and sciences through an analysis of the work of key figures, including Dante, Piero della Francesca and Kepler, as well as Galileo. In his final chapter, Peterson contends that an Oration delivered by one of Galileo’s students was largely written by Galileo himself. While I cannot personally evaluate the likelihood of this claim, it provides an opportunity for Peterson to bring together various strands.

Peterson’s scientific background (he is a professor of mathematics and physics) makes him well equipped to identify and elucidate mathematical ideas present in works which are often considered in purely artistic terms. I particularly enjoyed his essay on Dante, in which Peterson demonstrates that Dante’s ’Paradiso’ contains a construction of the 3-sphere and climaxes with an extended metaphor encapsulating Archimedes’ proof of the area of a circle.

Peterson’s writing style is very readable, and is effective in conveying his enthusiasm. Not only is the book itself self-contained, providing a comprehensive overview of all topics featured, but the sections can be read in any order, making it easy to ’dip into’. I found the book very thought-provoking; it gave me a new appreciation of how the Renaissance was not simply a ’flowering’ of cultural activity, but represented a significant step-change in modes of thought. Overall, I would thoroughly recommend this lively and stimulating book to anyone interested in the history of ideas.

Sophie Huczynska
University of St Andrews

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Carl Friedrich Gauß: Biographie und Documente by Hans Wußing, 2011, EAGLE 051 Leipzig, 279 pp, €26.50, ISBN 978-3-937219-51-6.

(Review published online 2 December 2011)

This book first appeared in 1974 in a series of popular short biographies of major scientists published by the Leipzig house Teubner. It consisted of ten chapters that ran efficiently through the main features of the life and especially the work of its subject, and ended with a timeline and a bibliography, mostly of historical literature. It was reprinted four times, to 1989; but this edition is substantially different. The chapters read more or less as before, but the text contains far more illustrations, especially likenesses of Gauss and others, and title pages of some publications and manuscripts.

The first main change is the addition of 25 short ’documents’ concerning ’Gauss in his intellectual and private circumstances’. The author, often quoting contemporaries or historians, writes several of them; the rest are passages photoreproduced from historical writings, ending with one by the author.

The second change is a further 25 documents on contacts between Gauss and Teubner, prepared in part to celebrate in 2011 the bicentenary of the house. Although they did not publish Gauss’s own books, they dominated the publication of Gauss scholarship, which began soon after his death in 1855. Gauss’s compatriots produced a fine edition of his Werke in 12 large volumes between 1863 and 1933, including many manuscripts as well as his publications (which are far more numerous than is suggested by his famous conceit ’pauca sed matura’). Large supplements to volumes 10 and 11 constituted (some reprints of) articles and monographs on aspects of his work that were produced in a project directed by Felix Klein. In addition, there have been editions of the main (and massive) correspondences with major colleagues, especially with astronomers. Some of these texts came out from another Leipzig house, Engelmann, including several Gauss items in their important series of source books ’Ostwalds Klassiker’, as is duly noted in 20 pages of reproduced title pages of ’Leipzig reference books’. Since 1962 the Gauss Gesellschaft has published a useful series of slim Mitteilungen. All of this distinguished historical scholarship is at least noted bibliographically in this sequence of 25, much of which is little known to those Gauss enthusiasts who do not read the language; again some short passages from historical texts are photo reproduced.

This half-century of additions more than doubles the length of the book; while the order somewhat resembles a random walk, much useful information on Gauss is made available. The book ends with illustrations of Gauss, especially in stamps (where the author draws upon his own splendid collection for sciences and scientists), banknotes, coins, medals in Gauss’s honour, and statues, and the lifeline. One hopes that someone will be inspired to draw upon all these resources and produce or edit the definitive Big-Book(s) biography that Gauss deserves.

The preface of this book is dated February 2011; as is noted on the next page, Wussing died in April after a long fight against cancer. A leader of the history of science in the German Democratic Republic for decades, he was also a major figure in the history of mathematics, receiving the May Medal in 1997 for his services to the field. He was also a substantial author in the popularisation of science and especially mathematics; for example, this book was one of several that he contributed to Teubner’s series.

I. Grattan-Guinness
Middlesex University

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Cows in the Maze and other mathematical explorations by Ian Stewart, 2010, Oxford University Press, 320 pp, £8.99, ISBN 978-0-19-956207-7.

(Review published online 2 December 2011)

Cows in the Maze is a collection of 21 articles from the author’s Mathematical Recreations column in Scientific American magazine. As one might expect, they are pitched at the intelligent, but not necessarily mathematically educated, reader.

The articles cover a variety of topics, most of them well within the traditional scope of "recreational mathematics", but none the worse for that. The stock-in-trade of the genre (from knots to knight’s tours, and magic squares to tilings) is all present, but the exposition here is outstanding, and in many cases the author finds a new slant to interest even those familiar with the basic ideas. For example, while the counterintuitive consequences of Bayes’ theorem have been covered many times before, the presentation here features a fascinating discussion of its implications in the legal world, with well-researched reference to real-life cases.

Three articles in the middle of the book make a detour into theoretical physics, forming a short(ish) story in which the characters discuss the possibility of time travel in a relativistic universe. A number of the other articles also take the form of dialogues or short stories. One can see how this would draw in the casual reader of a non-specialist publication, but the willing purchaser of an entire volume on mathematics can probably do without this kind of insulation from the subject matter, and might prefer to see things presented in a more straightforward way.

The only topic to leave me completely cold was that referred to by the title: an extraordinarily complicated self-referential "logic maze". For me, elegant and simply stated mathematical problems naturally cry out for attention, while more technical ones have to earn their crust by virtue of some external importance. This seems to be a problem crafted for the specific purpose of being convoluted and difficult to understand, and I couldn’t summon up the enthusiasm necessary to get to grips with it. That said, the prominence given to it suggests that the author feels it is a strong selling point of the book, so perhaps I am missing the point and everyone else will love it!

While the articles themselves are outstanding, the volume as a whole is perhaps not quite the sum of its parts. Each article is simply presented, with minimal editing, as a chapter. The only significant additions to each are a brief discussion of feedback received by the author after publication and a list of websites for further reading. A general lack of progress or development between chapters inevitably makes the book feel disconnected, and a little unsatisfying to read from cover to cover. On the other hand, the format (the only edition available seems to be a paperback with black and white illustrations) does not naturally make it the kind of book one would treasure and dip into from time to time, or keep out on a coffee table.

In summary, the articles collected are undoubtedly a masterclass in mathematical writing for the casual, non-specialist reader, but I am not persuaded that simply collecting them in a paperback volume does them justice.

Mark Kambites 
University of Manchester

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The Psychology of Learning Mathematics: The Cognitive, Affective and Contextual Domains of Mathematics Education by Paul Ernest, Lambert Academic Publishing, 2011, 164 pp, ISBN-13: 978-3-8443-1306-2.

(Review published online 4 November 2011)

Being interested in the study of how mathematics students think mathematically, I hoped this book might help me understand these thought processes better. I was not disappointed. The Psychology of Learning Mathematics is a study of current research into how mathematics is learned and taught. The first half of the book deals with the ways in which children learn mathematics and demonstrates how teachers can raise attainment by improving their own understanding of the learning process. There is a middle chapter on the theories of constructivism which leads nicely into the remaining section dealing with problem solving set partly in the context of teaching mathematics in Higher Education. It was this final section that I found the most helpful but the book as a whole has helped me increase my understanding of how mathematics is learned and, I hope, has brought fresh inspiration into my teaching this year.

The sub-title of the book is important. This is not just a study of how mathematics is learned cognitively but includes the affective domain; the influence of people’s feelings, attitudes and learning context on their mathematical learning. The book provides a comparison between the attitudes of 11- and 15-year-olds to mathematics and shows how enjoyment and ability to see the use of the subject diminishes as learners get older. There is some discussion concerning the differences between the genders and the relationship between attitude and achievement.

As someone who likes to do mathematics with some sort of writing implement in my hand. I found the example of the mental maths skills of street vendors in Brazil quite absorbing.  Researchers found that these children can perform some quite complex mental arithmetic in the market place but do not perform nearly as well on similar questions on paper. The author goes on to examine the differences between written algorithms and mental methods, understandably concluding that both are important and necessary in the learning of mathematics.

The last few chapters concern problem solving, a skill that we frequently tell our students is their main asset when it comes to future employment. Ernest makes the point that students can only acquire a finite number of examples (despite their insatiable hunger for more) and it is from these that they have to be able to form patters of heuristics that provide the basis for transferring to real world problems. This section of the book has helped me to realise why my students find this hard and has encouraged me to find new ways to persuade my students to think rather than solve problems routinely. There is much reference to work by Pólya, Burton and Mason in the context of general problem-solving strategies and ideas as to how this can be implemented in Higher Education.

This is a fascinating look at how mathematics is learned and cannot fail to influence the teaching of any reader involved in the teaching or learning of mathematics at any level.

Noel-Ann Bradshaw
University of Greenwich

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How to Fold It: The Mathematics of Linkages, Origami and Polyhedra by Joseph O’Rourke, 2011, Cambridge University Press, 180 pp; pb £19.99, ISBN 978-0-521-14547-3; hb £50, ISBN 978-0-521-76735-4.

