Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.

George Polya, or Pólya György, was a Hungarian mathematician who spent much of his career at the Federal Institute of Technology, Zurich (ETH Zurich) and at Stanford University. His is a family which suffered, or courted, difficulty. His father, perhaps best known for his Hungarian translation of Adam Smith’s The Wealth of Nations (Polya, 2006), was a lawyer whose academic ambitions were thwarted by institutional anti-Semitism, until he converted to Roman Catholicism. His brother Eugen (Jenö) was a surgeon, famous for a type of gastrointestinal bypass surgery for the treatment of stomach ulcers, who died at the hands of the Nazis; another gifted brother, Laszlo, was killed in the First World War. His great-nephew is the controversial artist and biochemist, Gideon Polya. George seems to have limited his misfortunes to an early punch-up with a student with royal and high level political connections in Göttingen: fortunately for mathematics, this incident partly led to his appointment at ETH Zurich.

Polya performed poorly in mathematics at school, which he later attributed to bad teaching. It was not until he was at university when, following qualifications in literature, he began to be interested in philosophy, that he began his mathematical studies. Despite this late start, he went on to make contributions in a range of topics so diverse as to call to mind the polymaths of ancient and mediaeval times, in a career that extended well beyond his retirement in 1953: remarkably, he was still teaching at Stanford University in 1978, at the age of 91. From the first, his real interest in mathematics seems to have lain in proof and mathematical discovery (Albers & Alexanderson (1985), in O’Connor & Robertson (2002)). Indeed, it is arguably that this underlies his contributions to mathematical education, for which he is most remembered. This legacy comprises a series of books on problem-solving in mathematics, the first of which, How To Solve It, is the subject of this review.

How To Solve It is an unusual book. There are in fact only 36 pages in all which outline Polya’s method: these form Parts I and II of the book, In The Classroom and How To Solve It – A Dialogue, respectively. These chapters are preceded by a two-page outline of the method, presumably intended as an aide memoire, and an Introduction, which outlines and explains the book’s rather unique structure. Part III, the largest section of the book, is entitled a Short Dictionary of Heuristic, essentially an appendix expounding in detail on the problem-solving techniques and ideas mentioned in Parts I and II, together with potted biographies of a few mathematicians and definitions of a few pertinent terms. The final section, Part IV, is the aptly-titled Problems, Hints, Solutions. Throughout the book, subsections within the chapters are numbered, rather more like a textbook or business document. Initially a little disturbing to the reader expecting something a little more literary, this does however reinforce the fact that this book is very much a guide: a textbook in mathematical guidance.

Of the book, Part I – In The Classroom – is the most important. It is this chapter that sets out the purpose of the book: to help improve and develop problem-solving in students. Five main issues are raised and explored briefly:

• Helping the student
• Questions, recommendation, mental operations
• Generality
• Common sense
• Teacher and student. Imitation and practice.

The emphasis is very much on finding problems suited to, yet challenging at, the student’s level of knowledge and ability, and then on unobtrusively guiding the student to a solution. Polya then explains that his model, for reasons of convenience, is split into four phases: understanding the problem; making a plan; carrying out the plan; and looking back. Each phase is then examined in detail, with a running example. Polya recommends a method of a quasi-Socratic, positive questioning – posing a variety of leading questions, in a variety of ways, to steer students gently towards finding ‘their own’ solution. On completion, the solved problem can then be examined – mined, as it were – in the same manner to explore connections to other problems and concepts under the guise of checking, incidentally reinforcing the learning that has just occurred. 

A significant difficulty with the book is the language in which it is written. Originally published in 1945, it has something of the flavour of F. Scott Fitzgerald and the more serious works of P.G. Woodhouse and Dorothy Parker – a convoluted phrasing, an emotionally-detached remoteness, that makes for difficult reading. Polya’s literary and philosophical background shows strongly, although the fact that the text is readable at all by the layman indicates that he must have tried to play both down considerably. Some of the mathematical terminology is outdated: in the first running example, he uses a problem concerning a parallelepiped. Fortunately, I vaguely remembered seeing the –epiped suffix somewhere as meaning a 3D figure, so I did not have to interrupt my reading to look up a dictionary. There is a diagram, but this appears later in the exposition of the problem-solving phases – perhaps too late for another reader. However, I had a difficulty with the printed text: the suggested guidance questions were identified by italicised font, but unfortunately, so are all Polya’s emphases. For the teacher skimming through for ideas, this is unhelpful. Since the edition I read was published in 1973, it is odd that some alternative was not found.

