Stewart, I. (1997) Nature’s Numbers: Discovering Order and Pattern in the Universe. 2nd ed., London: Phoenix.

Ian Stewart has recently retired as a Professor of Mathematics at the University of Warwick: he has been made an Emeritus Professor and Digital Media Fellow, with special responsibility for raising public awareness of mathematics (Dudhnath, 2009). He has been awarded the Royal Society’s Michael Faraday Medal (Buescu, 2004), and the Christopher Zeeman medal (The Guardian, 2009), for raising public awareness of and engagement in mathematics. But of his many accolades, arguably the most important to his many readers is his 1999 appointment as an Honorary Wizard of the Unseen University for his collaborations with that great sage of our times, Sir Terence Pratchett (University of Warwick, Public Affairs Office, 1999). He has written around 140 scientific papers and 70 books, of which about one-third are popular science: he claims his secret is that he ‘writes fast’ (Buescu, 2004). This review concerns one of those popular science books, Nature’s Numbers.

The prologue opens with a dream in which a Yahweh-like narrator issues Genesis-style commands that create a universe. This segues into a sequence reminiscent of the virtual reality computers in Minority Report - though not without a hint of Homer Simpson's Halloween trip to 3D-land - as the narrator manipulates his universe. The dream is then revealed to be but an average morning's work for a mathematician. Stewart explains that, with or without the help of advanced computers, this dream is how mathematicians 'see' their subject. Moreover, he promises to try to show his readers the universe through mathematicians' eyes.

Regardless of the dream's authenticity in the real world, the prologue sets the tone for the rest of the book: Stewart uses visual language throughout, deriving imagery from art and music, geography and - of course - nature, via household appliances, to make his points. The overall impression is of a book written for people who consider themselves to be 'arty', more interested in the humanities or the softer social sciences, or simply 'regular joes', for whom the word mathematics conjures up the worst memories of schooldays, with the word science not far behind. To allay further any fears, there follows a chapter on patterns in the natural world: with plenty of examples from stars to starfish, the idea of mathematics as a tool for “recognising, classifying and exploiting patterns” is slipped in almost incidentally in a discussion of snowflakes (Stewart, 1997, p1). The two subsequent chapters explain what mathematics is, and what it is for. Here mathematics is described as a tree, a landscape, a movie, even knitted fabric: clearly Stewart is aiming for the widest possible maths-phobic audience. Thus ends the first third of the book.

Chapter 4 begins the assault on real mathematics, out of our cosy trench into the no-man-in-the-street's-land of calculus. Having been coaxed over the top with propaganda about a hippy Newton dabbling alchemy, however, the promised enemy seems to have deserted the fray, taking most of their armaments: the word calculus itself appears only five times, two of those concealed in a caption to our first diagram. Nonetheless, it is a fairly painless introduction for the layman.

The same cannot be said for the chapter on symmetry (or rather breaking symmetry), a research interest of Stewart's – surprisingly, since symmetry in nature is so visual. I usually experience little difficulty in forming mental images, but I got lost in the prose: a few diagrams would have avoided this. Luckily, I recently watched a television programme about the Belousov-Zhabotinskii reaction, the memory of which helped. However, the chapter is hard to follow: too much is packed in, and the leap from symmetry to what appears closer to chaos (another of Stewart's research interests) is too long for this longest chapter in a short book.

Chapter 7, The Rhythm of Life, is essentially an annotated list of natural rhythms determined by a hypothetical neural oscillator circuit. Principally concerned with control systems in animal gait and firefly flash synchronisation, the question asked early in the chapter regarding why systems oscillate at all reminded me of the motor stereotypies displayed by people with a variety of behavioural and developmental disorders – rocking, tapping, drumming, and so forth. Presumably a similar oscillator circuit causes these behaviours. The remainder of the book continues the theme of occurrences of mathematical concepts such as chaos theory, quantum dynamics, and number sequences in nature. The epilogue is, oddly, a polemic calling for a new type of mathematics, another ‘dream’ called morphomatics. Stewart proposes that this new way of thinking will shed light on how nature’s patterns derive from simple rules, yet arise through networks of great complexity. It is not unusual for academics to introduce controversial ideas surreptitiously – via their PhD students’ theses, for example – but a popular science publication seems a strange choice.

Nature’s Numbers is certainly an enjoyable read, painting a comprehensible word picture of a wide range of mathematics. The language draws heavily on art, music, and animal life: a seemingly deliberate choice, to engage those who self-identify as artistic and see mathematics as entirely separate from their world and interests. Stewart has said in interview that his strategy is “don’t show the public a calculator or formulae” (The Guardian, 2004). True to this, there is only one barely-recognisable formula in the book, expressed in words rather than mathematical notation (Stewart, 1997, p62). This may suit the general public, but I found the overt avoidance of mathematical notation a little annoying: many lengthy descriptions could have been replaced with a few carefully chosen diagrams, or even a few worded equations such as that on page 62. Finally, much of the content reflects Stewart’s research interests over the years, and has been covered in greater detail and rigour elsewhere. For these reasons, I would recommend reading Science of the Discworld instead. The contrast between ‘Roundworld’ and Discworld makes the lack of equations less obvious, and provides a few good laughs along the way.

References
Stewart, I. (1997) Nature’s Numbers: Discovering Order and Pattern in the Universe. 2nd ed., London: Phoenix.
Dudhnath, K. (2009). The Interview: Bookbag Talks To Ian Stewart. [online]. [cited Sunday, 7 February 2010].
Buesco, J. (2009). An Interview with Ian Stewart. [online]. CIM Bulletin No. 16: Portugal, Centro Internacional de Matemática, June 2004 [cited Sunday, 7 February 2010].
Shepherd, J (2004). The magic numbers. The Guardian: London, Guardian News and Media Limited, Tuesday 8 June 2004 [cited Sunday, 7 February 2010].

University of Warwick, Public Affairs Office (1999). Terry Pratchett Receives Honorary Degree from University of Warwick. [online]. [cited Sunday, 7 February 2010].
Pratchett, T., Stewart, I., and Cohen, J.S. (2000). The Science of Discworld. London: Ebury Press.