(Review published online 3 October 2011)

With its origins in a monograph by the author (with Erik Demaine) How to Fold It takes a mathematical approach to a selection of topics relating to ’folding’ – whether these folds are found in the hinges of linkages and robot arms, the bond angles in proteins, or in paper shopping bags. Each topic is treated in an analytical way with a traditional theorem-and-proof approach, and is presented with motivating discussion, photographs and exercises to develop understanding. Animations and templates are provided at www.howtofoldit.org to help with visualisation and encourage the reader to print, fold and experiment.

The major theorems presented are remarkable – results that may surprise the reader include the fact that, with the right folds, any shape or collection of shapes (even ones with holes in) that is composed of straight lines may be cut out from a sheet of paper with just a single cut. For those that are motivated more by abstract shapes than paper chains, there is a worthwhile discussion of general results regarding whether a convex polyhedron can be sliced along its edges to unfold into a net – and conversely which polygonal nets can be folded up to form which polyhedra (yes, in many cases more than one!). My personal favourite shows why a paper shopping bag stays open while it is being filled, but folds away neatly afterwards – with the answer lying closer to graph theory than to physics.

The author aims to assume only ’high school’ mathematics – ’a little algebra, trigonometry and geometry’ – but still manages to reach the boundaries of knowledge by describing open problems. Inevitably, some of the fine detail of the bigger theorems is glossed over, but this is not unwelcome as the sketches provided give a good idea of how the proofs would look.

Although time is taken to explain the process of proof by induction (and even the notion of using Greek letters to label angles) I was still left wondering how far this book would suit a non-mathematician. The proofs demand concentration and careful reading, and the rigour of the approach may well scare off those who are not already familiar with a thorough theorem-and-proof approach. Those most suited would be undergraduates and postgraduates who have yet to be convinced that the topic of folding can produce any ’proper’ mathematics – this will surely leave them in no doubt.

Martin Smith

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The Best Writing on Mathematics 2010 edited by Mircea Pitici, Princeton University Press, 2010, 440 pp, £13.95, $19.95, ISBN 978-0-6911-4841-0.

(Review published online 5 September 2011)

The Best Writing on Mathematics 2010 is a diverse collection of thirty-five articles on and about mathematics appearing in 2009. Whilst a good number of them are taken from familiar publications such as Mathematical Intelligencer and Notices of the AMS, the majority appear in places I (for one) would not normally look, or even have heard of.

Initially approaching this book as something of a bag of sweets, I was a little disappointed not to find more articles of the kind found in the section called Mathematics and Its Applications, short pieces about some interesting bit of mathematics. For example ’Knowing When to Stop: How to Gamble If You Must – The Mathematics of Optimal Stopping’ is a very nice exposition by Theodore Hill around the Marriage Problem (or Secretary Problem). However, one could reason that this sort of article is readily available and that we don’t really need a guide to locate them.

In a similar vein to Mathematics and Its Applications, although about mathematics in general rather than specifics, are the sections entitled ’Mathematics Alive’ and ’Mathematicians and the Practice of Mathematics’.  These consist of articles which often defy classification, ranging from ’What is Financial Mathematics’ to the very promising sounding ’If Mathematics Is a Language, How Do You Swear in It?’, and from a short description of Tim Gowers’ recent ’Massively Collaborative Mathematics’ project to the somewhat rambling ’Birds and Frogs’. These pieces really are varied, and contain some thought-provoking stuff together with the slightly disappointing.

The bulk of the book lies in the sections on education, history and philosophy. This is where things get more serious, and is perhaps where the book is most useful. ’Mathematics Education’ contains well-chosen journal articles covering very diverse aspects of the area. I enjoyed all of these pieces and so am loathe to pick out one or two for mention, but they encompass adolescent learning, textbooks, special needs teaching, aesthetics, cognition, mathematical writing and the use of paper models in geometry. Six out of seven of these articles I would not otherwise have read or indeed have known about had they not been served up to me on a plate.

’History and Philosophy of Mathematics’ also consists mostly of scholarly pieces, and includes articles on Kronecker, Rota, inconsistent mathematics and mathematical belief.

As a mathematician engrossed in my own area, but who will read Mathematical Intelligencer if it is on the table of a common room, I’ve been delighted to have this book in my house. One inevitably will not agree with every choice of work for inclusion, but it would be a dull book if it simply presented us with what we like. What is important is that it is varied and balanced, and contains the odd surprise.

Charles Eaton
University of Manchester

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Mathematics of Life: Unlocking the Secrets of Existence by Ian Stewart, Profile Books, 2011, 320 pp, £20, ISBN 978-1-8466-8198-1.

(Review published online 4 July 2011)

Mathematicians and physicists have long fought over the big names. Newton? He was one of us, the physicists claim, with his studies of gravity, light and motion. But he invented calculus, the mathematicians reply, he worked on geometry and harmonic series. And what about Stokes, Dirac and Einstein – were they motivated by the abstract ideas of mathematics or the natural wonders of physics?

If a rapidly developing trend continues, mathematicians and biologists may soon be having similar arguments. Ian Stewart’s latest book guides us through the recent collision of these two fields. This is not a book about mathematics with a bit of biology sprinkled on afterwards – Mathematics of life weaves a history of biology with examples of how mathematics can help solve the unanswered questions that were created along the way. Mathematics, Stewart argues, will be the next biological revolution.

In fact it’s already happening. In the last couple of decades mathematicians have increasingly become involved in everything from ecology to genetics – the tools at their disposal a valuable addition to existing biological methods. Science is evolving, with new levels of cooperation allowing us to do things that were once unachievable.

Of course, mathematical biology covers a bewilderingly large variety of research. Many of these topics would fill books on their own, so instead Stewart gives us an overview of a few key research areas over the course of nineteen chapters. Well-chosen illustrations allow the reader to get their head around the biological background, and analogies are provided to help them understand the systems that mathematicians are working to explain. The topics covered are wide ranging: how symmetry plays a role in virus structure, the split of a single species into two and even animal coat patterns; why game theory – famously used in A beautiful mind to ensure that everyone gets a date – can help find out which evolutionary strategies are best; what problems in genetics can be understood using probability.

Stewart is a stalwart of the popular maths genre, having previously written accounts of mathematical subjects as diverse as chaos theory, symmetry and probability, and his engaging, accessible style is also present here. In fact, this book doesn’t contain much mathematics in the shape of formulae and calculations, but this is precisely Stewart’s point about mathematical biology – the puzzles should come from the biologists, rather than biology just being another area of application for existing mathematical results.

As well an author, Stewart is also a researcher, and his work on how animals walk inevitably gets a mention, sandwiched between sections on the brain and leech heartbeats. The book’s breadth, ambitiously aiming to give the reader a gist of many different corners of such a big research field, makes it an interesting read but inevitably creates a weakness too – many topics are omitted or mentioned only too briefly. However, by including a solid biological backdrop for the problems he does cover, Stewart gives the book a nicely rounded feel, even if some chapters leave the reader wanting more. As an overview, it provides an entertaining and up-to-date insight into this exciting field.

Despite its title, this is a book for fans of biology as much as for those interested in mathematics. In many ways it reflects the increasingly blurred boundaries between the two subjects (I know mathematicians that work in zoology departments, medical schools and even for government health agencies), and gives an absorbing introduction to one of the fastest growing areas of modern science. According to Aristotle, "in all things of nature there is something of the marvellous". Now mathematics can help us find it.

Adam Kucharski
University of Cambridge

A version of this review first appeared in Plus magazine (http://plus.maths.org).

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The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions and Commentary by K. Kedlaya, B. Poonen, R. Vakil, MAA, 2011, 352 pp, £37, US$59, ISBN 978-0-8838-5827-1.

(Review published online 4 July 2011)

The William Lowell Putnam competition is considered to be one of the hardest mathematical competitions for undergraduate students in the world. It is open to university students in USA and Canada and held in two sessions on the first Saturday of December with six problems for each session. The problems can all be solved with basic undergraduate mathematics but all need a non-standard approach or idea. The Putnam is notoriously difficult: in the first place the problems are hard; in the second, a partial solution carries very little or no credit. As a result, most of the participants receive a total score less than five (out of 120). Another interesting rule is that the ranking of the top five scoring students (the Putnam Fellows) is not provided.

The Putnam competition has been running since 1938 and this book, which is the third in the series, covers the period between 1985 and 2000. The most striking feature is the detailed solutions section, which occupies the majority of the book. Multiple solutions are given when known, and provided they are sufficiently different and/or illustrate different approaches to the problem. Some of these solutions were found by the competitors themselves in the limited time of the contest. Connections with general theories and current research are frequently given, which invite the reader to think about new and old open problems in Mathematics. There is also a detailed list of results, trivia facts and an entertaining article by Bruce Reznic on the selection of the Putnam problems and what it means to be a member of the Putnam committee. The book will be invaluable for anybody with an interest in mathematical competitions or just trying his or her hand at challenging Mathematics at university level. All theories begin with simple, easy to state and beautiful problems and this book has an abundant supply of those.

Nikolai Nikolov
Imperial College London

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Why Beliefs Matter: Reflections on the Nature of Science by E. Brian Davies, Oxford University Press, 2010, 272 pp, £25.00, ISBN 978-0-19-958620-2.