In all, this is an immensely useful book which fully repays the endeavour of reading, despite the difficulties of language. Polya makes a heroic attempt to explain, in albeit condensed and simplified terms, a difficult task. As anyone who has ever attempted to solve a difficult mathematical problem can attest, the thought processes that lead to an answer are proof (sic) against analysis: occasionally lightening fast, at other times tortuously slow, and then again, suddenly productive after perhaps weeks of drought. Polya takes this amorphous thing, and supplies a structure, a logic, and a process from which something might reliably emerge. It is not a book to be read through, short as it is. It is a book that should be kept on a convenient bookshelf, to be referred to, dipped into, and mulled over. Frequently.

References
J J O'Connor, J.J., & Robertson, E.F. (2002). MacTutor: George Pólya. [online]. University of St Andrews: MacTutor. [Cited 26/03/2010].

Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.

Polya, G. (no date). Personal Profile. [online]. Media With Consciousness News . [Cited 26/03/2010].

Polya, G. (2006). Global Avoidable Mortality. [online]. Blogger. [Cited 26/03/2010].

Who Named It? contributors (no date). Eugen (Jenö) Alexander Pólya. [online]. Oslo: Who Named It? [Cited 26/03/2010].

What is this thing called Mathematics?

There is an apocryphal tale of the French mathematicians who published collectively under the name Nicolas Bourbaki, who apparently wrote 200 pages on the number one alone (Bergamini, 1972). I cannot say I believe this story completely, having heard similar anecdotes of topics broad (psychology) and narrow (the modern Italian historical novel). The experts shake their heads and chuckle wryly at the naivety of asking for a definition. Thus forewarned, I shall not fall into this trap: I shall say only what mathematics means to me. Mathematics is, to me, the science of number; its uses, both practical and theoretical (Courant, Robbins & Stewart, 1996); and associated notations. I will explain my take on these in reverse order. Notations I do not regard as especially important: Shakespeare is still Shakespeare, whether in the original Elizabethan English, translated into Norwegian, or performed in British Sign Language. It is unfortunate that notation is often the first thing people think of on hearing the word ‘mathematics’. Perhaps more effort is needed in schools to describe this as a code, shorthand, or language.

The uses of mathematics I regard as practical, meaning applied or descriptive, or theoretical. The practical element is relatively easy to explain: a shepherd counts his sheep to keep track of them, plan grazing, and so on; that number is also a handy descriptor for others, rather than taking them to the hills to see for themselves! This practical element also elucidate processes and phenomena which cannot be observed or comprehended directly, such as the detection of orientation in vision (Anderson, 2001) or the origin of the universe. The phrase ‘theoretical uses’ is something of an oxymoron - perhaps another synonym of ‘use’ would be better here – service, practice, exercise… However, I am inclined to think that theory – the study of a subject purely for the subject’s sake, as it were – is a ‘use’ whose day is not yet come. Reading a popular science book on mathematics some years ago, I was amazed (briefly, and I really shouldn’t have been. It won’t happen again.) that so many ‘pointless’ theories were discovered to have practical applications, some discovered long after the theory was first promulgated (Stewart, 1998). A true application may appear later, or suggest itself immediately: the effort is not wasted.

I define number very broadly and inclusively: the integer 6, a triangle, or the algebraic notation x², are to me number. Similarly, the word ‘happy’, Hals’ Laughing Cavalier, and a smiley icon convey the same emotional information - in informational, pictorial/geometric, and symbolic forms. Where does number come from? While I don’t want to come over all Pythagorean, I feel that number is present in the structure and ordering of the Universe. Not as a separate and discoverable entity in itself, but as our code for the otherwise incomprehensible and ineffable. It should not have surprised me that the Fibonacci Sequence can be seen in the different petal arrangements on flowers – I should have been asking how could petal arrangements be modelled, and what are the biological processes underlying the Fibonacci Sequence’s fit to the data. God may or may not play dice, but is almost certainly a mathematician.

References
Anderson, R. (2001). Detection and Representation of Oriented Contours in Human Vision. Ph.D. Thesis, University of Birmingham.
Bergamini, D. (ed.) (1972). Mathematics. 3^rd ed., Netherlands: Time-Life International.
Courant, R., Robbins, H., & Stewart, I. (1996). What is mathematics?: an elementary approach to ideas and methods. 2^nd ed., Oxford University Press Inc.
Devlin, K., (2000) The Maths Gene, Why Everyone Has It, But Most People Don't Use It. London: Weidenfeld & Nicolson
Richer, É. (no date). PlanetMath [online]. [cited 12^th January 2010]. http://planetmath.org/encyclopedia/NicolasBourbaki.html
Stewart, I. (1998). Nature's Numbers: Discovering Order and Pattern in the Universe.London: Phoenix.

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.