(Review published online 6 June 2011)

This is a fascinating book on the nature of science, and its relationship to our beliefs about the world. The main scientific areas covered are mathematics and physics (plus a little biology), with topics including the scientific revolution, the Platonic idea of mathematical reality, and the relationship between science and religion. Davies is arguing on several fronts, so it’s difficult to sum up his ideas briefly, but broadly speaking he argues in favour of a Pluralist approach, arguing that the science that we, as humans, develop is inevitably determined by our biological nature. He posits that a purely reductionist approach to understanding the human condition will never succeed, and that with our limited intellectual capacity we will always need more than one explanation for why higher-level facts are true: for example, we should never expect physics to cast light on ethical questions, even though the two areas can occasionally intersect.

On the mathematical front, he argues strongly against Platonism, mocking the idea that when we do research we are stumbling blindly around a world of ideal mathematical forms, eventually deducing something about their shape which we then bring back to our physical universe and christen a theorem. Instead, he advocates a Pluralist approach, rejoicing in intellectual ingenuity, seeking constructivist proofs where possible, and (possibly a big ask!) retaining a clear idea of which parts of mathematics can be proved constructively and which cannot.

Standard theories of science present it as developing from hypothesis, to observation, to either rejection of, or increased confidence in, the hypothesis. Eventually the hypothesis is either disproved or becomes an accepted fact. Davies argues that this description is flawed in several respects. The practice of science depends strongly on both intuition and the availability of technology, and it is manifestly not the case that a single observation which ran contrary to the predictions of a well-known theory would make one throw away the entire theory. Davies also disagrees vehemently with those postmodernists who argue that science is just a cultural phenomenon: "science is important because it works, not because it has advocates in high places". 

On the relationship between science and religion, Davies surveys a huge range of thinkers, from Dawkins, to Swinburne, to Keith Ward. He concentrates on Christianity, but most major religions are mentioned. A description of the religious beliefs of ten eminent scientists shows conclusively (and unsurprisingly) that it is impossible to generalise about scientists’ attitude to religion. More unexpectedly, a discussion of some leading Christian thinkers goes on to conclude that it is difficult to find any common core beliefs amongst them, either! Ultimately the chapter is inconclusive, and is clearer about what Davies disagrees with than what he agrees with, but this does not feel like a weakness.

Davies is not afraid to pick a fight, and he attacks both individuals such as Feyerabend, and whole disciplines such as multiverse theory, with verve, wit and not a little acidity. For example, in the middle of a discussion of determinism we get the lovely sentence "Replacing the phrase ’in principle’ by ’not’ often makes a sentence correspond more closely with reality", which I’ve been mentally applying to news broadcasts ever since! Although some of the ideas in the book are complex, the presentation is both lucid and entertaining. It has made me re-evaluate my own beliefs about the nature of mathematics. Davies raises more questions than answers, and I strongly recommend to you this thought-provoking book.

Colva Roney-Dougal
University of St Andrews

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Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice by Sasha Borovik, AMS, 2010, 317 pp, US$59, £43.50, €50.00, ISBN 978-0-8218-4761-9.

(Review published online 5 May 2011)

Announcing that you are a mathematician in polite company is likely to be met with awe and terror in equal measure, as well as the conviction that the mathematical brain is a curious and unusual beast. So this book is a welcome attempt to try to de-mystify the processes that underly mathematical cognition.

The point of view is that of a mathematician interested in talking to neurophysiologists and psychologists, trying to give insights into the basic structures and mechanisms necessary for higher mathematical thought. The idea being that while there has been a good deal of work done on mathematical cognition insofar as it relates to counting and quantification, there has been much less work on higher-level mathematics.

The book draws on a staggeringly eclectic array of cultural references – from Vermeer’s The Astronomer, to the sitcom Father Ted, to Bulgakov’s The Master and Margarita – by which the author entertains and amuses while never straying too far from the topic. It also contains a wealth of mathematical examples and puzzles, and the reader should be prepared to be challenged by an array of Olympiad-style problems, as well as Coxeter Theory and a particularly tricky Sudoku.

The author’s main thesis is that mathematics is ’vertically integrated’, and mathematical cognition consists of various simple processes including visual and geometric intuition, parsing (formal manipulation of symbols) and reification (encapsulation of a mathematical technique as an object). However, while all of these points and many more besides are amply illustrated by a variety of mathematical problems, anecdotes and history, I found it hard to come away with a coherent picture of the author’s view. This is perhaps unsurprising given the difficulty of the subject and the daunting task facing an experimenter.

Also, to name a criticism which both the author and a potential reader would anticipate, I was never convinced that what was being communicated was much more than an insight into the author’s mental processes whilst doing mathematics. I am not sure, for instance, that Coxeter’s proof of Euler’s Theorem – a geometric proof that an orientation-preserving isometry of Euclidean 3-space fixing a point is a rotation – is really a slam dunk in demonstrating the prominence of a certain kind of geometric intuition in mathematical thinking. While I myself found it convincing, that is a world away from saying that I’d expect all mathematicians to find it so. And given the fallibility of geometric intuition in certain contexts (all the false proofs of the Poincaré Conjecture, for instance), one needs to be doubly careful.

I am equally skeptical of the stated aim to communicate to neurophysiologists and psychologists using "simple examples". For instance the first, and by no means most difficult, such example is the iteration of the function ||x| – 1|. Having spoken to some of my reasonably numerate colleagues from the biological sciences, I am reasonably certain that they would not find such examples easy to deal with at all, and certainly not illustrative in the way the author expects.

Despite all these criticisms, this book is nonetheless a joy to read. The fact that it falls short of its own impossibly high ambition is not something I can hold against it, as the delight in mathematics is both present on every page and infectious in the extreme. This book contains some of the most interesting mathematical anecdotes and puzzles I have ever encountered, all decorated with a variety of diagrams, pictures and photos of mathematicians from when they were children, making for an excellent and entertaining read.

Armando Martino
University of Southampton

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The First Six Books of the Elements of Euclid, with explanatory booklet, by Oliver Byrne, Taschen, 2010, 268pp + 95 pp, £34.99, ISBN 978-3-83651-775-1.

(Review published online 5 May 2011)

In the 19th century Euclid’s Elements were the basis of school mathematics. In 1847 a radically different edition of the first six books was published by Oliver Byrne, a mathematics teacher who went on to become "Surveyor of Her Majesty’s Settlements in the Falkland Islands". His edition, "in which coloured diagrams and symbols are used instead of letters for the greater ease of learners", is the best known of his works. Byrne claimed that its use could reduce the time needed to master the material by two thirds.

Byrne’s novel idea was to replace the usual symbols for lines, triangles, etc, by coloured shapes. He had four colours at his disposal and so a line became a coloured segment: — , a triangle a set of (vari-coloured) lines, an angle a solid coloured shape like   and so on. Then, in the proofs, the various geometric entities were represented by symbols corresponding to parts of the original diagram.

To one accustomed to the usual way of writing out a proof this does not seem like much of an improvement, though one must admit that, for example, in the pons asinorum or Pythagoras’s Theorem the proof becomes an attractive riot of colour.

Byrne had experimented with this in schools and had evidently decided that the schoolteacher’s approach would be better described by his method than in the usual way using A, B, C, ... . He says:

... the use of coloured symbols, signs, and diagrams in the linear arts and science renders the process of reasoning more precise, and the attainment more expeditious ... the retention by the memory is much more permanent; these facts have been ascertained by numerous experiments by the inventor.

There were, however, severe implementational difficulties. In the 1840s accurate colour printing was rare, and Byrne was lucky to find a publisher who was prepared to take on the task. He found William Pickering, who produced a masterly version of Byrne’s vision. Alas, such a production was bound to be expensive and consequently sales were limited: the production costs of this work contributed to the bankruptcy of Pickering a few years later. This means that only a few copies of Byrne’s work found their way into circulation and hence, by the peculiar irony of the second-hand book trade, those that did are now extremely valuable.

Of course this meant that Byrne’s dream of seeing his method replace the usual way of presenting Euclid was never realised. Cajori, writing eighty years later, dismissed the work as "a curiosity" and echoed the poor opinion that De Morgan had expressed a couple of years after its publication.

However, Ruari McLean, writing in Victorian Book Design and Colour Printing (1972) calls it "one of the oddest and most beautiful books of the whole century". Historians of art have seen the geometric works of Mondrian pre-figured in Byrne’s diagrams – though Mondrian admitted to never having seen the work! It is easy to leaf through the volume and appreciate it as a fine work of design rather than as a mathematical text.

The facsimile edition by Taschen is beautifully produced, cloth-bound in its own hard case. It comes with a booklet which actually adds rather little to the book itself. An historian of mathematics rather than of art will probably not covet this sumptuous reprint, but would certainly appreciate the web pages put up by Bill Casselman (www.math.ubc.ca/~cass/Euclid/byrne.html).

I particularly recommend Book II Propn VI to appreciate just how lively Byrne can make a result as dry as:

If a straight line be bisected and produced to any point, the rectangle contained by the whole line so increased, and the part produced, together with the square of half the line, is equal to the square of the line made up of the half, and the produced part.

John O’Connor
University of St Andrews

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The Shape of Inner Space by Shing-Tung Yau and Steve Nadis, Basic Books, 2010, 378 pp, US $30.00, £20.00, ISBN: 978-0-4650-2023-2.