Charles Seife (2000). Zero: The Biography of a Dangerous Idea. London: Souvenir Press Ltd. ISBN-13 978 0 285 63594 4

As I settled to read this book, my son came running up to me, a look of mock consternation on his face: “Mommy, I have got only NINE fingers!” He proceeded to demonstrate, uncurling each in turn - and indeed, there were only nine fingers. I was called upon to locate the missing finger, which I did. But he counted again, and lo! Only nine fingers! It was all terribly confusing. He then ran off laughing at my befuddlement, shouting “Only joking, Mommy”.

This scenario has been repeated for weeks, ever since he learned about zero – or 'zewo' – at nursery. He happily counts forward and backwards, from and to zero – the latter often followed by “Blast-off!” He sings songs about zero, plays tricks on his poor old parents with zero, returns his plate to the kitchen when it has 'zero food' on it. If a 4-year-old is so comfortable with the idea of zero, why should anyone go to the trouble of writing a book about it?

Charles Seife clearly thought it worthwhile. Currently a professor of Journalism at Columbia, having written for some of the best-known science publications, he holds a Mathematics degree, and studied with Andrew Wiles, the mathematician who solved Fermat's Last Theorem (twice!). He has also done research relating to mathematics, so one can assume that he knows what he is talking about – and has something to say.

However, to the modern reader, there seems little point in writing a book on the subject. Zero, nought, nothing, nada, are all part of the common vocabulary. Yet, as Seife points out early in the book, zero is not a common number. Most of us can go through life without thinking about it. Palaces and pyramids have been built without it, inheritances divided, taxes reckoned, routes planned. Apart from the inconveniently empty state of bank accounts at the end of the month, we might never need to think about zero at all. So, why zero, and why is it dangerous, aside from the exorbitant bank charges for letters to tell us we are out of money?

I came to this book with what I considered a fair knowledge of zero. In addition to its mathematical uses, I was aware, for example, that the Romans and Greeks did not use zero, that it was introduced via Islam from India but probably pre-dated these civilisations, that its absence in the early Christian church is the reason for the arguments over the date of the millennium and more recently over the end of the decade - in short, I knew more than most. Seife covers this and more in a mathematics-free introduction. Writing clearly and with occasional flashes of Brysonesque humour, he uncovers the beginnings of human counting in a 30,000-year-old wolf-bone tally, and talks us through the development of written mathematical notation, taking in religion, mysticism, philosophy, the calendar, art, a smidge of comparative linguistics (zero and cipher the same word!), computer programming, astronomy and both classical and quantum physics along the way – including one of the nicest lay explanations of the uncertainty principle I have ever seen. No potential kittens in thought-boxes here - just the difficulty of measuring the length of a pencil (p 170). Amazingly, this is achieved with very few equations – the first appearing on p 95, almost halfway through the book. What little the author cannot explain in words is conveyed through well-chosen diagrams and images, many derived from contemporaneous sources, but even these support rather than elucidate the text.

The part of the book that worried me most was the section on Newton’s development of calculus (pp 114-126). This was clearly signalled in early chapters, adding to the apprehension. I had a niggling memory about strange notations; and while I have had little difficulty in using and understanding calculus, I was concerned about understanding proofs of the topic from first principles. However, I was pleasantly surprised. Not only is Seife’s explanation clarity itself, I actually recognised the notation and the vocabulary of fluxions and fluents, and even Newton’s ‘dirty trick’ of expunging the infinitesimals that allowed him to develop his proofs. I also seem to have learned L’Hôpital’s rule, though not by name: clearly, my maths teacher must have passed on more information than I remember!

The following chapter, Infinity’s Twin, is, I feel, the least successful part of the book, particularly the section on projective geometry and the complex plane. I came unstuck on the discussion of Riemann spheres, unable to visualise the complex rubber universe of his imagination. However, this may yield to a closer examination of the text.

The remainder of the book deals with zero – and infinity – in what is for me the more comfortable domain of physics: thermodynamics, relativity, quantum mechanics – nothing too strenuous. However, here I found one of the great surprises of the book. I have been myopic most of my life, but was not diagnosed until I was twelve because I had developed excellent coping mechanisms, one of which was looking through a ‘pinhole’ created by pinching my fingers together. I had always been aware of faint stripes when looking through this pinhole, for which I vaguely blamed my eyelashes. Thanks to Seife, I now know these stripes are interference!

In sum, I feel I could recommend this book to almost anyone who is interested in the history of mathematics, regardless of his or her subject knowledge. The mathematical treatments are light: there are few equations to scare off the layperson, and even these are offset by some truly outstanding explanations and examples. For those with an appetite for more, there is a wide-ranging selected bibliography to chew over. The appendices, usually the repository of the esoteric and arcane, are amusing little pieces – build your own time machine; why Winston Churchill is a carrot, and so on. Seife’s real gift is telling the stories behind the mathematics: his book is worth reading just for the image of Pythagoras as some crazed antediluvian guru, railing against the infamy of beans.