(Review published online 5 April 2011)

This book tells the fascinating story of strange geometric objects that have achieved some fame outside of mathematics: Calabi–Yau manifolds. Inspired by an open question in geometry, Shing-Tung Yau goes in search of a weird multi-dimensional object he thought didn’t exist, finds it, and wins the Fields medal for his efforts. A year later theoretical physicists noticed that the object he found, or rather objects, for there are many, are just what they needed. String theory claims that we live in a ten-dimensional universe. Since we can only perceive four of these dimensions, the other six must be hiding somewhere. As it turns out, the kind of object that can harbour the six extra dimensions and cater to other requirements of string theory is a Calabi–Yau manifold. String theory claims that every point in the 4D space we can perceive is in fact a tiny little 6D world with the structure of a Calabi–Yau manifold.

The story is told by Shing-Tung Yau himself, with the help of the science writer Steve Nadis. On a recent visit to London, Yau was adamant that mathematics should be brought to the masses without dumbing down or glossing over the tricky parts. And this is just what this books sets out to achieve. While aimed at a general audience, it doesn’t just tell the story of Calabi–Yau manifolds, but explores their mathematics in detail.

The book takes a broad approach, starting with a look at the intertwined history of geometry and physics. This sets the scene to explain the question, first asked by the mathematician Eugenio Calabi, which eventually led Yau to the famous manifolds. True to Yau’s conviction, the mathematics is at the forefront throughout the book. Every single technical term in Calabi’s conjecture is explained, and there’s a chapter devoted to Yau’s proof as well as the mathematical machinery developed for it. Yau and Nadis explore the manifolds’ relevance to string theory, but also another interesting twist to the story: while geometry boosted string theory, string theory in turn revived a nearly forgotten area of geometry, concerned with counting the number of rational curves that can fit on a given manifold. Techniques from string theory eventually provided the answer to a question first posed by Hermann Schubert in the 19th century involving quintic manifolds. The book wraps up by exploring how and if all of this is relevant to the real world and pondering the connections between mathematics, beauty and truth.

The collaboration between a mathematician and a science writer has worked wonders in this book. It’s crowded with beautiful metaphors that clarify complex ideas and provide a peek into higher-dimensional worlds. Personally, I already knew a little bit about some of the mathematics involved, yet I had several penny-dropping moments I wish I’d had when I was first studying it.

One thing that comes through on every page of this book is the beauty of the mathematics and its power to shed light on the secrets of our Universe. This, therefore, is a great book to while away those dark winter evenings.

Marianne Freiberger
Co-editor, Plus

A version of this review first appeared in Plus magazine (http://plus.maths.org). Plus has also published an article based on an interview with Shing-Tung Yau; see http://plus.maths.org/content/node/5388.

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Is God A Mathematician? by Mario Livio; Simon & Schuster 2009, 320 pp, $26, ISBN-10: 074329405X.

(Review published online 9 February 2011)

My father told a story of, when he was a boy in the 1920s, an old man who couldn't believe that the music he heard on his new "cat's whisker" wireless "has been floating around in the air all these years and I've never heard it". The similar puzzle 'do we invent mathematics or discover it?' prompts the further question 'is God a mathematician?' – the title of this thought-provoking book. This generalises beyond the realm of mathematics and, in the author's words, "did God create humans in his own image, or did humans invent God in their own image?"

This is challenging stuff, and easy answers are too much to expect, but Mario Livio, like a good philosopher, has justified the case for such questions in the context of the history and development of mathematical thought and practice. He has produced an erudite but highly readable book, bringing to life something of the personalities and vision of many of the great masters of our subject. Though inevitably selective, the book nevertheless captures in a series of vignettes many of the great advances which have shaped our familiar mathematical landscape, and builds a convincing picture of Wigner's "unreasonable effectiveness of mathematics" in describing and explaining the workings of the universe.

We are introduced to the "mystics" of the great Greek flowering of philosopher/mathematicians – Euclid's geometry, Plato's "self-evident mathematical truths", Pythagoras' "natural laws" – who first identified those mysteries which form the philosophical basis for the rest of the book with its insistent question. There follow the "magicians" – Archimedes, Galileo, Newton, Gauss – who were able to pull the "rabbits" of mathematical theory out of the "hat" of the physical world. Then, moving towards the present day, the statisticians and probabilists, who demonstrated the power of mathematics in understanding and predicting apparently uncertain or even random behaviour, the geometers, with their departure from the constraints of Euclid to a world apparently unencumbered with the requirement of physical reality, but later providing the essentials to understanding, or even describing, some of the most fundamental aspects of particle physics or cosmology. Finally the logicians, whose search for formalism in mathematics was halted in its tracks by "the stake through the heart" of Gödel's incompleteness theorems.

Each of these chapters lays a fascinating historical trail, illuminated at each turn by anecdotes which bring the characters to life as real human beings of every nature and habit, influenced by their peers and the times they inhabited, but so often ahead of them in their vision and invention (or discovery?). The stage is set for the final assault on the summit – is God indeed a mathematician? Here Livio has assembled an impressive collection of views on the existence (or not) of a mathematics independent of the human mind, quite startling in their variation, but leaving me sitting on the fence kindly provided by his suggestion that mathematics is partly created by us (concepts) and partly discovered (relations among those concepts). Judgment on God's role is left as an exercise for the reader, who might bear in mind that, even in the 17th century, Spinoza pondered "whether we say that God has eternally willed and decreed that the three angles of a triangle should be equal to two right angles, or that God has understood this fact".

Every serious mathematician should read this book. It clothes its erudition (the notes and bibliography alone are almost worth the price!) with humour and humanity, and encourages that deep contemplation of our subject that is so often threatened by the pressures of the modern world.

John Brindley
University of Leeds

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Driven to Innovate: A Century of Jewish Mathematicians and Physicists by Ioan James; Peter Lang 2009, 312 pp, £25, 978-906165-22-2.

(Review published online 9 February 2011)

It is well known that many notable mathematicians and physicists have been of Jewish descent, but in what way did this affect their lives and contribute to their success? In Driven to Innovate, the distinguished mathematician (and former President of the LMS) Ioan James examines the lives of thirty-five Jewish mathematicians and physicists born in the nineteenth century. Whilst many suffered as a result of anti-Semitic prejudices or circumstances, James also shows how their Jewishness played a positive part in their achievements.

The book begins with a brief but essential overview of Jewish history that contains background concerning the community and ideology. James expresses his views and those of others as to why there have been so many prominent Jewish scientists, and explains what this collection of intellectuals has in common.

The biographies are grouped together chronologically with the majority from the second half of the century. They are written in a pacey popular style that is easy to read, and contain a wealth of appealing anecdotes. As well as discussing the area of mathematics or physics that each person is noted for, James comments on their relationships with other mathematicians of the time, their family life and, where applicable, how they were received as a teacher. It is these insights into the characters of the subjects that I think students in particular will enjoy, and I would urge them to read this book.

One profile I particularly enjoyed, possibly because it concerns the only female mathematician in the book and one of a total of three women in all (something James is quick to apologise for) is that of Emmy Noether. Having previously read only brief accounts of her mathematics, I was intrigued by this account of her life and the perceptive comments on her personality given by her colleagues and students. Amongst other things, we learn that she had a great rapport with her students and enjoyed spending time with them on long walks discussing mathematical topics.

James' style, as I have said before, is popularist; he is not setting himself out to be an historian and thus it would be unfair to judge him as one. But readers might be entitled to expect more references and details of sources. In many of the chapters there are sizable quotations, which give illuminating and fascinating details about the character in question, but sadly the specific source for these quotes is sometimes lacking.

Despite this, it is a cleverly written book riddled with Jewish history which gives it both structure and purpose. James' view, that what these characters have in common is that they have all been in some way Driven to Innovate, seems valid and as one who is (in James' eyes) partially Jewish this book has helped me to understand something of my own Jewish heritage.

Noel-Ann Bradshaw
University of Greenwich

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The Num8er My5teries by Marcus du Sautoy, Fourth Estate, 2010, 320 pp, £16.99, ISBN 978-0-0072-7862-6.

(Review published online 1 February 2011)

Imagine riding a train, and the man opposite you engaging you in conservation. He does so by inviting you to ponder a few mysteries: the reality of climate change, the security of the internet, the stability of the solar system. Then you notice something curious. While it seems like he is putting together random ideas, all of his stories culminate in a fascinating fact about the next station. This sense of going on a journey with a brilliant and entertaining companion is the strongest impression I got from this lovely book.

Du Sautoy has genuine gifts: for coming up with new ways of illustrating old ideas, and for telling old stories in fresh ways (correcting outmoded notions of mathematics as a progression of white male protagonists). I never thought I would see Mersenne primes and the Riemann hypothesis linked with dragon noodles from Mr Chang’s restaurant in Taipei.  The extent to which Du Sautoy links mathematics not only to things in the world, but to music, to movies, to art, is exhilarating.

The supporting structure of the book is composed of five of the seven Millennium Prize Problems: the Riemann hypothesis, the Poincaré theorem, the P vs NP problem, the Birch & Swinnerton–Dyer conjecture, and the Navier–Stokes problem receive one chapter each. (It bothered me that the Hodge and Yang–Mills conjectures were omitted, until I read that the book grew from five Royal Institution Christmas lectures.) Each chapter progresses via serpentine stories, puzzles, and illustrations which seem to be mere detours, until you realise that they were essential for allowing the final ’my5tery’ to be discussed at the end. While this sounds like a recipe for disaster, it works.

Distributed throughout are barcodes which can be photographed by a smartphone, linking to websites with additional information. Luddites like me – whose only phone is connected to the wall – are not left out: web addresses are there too. The websites contain games, things to print out and build, movies, and more besides. I think that my favourite was one which automatically generated artwork in the style of Coldplay’s X&Y album art. This was related to the discussion on codes – and the pitfalls of using them on album art covers.

The Christmas Lectures are aimed at children, and certainly many of the games and puzzles are aimed at that category, not to mention the book’s hyperlinked nature. But what would they make of references to "the head on your Guinness" (in a discussion of the Poincaré Theorem) and the drug ecstasy (in the build-up to the P vs NP problem)? The cover claims that the book is aimed at "ages 1–101", which might be a tad optimistic, but is broadly true.

If I have to find one area for improvement it would be in the figures. On the plus side, although all black and white, they are very varied, with little purpose-made quirky works of art, photos, diagrams, sketches. But it seems that the publisher did not want to pay for reproductions of many images, so for example chapter three has to describe, rather than show, Dürer’s Melancholia, a Roman 20-sided die, and the Sagrada Familia cathedral in Barcelona. In fact, I could only count three images in the book not produced by one of the team. But this is a small quibble.

Telling old stories with new twists (just how big is that pile of rice on the chessboard?), bringing new illustrations to bear on old problems (what is the history of the tetrahedral tea bag?), asking insightful questions (why can’t you blow a cubical bubble?): in all of these du Sautoy excels. The book contains some of the clearest and most remarkable explanations I have ever read of some of the deepest questions in mathematics. It illustrates like no other the beauty, power, fun, ubiquity, and compulsive nature of mathematics.

Phil Wilson
University of Canterbury, New Zealand

A version of this review was published in Plus magazine (http://plus.maths.org) in October 2010.

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Nets, Puzzles and Postmen: An Exploration of Mathematical Connections by Peter M. Higgins, Oxford University Press, 2009, 256 pp, £8.99 pbk, ISBN 978-0-19-921843-1.

(Review published online 1 February 2011)

Networks are everywhere where there are interconnections between objects – road, rail and airline networks, electrical circuits, neural networks and the world wide web. They also have their recreational side, in problems such as the Königsberg bridges problem and the four-colour problem. In view of their importance and accessibility, it is somewhat surprising that there are few popular mathematics books on the subject.

This book under review is a largely successful attempt to fill the gap. The opening chapter discusses networks in general, with particular reference to trees, chemical isomers and truthteller-and-liar problems. Here it is a shame that there are no pictures of real-life networks to set the scene – the first picture depicts the trees with six nodes, which is hardly likely to excite the reader.

Later chapters deal with Games of logic, Connection problems, Colouring and planarity, Traversing a network, One-way systems, Spanning networks, Network flows, and various recreational applications – the sort of topics that appear in many introductory graph theory courses. Throughout, the writing is clear, the examples are interesting and well chosen, and there is also a good balance between the algorithmic aspects of the subject and the ’fun’. The text is well written and easy to read and the book is inexpensively priced, though the print is too small. 

Unfortunately, inaccuracies abound. The author’s map of Königsberg shows two islands when there was only one. He shows the 4-vertex graph that Euler is supposed to have drawn to solve the Königsberg problem, but Euler drew no such graph. (It first appeared 150 years later.) In the discussion of the four-colour problem Alfred Bray Kempe appears as A.P. Kempe, Percy Heawood is described as an American mathematician (he was English), and the author claims that Kempe was admitted to the Royal Society on the strength of his paper on the four-colour problem (it was only one of eight papers cited – the other seven were on linkages in which Kempe had done excellent work). The Appel–Haken proof was a proof and not a verification, and the Robertson et al. proof appeared before 1994 and not in 1996.

The book concludes with a section on further reading. It would also have been useful to have a proper bibliography that includes current editions of the books he cites, rather than earlier editions that are no longer available.

It is a shame that such blemishes tend to mar what would otherwise have been an excellent book. It is to be hoped that a revised and corrected edition will eventually be available.

Robin Wilson
The Open University

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A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics by Peter Hilton and Jean Pedersen, illustrations by Sylvie Donmoyer, Cambridge University Press, 2010, 306 pp, £19.99 pbk, ISBN 978-0-52-112821-6, £60  hbk, ISBN 978-0-52-176410-0.

(Review published online 10 January 2011)

This book is dedicated to the late Martin Gardner, and the first chapter could have been written by him as an extension to one of his famous subjects, hexaflexagons. Although it is, in part, marketed as a recreational mathematics book, it is far more and can be read at many levels. The word tapestry in the title arises because it weaves a number of threads which the authors specify as: paper folding, number theory, polyhedra, geometry, algebra, combinatorics, symmetry, group theory, history, so the "beautiful unity of mathematics" in the title is very appropriate. There are also many other loose threads like topology, although this one is not easy to follow through the index which is generally good.

The foundation of the book is folding strips of paper. This might seem trivial mathematics, but it develops into some intriguing number theorems as well as many interesting paper models. For example, what they call the quasi-order theorem enables you to determine for any given odd number b ≥ 3, using an algorithm that involves only subtraction and division by the number 2, the smallest power k to which 2 must be raised in order that either 2k – 1 or 2k + 1 is exactly divisible by b. The folding is mostly quite approachable by a teenager but, towards the end, the number theory can get quite heavy; however earlier on when it is directly related to the folding it is very easy to understand. What I particularly liked was that, as a kind of aside, there are many parlour tricks with numbers that emphasise the understanding.

The strongest thread follows Jean Pedersen’s discovery that by folding strips of paper some regular polygons could be constructed to any desired degree of accuracy. Working with Peter Hilton, she developed this idea into a systematic algorithm for producing any regular b-gon for b ≥ 3. This includes the polygons that cannot be constructed by straightedge and compass methods. Gauss’s work linking ones that could to certain Fermat numbers appears in the number theory thread, as do Mersenne numbers. The polygons give rise to polyhedra, of course, but for the recreational mathematician there is a wealth of options to explore, since they are created from strips. As well as the Platonics, there are star polyhedra and rotating rings, giving rise to a topology thread and going beyond Euler’s formula to the concept of a genus and Euler’s characteristic. The simple concepts soon develop into the weightier number theory, but then the next chapter goes back to another simpler one; the practical is interspersed with the theory which makes it easy to assimilate. So symmetries, combinatorics and group theory threads provide light relief. Polyhedra are also constructed by weaving the strips and there are polyhedra which collapse. All of this is hard to review because you need to see the excellent illustrations and follow the threads to appreciate what a well-organised book this is.

The book would be an invaluable tool for teaching, especially as it covers so many areas of mathematics and interweaves them. The development of ideas like approximation and convergence, as well as the threads mentioned above, are dealt with in a concrete way and is a very good way to learn. Too much teaching does not involve the hand and eye as well as abstract thought. The treatment of group theory is excellent in this respect. It also shows how basic mathematics can progress to advanced number theory.

As I finished this review, the news came in that Peter Hilton had passed away at the age of 87. This book is a fitting memorial to his work as a mathematician. One can see, from the way the book produces so much mathematics from a simple start, why his work at Bletchley Park in the war was so successful. He was a member of the LMS, and an obituary of him will appear in a future Newsletter.

John Sharp
London Knowledge Lab

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Mathematics and Music by David Wright, Mathematical World series, volume 28, American Mathematical Society, 2009, 161 pp, £26.50 (paperback), ISBN 978-0-8218-4873-9.

(Review published online 10 January 2011)

The connections between mathematics and music can be traced back 2500 years to the Pythagoreans, who linked musical intervals with ratios of whole numbers. Later, the European universities of the Middle Ages and the Renaissance taught music as one of the four ’mathematical arts’ of Ancient Greece (the quadrivium), while mathematicians such as Kepler, Mersenne, Newton and Euler investigated various connections between the two subjects. On the musical side many composers, from Josquin, Bach and Haydn to Schönberg, Bartok and Xenakis, have employed a range of mathematical devices in their compositions.

Recently there has been a resurgence of interest in the links between these two disciplines. A journal has been founded, and increasing numbers of conferences feature these cognate arts. Among the books on the subject are technical tomes on the mathematical aspects of music theory [1, 2], collections of accessible articles [3, 4], and mathematical accounts of particular musical topics, such as temperament and twelve-tone scales.  

On the educational side, several UK universities offer joint honours degrees in mathematics and music, while the subject has taken on a new lease of life in the USA as enrichment material in high schools or as an attractive course for liberal arts students needing to fulfil a ’math requirement’. To meet this need, several textbooks have been published, ranging from those that assume prior knowledge of such topics as Fourier series (such as [5]) to those at a more basic level ([6] and [7]).

One of the best introductory books is the one under review, which was designed for a one-semester course on mathematics and music at Washington University in St Louis and  covers the usual subjects of intervals and scales, tuning and temperament, timbre and Fourier series, modular arithmetic in music, and much else besides. All the necessary mathematical and musical theory and notations are introduced as they are needed, and the book is clearly illustrated, easy to read and learn from, and certainly to be recommended. 

Indeed, my only quibble is that, having taught such courses in an American liberal arts college where many first-year students still have difficulties with basic arithmetic and algebra, I considered his prerequisites (set theory, functions and relations, etc.) somewhat unrealistic. But apart from this, the author’s treatment of the subject is down to earth and realistic, and I certainly plan to use the book in future courses that I teach on the subject.

Robin Wilson
Open University

References  [Back to the article]

  1. D. Lewin, Generalized Musical Intervals and Transformations, Yale University Press, 1987.
  2. G. Mazzola, The Topos of Music: Geometric Logic of Concepts, Theory and Performances, Birkhäuser, 2002.
  3. G. Assayag, H. G. Feichtinger and J. F. Rodrigues (eds.), Mathematics and Music: A Diderot Mathematical Forum, Springer, 2002.
  4. J. Fauvel, R. Flood and R. Wilson (eds.), Mathematics and Music: From Pythagoras to Fractals (paperback), Oxford, 2006.
  5. D. J. Benson, Music: A Musical Offering, Cambridge, 2006.
  6. T. H. Garland and C. V. Kahn, Math and Music: Harmonious Connections, Dale Seymour, 1995.
  7. L. Harkleroad, The Math behind the Music, Cambridge/MAA, 2006.

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Numbers Rule by George Szpiro, 2010, Princeton University Press, 248 pp, £18.95, $26.95, ISBN 978-0-691-13994-4.

(Review published online 1 December 2010)

The author’s intention is to present the relationship between mathematics and collective decision making, progressing from the earliest attempts to establish a ’fair’ method of aggregating individual preferences, and indeed to establish what is meant by a ’fair’ voting system.

Each chapter is presented as a history of a development in the mathematics of collective choice, and a personal biography of the key protagonists. Each chapter has an appendix providing extra details of the personalities involved, or explains some of the mathematics in more depth.

Chapter 1 describes Plato’s discourse on government in Republic and Laws. This chapter is the exception of the book, as it describes Plato’s preference for a well managed, but non-democratic society; other chapters describe the respective protagonists’ search for a fairer democracy.

Chapter 2 covers Pliny’s attempt to establish a fair method of decision making by juries, and introduces the problem of making a decision amongst more than two options. The problem of knowledge, or anticipation of the preferences of others, and strategic voting are also introduced.

Chapter 3 introduces Ramon Llull and his proposal for pairwise comparison of candidates (essentially what is known today as the Condorcet system).

Chapter 4 presents an early description of the Borda method for electing Popes, as proposed by Nikolaus Cusanus. Chapter 5 goes on to explain the work of Borda himself and his discussion of ’circular’ rankings in aggregated group preferences. Chapter 6 introduces Condorcet and provides some background on his interactions with Borda.

Chapter 7 briefly describes Laplace’s contribution to election theory and his proposal for run-off elections (used in many presidential elections).

Chapter 8 introduces a variant to the Borda count proposed by Dodgson, in which candidates can be awarded equal points if ’equally preferred’.

Chapters 9, 10 and 12 depart from election systems and instead cover debates over the fair apportionment of seats amongst a population in a representative assembly.

Chapter 11 describes the background to Kenneth Arrow’s contribution, demonstrating the impossibility of obtaining a ’fair’ voting system, without entering into too much of the mathematical detail. Finally, Chapter 13 illustrates the preceding chapters with a number of case studies.

The treatment of Arrow’s Theorem is rather superficial, which is a little disappointing, given its importance to the material covered in the book. As I approached the chapter, I’d assumed the author was planning to spend some time working through the theorem in an accessible manner, and the absence of this does feel like an omission.

Curiously, the book also lacks a reflection on the importance of numbers to collective decision making, which is strange given the title. Chapter 8 touches on some of these issues, e.g. the way in which voters will act strategically based on the anticipated preferences of others and the ’rules of the game’ to get the best realistic result for themselves. However, this ’social compensation’ for the limitations of a voting system doesn’t receive much attention in the book, which might benefit from a discussion of whether it is such a bad thing in the first place!

Having said that, the book is thoroughly entertaining. The writing style is lively and the social narrative adds much colour to what could otherwise be an extremely dry topic. I would happily recommend it either as an enjoyable holiday read or a great gift for a friend or relative.

Tim Storer
University of Glasgow

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What’s Luck Got to Do with It? The History, Mathematics, and Psychology of the Gambler’s Illusion by Joseph Mazur, Princeton University Press, 2010, 296 pp, £20.95, $29.95, ISBN: 978-1-4008-3445-7.

(Review published online 1 December 2010)

Gambling has long fascinated the human race, but at no time in history has it been more visible than now. Widespread internet access has resulted in an explosion of online gaming sites over the last decade. Televised poker, and shows such as Deal or No Deal, are increasingly popular. The lottery industry has shown an immunity to the global economic crisis, and it seems even to have grown in recent years.

The book under review is a timely account of our obsession with gambling. Written in three (more or less independent) parts, the book examines the related questions of why we are willing to place bets that have negative expectation, and what makes us feel that we are in control of our own fortune. The first part is a brief history of gambling that begins (somewhat tenuously) in pre-historic times and continues all the way through to the aforementioned economic crisis. The second part presents a mathematical analysis of some of the popular games of chance. Part three is a discussion of the psychological aspects of gambling. The book as a whole is nicely written in a friendly, conversational style. The reader is furnished with a wealth of examples and anecdotes, and with many literary and historical references.

In the ’mathematics’ part of the book, Mazur discusses at length the theory relating to Bernouilli’s weak law of large numbers. He then analyzes the mathematics that govern various standard games of chance, such as roulette and blackjack. It is hard to say what a typical reader will gain from his analysis: the mathematically mature reader will not find too much of interest, while one not so fluent in mathematical notation will probably have rather a tough time of it.

The author acknowledges an original motive to "write about the follies of ambitious belief in windfall"; a desire to educate and caution. (Any mathematician who has observed the sad cases arrayed on endless banks of Las Vegas slot machines would surely share his desire!) Most adroitly he explains why those who bet against the house will lose (and lose big) if they continue to play for a long period of time. As Mazur observes, few gamblers have even heard of the law of large numbers, and fewer still interpret its meaning correctly; diligent readers of his book will no longer feel compelled to bet on red after seeing a long string of black!

Although Mazur claims ultimately not to sermonize in his book, by his choice of topics he convinces the reader that gambling is folly. This is clearly so for the various games he analyzes, but he misses opportunities to present a more balanced view. It is a shame, for example, that the book gives such short shrift to poker, the game that has done most to popularize gambling in recent years. There are legions of mathematically astute online poker players who are using their own "laws of large numbers" to devastating effect; in a meaningful sense, the best of these players actually are in control of their own fortune. Add to that the mysterious blend of mathematics and psychology that influence the individual decisions of all good poker players and you have a wonderful setting for many of the themes in this book.

Minor criticisms aside, however, Mazur has written an enjoyable and very readable book. His diverse choice of perspectives provides something of interest for most readers, and makes the book quite unique.

Peter Brooksbank
Bucknell University

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You Can Count on Monsters by Richard Evan Schwartz, A.K. Peters Ltd, 2010, 244 pp, £20.00, US$24.95, ISBN 978-1-5688-578-7.

(Review published online 1 December 2010)

With the festive season approaching, many of us will be seeking a creative solution to the problem: what to give the small children in our lives, which isn’t made of fluorescent plastic and doesn’t emit incessant beeping noises? The mathematical children’s book You Can Count on Monsters could be a promising candidate.

The author, Richard Evan Schwartz, is Professor of Mathematics at Brown University and is active in the area of geometry and topology. As well as research publications, he has written a few works inspired by his own children. You Can Count on Monsters was apparently produced to help teach his daughters about prime numbers and factorization. The publisher’s suggested age-range is 4–8 years; more generally the back cover states that it should give "children (and even older audiences) an intuitive understanding of the building blocks of numbers and the basics of multiplication".

The basic premise is that each prime number is represented by a monster, which exhibits some property relating to its number (e.g. the 3-monster has a triangular face, etc). In the main part of the book, each number from 2 to 100 is represented as a scene in which the monsters from its prime factorization are intermingled and combined. This is preceded by a brief description of multiplication in more traditional terms, and a child-friendly definition of prime numbers; it is followed by a discussion of the Sieve of Eratosthenes and Euclid’s proof that there are infinitely many primes.

The book is beautifully designed and produced, and is more stylish than the general run of children’s books. The graphics are angular and brightly coloured (variously reminiscent of Picasso in the 1930s, 1950s advertising, and the work of Eduardo Paolozzi). My assistant reviewer, aged just below the suggested age-range, enjoyed looking at the monsters and working out which of their attributes related to their numbers. I also found it a good starting-point for discussions with him about various aspects of numbers, addition and multiplication.

However, I have a couple of reservations about the book. Firstly, I am not convinced that it genuinely gives an "intuitive understanding" of multiplication and prime decomposition. Putting several prime-monsters together in a picture does not seem really to capture the essence of multiplication; if anything, it seems closer to representing addition. There is an attempt to ’morph’ the monsters together to address this, but overall I feel that that the 2 x 3 array of dots in the introductory page conveys more about multiplication than the picture of the 6-monster made up of the 2- and 3-monsters. My other reservation is the age-range: the age-range during which examining pictures of monsters appeals may not overlap very much with the age-range during which the concept of prime decomposition makes sense!

Overall though You Can Count on Monsters is an enjoyable and lively children’s book, which is good as a starting point for discussions, and introduces key concepts in an unobtrusive way. Furthermore, it isn’t made of plastic and doesn’t make any noise!

Sophie Huczynska
University of St Andrews

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INCLUDED IN THE NOVEMBER 2010 NEWSLETTER:



 

Femininity, Mathematics and Science, 1880–1914 by Claire G. Jones, 2009, Palgrave Macmillan, 280 pp, £55.00, ISBN 978-0-230-55521-1.

(Review published online 29 October 2010)

The title of this book reminded me of a dinner which occurred shortly after my arrival in Cambridge. I was sitting beside the wife of an eminent mathematician. On discovering I was also a mathematician she offered the following advice: I should never ignore my femininity. "So many female mathematicians do!" she declared. Although taken aback, I did, at some level, understand her. To exist in a man's world women often play down their femininity. However the opposite also occurs, and women are known to become more feminine in such an environment. What seems true is that it is difficult to just be oneself. This book is not just about being a woman in a man's world, it is about whether being a mathematician is a suitable career for a woman at all.

The author considers two case studies: one is Grace Chisholm Young (1868–1944), the other is Hertha Ayrton (1854–1923). Both studied mathematics at Cambridge, then Grace continued to work in pure mathematics whilst Hertha became involved in engineering.

The Cambridge mathematical tripos was of key importance in the fight for suffrage. Heralded as "the prestige degree for men", it was important for the movement that women succeeded in the tripos. However, "Towards the end of the century the examination lost much of its prestige ... due (in part) to [a] recognition that women were able to compete successfully alongside the men and, in some cases, surpass even the best of them".

With the usual options not open to them after graduation, Grace and Hertha had to pave their own way. Grace was awarded a doctorate from the University of Göttingen but then her career stalled. Hertha studied at the City and Guilds Technical Institute at Finsbury, one of three women out of the 118 men.

Ultimately both women married men in their fields. The working relationship between Grace and her husband was both practical and clearly unfair. Needing at least one of them to hold an academic position for financial reasons, they often submitted under only his name, thus promoting him (and demoting her). That they chose to pursue his career over hers was clearly a gender choice, as until then her career had looked more promising.

Hertha's husband William Ayrton had already been elected a Royal Society Fellow when they married. They were well aware of the "gendered interpretations given to collaborations of differing sex" and thus denied "any hint of collaboration or collusion in their work". However, they did in fact work closely together. The author comments that "Recent scholars have shown how women were inevitably assigned the role of 'assistant' in both-sex collaborations, whatever [their] contributions."

The sheer practicalities of laboratory work made it incredibly hard for women to pursue scientific careers. After Ayrton's death, Hertha found it impossible to find professional laboratory space and was forced to work in a make-shift laboratory in her house. However "Hertha's laboratory lacked credibility ... at a time when increasing emphasis was being placed on precise measurement and the use of manufactured instrumentation." On the contrary, the rather genteel pursuit of pure mathematics research makes it quite a respectable pass-time for women: certainly Grace had a much easier time integrating into academic society. But the image of a lone mathematical genius is not readily reconciled with femininity and this, subconsciously or otherwise, works against the female mathematician; certainly then, but maybe still?

It is impossible not to make modern comparisons. Huge advances have been made, and Herbert Spencer, who believed "that female ... intellectual evolution [stops] before man's in order to preserve vital organs for childbirth" would nowadays be deemed certifiable. However, I believe Grace and Hertha would be saddened by today's situation: the small numbers of women in academic positions in mathematics and engineering, and the even smaller numbers in positions of power: professors, editors of journals etc. "What would Hertha Ayrton, who nearly achieved 'FRS' after her name in 1902, make of the fact that women are still a tiny minority within the Royal Society,... hovering around just 5%." Even the leaps forward in female undergraduate numbers seem to be reversing, as a Guardian article (13.07.10) illustrates: "Women still favour 'feminine' subjects ... over engineering, sciences and mathematics, despite efforts to change this. Why?". The discussion rages on, and this book makes an interesting and valuable contribution.

Rachel Camina
Cambridge University

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Alex's Adventures in Numberland by Alex Bellos, 2010, Bloomsbury, 448 pp, £18.99, ISBN 978-0-747-59716-2.

(Review published online 29 October 2010)

This is an excellently researched and well-written book that distinguishes itself from the body of popular science books by interspersing and motivating the mathematics it contains using stories, interviews and conversations with a variety of people, ranging from mathematicians and linguists to mystics. The result is a mixture of journalism, travel literature and mathematical history that will have a much wider appeal than many other accessible texts on mathematics.

I must admit that my heart sank after reading the first page of Chapter Zero, where Bellos describes an Amazonian tribe only capable of counting to 5 (didn't I once read another book that started like this? See Gamow, One, two, three – infinity). However my initial scepticism was fairly swiftly beaten down: it is included not as a gimmick but because it is the research topic of one of his interview subjects, and the chapter grows into a discussion on our perception of numbers and quantities.

After the "pre-mathematics" of Chapter Zero, there are eleven chapters touching on a selection of topics, chosen not to present a wide spectrum but to provide glimpses into Numberland and (perhaps more importantly) how it can be related to everyday life. There is a lot in the first four or five chapters on number systems, counting and methods for basic arithmetic, while the later chapters tackle slightly more advanced material such as probability and countability. However, the mathematics never gets very complicated – the nearest we ever get to formal mathematics has been consigned to the appendices. This is no bad thing: Bellos isn't trying to teach mathematics, but "to communicate the excitement and wonder of mathematical discovery", and in this he does an excellent job. Bellos' own enthusiasm for mathematics is clear throughout, for example in Chapter Ten he ritually buys and weighs baguettes for 100 days solely to introduce the normal distribution.

Besides the mathematics, the other major aspect of the book are the people he meets. This is a fine idea – it transforms the book into a travelogue which seems to suit the popular mathematics genre very well. One of his stated aims is "to show that mathematicians are funny", which I don't think is really necessary and in any case I don't believe it. As it turns out, most of the eccentric people he meets are not professional mathematicians but "numerically obsessed lay-people" seeking magic or mysticism in numbers, or the golden ratio in everyday objects, or who are Zen masters of business card origami. Incidentally, I was surprised to learn that business card origami "is a winning way [...] to hand over your business card during mathematics conferences."

Still, the interviews are entertaining and complement the mathematics very well. Moreover, the people he meets are passionate about mathematics, and Bellos does an excellent job of describing this passion in a way that will not be lost on a general audience. Coupled with the numerous interesting facts and slices of history that appear throughout the book, this is a worthy newcomer to the popular science bookshelf.

Robert Brignall
The Open University

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INCLUDED IN THE OCTOBER 2010 NEWSLETTER:


  www.southviewimages.co.uk)
 
Ammonite, by Simon Williams 

MATHEMATICS IN NATURE

(Review published online 16 September 2010)

Exhibition of photographs by Simon Williams, Ice House Gallery, Holland Park, London, 14–29 August 2010.

Simon Williams, who trained as a botanist, has an eye for patterns in nature, and fascinating mathematical structures abounded in this beautiful collection of photographs displaying Fibonacci numbers and fractals, symmetry and spirals. Over 3000 visitors saw the exhibition in its fortnight in London, apparently including one mathematics phobic who loved the pictures but shied away in horror from the explanatory mathematics books on display. Williams’s subjects included fossils, broccoli and pineapple, ferns and orchids, and the one man-made artefact to feature, an impressive sculpture by Peter Randall-Page, was not at all out of place amongst these. The show particularly suited the interesting gallery space, but happily the images can also be seen online at www.southviewimages.co.uk. These images of the beauty of natural mathematical structures are inspiring and stimulating: on leaving the exhibition I found myself seeking out patterns as I walked through Holland Park. The photographs would look good – and provide food for thought for students – in a department of mathematics with suitable display space: an opportunity perhaps for a future showing of these fascinating images?

Tony Mann
University of Greenwich

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Decoding Reality: The Universe as Quantum Information by Vlatko Vedral, 2010, Oxford University Press, 229 pp, £16.90, ISBN 978-01-9923769-2.

(Review published online 16 September 2010)

Vlatko Vedral takes on quite a job: his aim is to unify our understanding of the nature of reality through the notion of quantum information. All this in a popular science book. It is unsurprising to find that – for me at least – he is not entirely successful.

The book is split into three parts. In the first, Vedral introduces Shannon's classical notion of ’information’. He then considers this idea in a variety of different settings – around questions of efficient communication (error correction, security), biology (DNA, evolution), physics (entropy), probability (betting) and sociology (social connectivity) – with a view to demonstrating how the notion of information can, in some sense, unify our understanding of these disparate areas.

Unfortunately the treatment is too much at the popular end of the popular science spectrum. Although I got the impression that everything discussed could be made rigorous, his unwillingness to use equations or diagrams of any sort left me in a fair fog.

In the second section he was substantially more successful – his explanation of the more general notion of ’quantum information’ was clear and interesting. His discussion of ideas around randomness and its importance in connection with the theory of evolution, and with notions of free will, was stimulating and well written. Again equations, and particularly pictures, would have helped a lot but, still, I got the drift.

In the final section come the "big ideas": that the notion of quantum information can unify the theories of quantum mechanics and gravity, and can explain how the universe emerged out of the void (creation ex nihilo). Unfortunately the fog was back. Although his explanation of theories around creation ex nihilo, in particular, was interesting, I was left frustrated. I really wanted to understand more!

I have two remaining criticisms. The first pertains to an aside in Chapter 5 where Vedral spends a couple of pages discussing global warming, and its inevitability in light of the second law of thermodynamics. I have no doubt that everything Vedral wrote here is entirely correct but of course his notion of global warming is not at all what is meant when we read the phrase in the popular media. Given the importance of global warming (as popularly understood), and given the vast amount of misinformation around the same, it seems vital that science writers treat this subject with particular care. Best not to muddy the waters with new definitions.

My last criticism is philosophical. At one point in the book, Vedral writes "physicists are not as witty as playwrights, but ... they probably have a deeper insight into the behaviour of the universe." Oh dear, thought I, here we must disagree.

If the physicist's role is to shed light, then the playwright's is to move through the darkness that remains. Their activities are complementary: one goes where the other cannot, both (potentially) bringing rich insight into the human situation. Crucially, too, both have an important aesthetic role in human life.

The aesthetic of the physicist is one of simplicity; hence the rhythm and movement of this book is ever towards ’reduction’: towards simpler theories explaining more phenomena. The reduction so described is impressive but – for me – Vedral's presentation lacked an appreciation of the aesthetic implications of such a process. I want my reality to be more than impressive, I want it to be beautiful, I want it to move me. Whilst Vedral's presentation is erudite and engaging it wasn't, for me, beautiful. Perhaps a playwright's feel for mystery and darkness is just what this book is missing.

Nick Gill
University of Bristol

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Bright Boys by Tom Green, 2010, A.K. Peters, 320 pp, £33.00, $39.00, ISBN 978-1-56881-476-6.

(Review published online 16 September 2010)

This book is about the development of the Whirlwind computer (one of the first large-scale electronic digital computers), the SAGE (Semi-Automatic Ground Environment) North American air-defence system, the young so-called ’bright boys’ who made them happen, and some of the surrounding Cold War American military history.

The Whirlwind project started life at the Massachusetts Institute of Technology Servomechanisms Laboratory in 1944, in an attempt to build a flight simulator using an analog computer, and was initially funded by the US Navy. The project later switched to building a general-purpose electronic digital computer, with the application to flight simulation seemingly abandoned. Unlike other digital computers of the time, the Whirlwind was designed for real-time computing, and so had to be fast and very reliable. These requirements eventually led to many firsts, including the design and use of random-access magnetic-core memory, and the use of graphic displays and ’light guns’ for human–computer interaction. However, Whirlwind was a huge and costly project, which was not fulfilling the original aims of the US Navy, and its future was threatened. Fortunately, Whirlwind was exactly the sort of computer the (relatively new) US Air Force needed in its plans for a complex air-defence system, urgently required after the Soviets tested an A-bomb unexpectedly early, in 1949, and the Whirlwind project was saved. In the end, Whirlwind technology was used in the computers (built by IBM) for the SAGE air-defence system, and the subsequent development of the SAGE system and its technology formed the pre-history of the internet.

All this should be a very interesting story, but I did not warm to the book. First, there is an annoying gung-ho style; for example when the author makes grand claims such as: "If the country had pinned its hopes on the Manhattan Project to end a war, its trust was pinned on the bright boys to help save a world. … they poked a gaping hole in the future and dragged into being the modern world of Information Technology". Also, there is little technical detail and no real mathematics. In fact, mathematicians "and their penchant for quickly factoring the previously impossible" (whatever that means), who want to use computers for stand-alone calculations, are sometimes cast as enemies of the MIT engineers designing and building their expensive, real-time, command-and-control computer. Perhaps the book I should have read instead is: Project Whirlwind: The History of a Pioneer Computer, by Kent C. Redmond and Thomas M. Smith, Digital Press, Bedford, MA, 1980, for a more technical and satisfying account of Whirlwind. (Redmond and Smith have also written From Whirlwind to MITRE: The R&D Story of the SAGE Air Defense Computer, MIT Press, Cambridge, MA, 2000.)

Leonard H. Soicher
Queen Mary, University of London

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Duel at Dawn by Amir Alexander, 2010, Harvard University Press, 307 pp, US$29, £21.95, €26.10, ISBN 978-0-674-04661-0.

(Review published online 16 September 2010)

If the title of this book does not make a mathematician of sense and sensibility wince, its contents will. It has a sort of subtitle ’Heroes, martyrs, and the rise of modern mathematics’. This is not a subtitle in the ordinary sense. It appears on only one of the three title pages; it is not printed on the cover of the book, though it appears on the paper jacket, down in the bottom right corner, well separated from the title. It seems to serve as a substitute for a preface and may help mathematicians and non-mathematicians alike to get some idea what the book is about.

According to the author the book belongs to "the new field of mathematics and culture" (p. 299). What is this new field? Is mathematics not a part of culture? He writes (p. 1):

the central argument of this book is simple and can be stated briefly: the duel that ended the life of young Galois marks the end of an era in the practice of mathematics and the beginning of another. In a word, it marks the birth of modern mathematics.

Is modern mathematics, then, such a well-defined concept that it can be said to have been born on 30 May 1832? Of course not. Mathematics evolves. Even its precisely formulated theorems are usually the product of a long period of evolution.

The author compares (p. 3) the story of Galois with those of Abel, János Bolyai, Ramanujan, Nash, Gödel, Grothendieck and Perelman:

Among modern mathematicians, it seems, extreme eccentricity, mental illness, and even solitary death are not a matter of random misfortune. They are, rather, almost signs of distinction, reserved only for the most outstanding members of the field.

So were Hilbert, Poincaré, Burnside, Hardy, Littlewood, Emmy Noether, Philip Hall, Hodge, Feit not among the most outstanding members of our field? And what about those who, with sanity intact, are still with us, such as Serre, Atiyah, Hirzebruch, Thompson, Wiles? Depending on the force of the word ’almost’ and the scope of the word ’only’ the logic of the above passage may have no such implication, but it comes perilously close to doing so. Besides, is not outstanding mathematical ability eo ipso a form of eccentricity? This passage is followed (p. 5) by

Remarkably, the new persona of the tragic mathematical misfit and the new practice of pure and insular mathematics came on the scene at precisely the same time [the early decades of the nineteenth century]. The central argument of this book is that this is no coincidence: the mathematical legend that appeared in the age of Galois is inseparable from the new mathematical practice that transformed the field in those years.

This ’new mathematical practice’ near the beginning of the nineteenth century is pure mathematics ’unsullied by the crass realities of the world around us’ (p. 4).

The principal characters treated in the book are d’Alembert, Galois, Abel, Cauchy and János Bolyai. Each of these gets a chapter to himself (with the titles "The Eternal Child", "A Habit of Insult: The Short and Impertinent Life of Évariste Galois", "The Exquisite Dance of the Blue Nymphs", "A Martyr to Contempt", "The Gifted Swordsman" – if this is culture then what is kitsch?). In between are three other chapters: "Natural Mathematics", on the Enlightenment and the thesis that all eighteenth century mathematics is based on providing a description of how the world works; "The Poetry of Mathematics", which compares mathematicians with other artists but underrates Shelley and overlooks Büchner, Rimbaud, Verlaine, for example, and fills a much-needed gap in the literature; "Purity and Rigor: The Birth of Modern Mathematics", in which ’Cauchy Reinvents the Calculus’ and Galois solves ’The Mystery of the Quintic Equation’. Preceding the eight chapters is an introduction summarising their contents, and following them is a conclusion entitled "Portrait of a Mathematician" in which portraits of various people, some mathematicians, some not, are discussed and related to the theses propounded in the main body of the work. Presenting the gentle sketch of Galois aged 15 that was first published by Paul Dupuy in 1896 the author focuses on the eyes (p. 256):

Dark and piercing, they burn with a fire that testifies to fierce passions within and reaches out to distant and profound truths. They look upon us with an ironic skepticism that belies their owner’s tender years, and they convey clearly that he is not truly interested in us, who stand before him. What he sees lies far beyond our horizons.

Really? That is not what I see there. And why does the author not compare with the other extant picture, a sketch made from memory by Alfred Galois in 1848, sixteen years after his elder brother’s death? To me that one shows a shifty-eyed, untrustworthy scamp. Oh dear!

The narrative of this book is based upon a small amount of mathematics and a considerable amount of history of mathematics. Neither is reliable. On pp. 202–206, for example, there is a horribly garbled account of Galois’ main contributions to the theory of equations. It is neither mathematically nor historically correct. As far as history goes, the thesis that Abel, Cauchy and Galois were men who introduced a kind of mathematics that was ’not derived from the physical world but was, rather, a world unto itself’ (p. 4) ignores the efforts of the many mathematicians of the two preceding centuries (and, indeed, of earlier times) who had developed much thoroughly ’pure’ mathematics in, for example, the theory of equations and the theory of numbers. It also ignores the fact that Cauchy, for example, contributed at least as much to our understanding of differential equations, mathematical physics and mechanics as he did to ’pure’ mathematics and its ways of thinking. Furthermore, insofar as Cauchy is credited with the construction of "a new kind of mathematics, strictly circumscribed, but pure and rigorous on its own terms" (p. 185), it belittles (pp. 187–191) the achievements of Cauchy’s predecessors and over-rates his own. He was a great mathematician, but he was just one contributor to a long-lasting effort to pin down what a function is, what continuity and differentiability are, what a real proof in Analysis is, that began in the early eighteenth century and progressed far beyond Cauchy’s own quite primitive ideas of rigour later in the nineteenth century.

To some extent the author distances himself from the mathematics and its history by examining the development of myths about his romantic heroes alongside his treatment of their lives and mathematical contributions. But it does not work. In my opinion there is little of any value in the book. I cannot recommend it.

Peter M Neumann
The Queen’s College, Oxford

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