During a recent consultation with an orthopaedic surgeon, he asked me what I did for a living. His response to my one word answer was: ‘Ah! So you dwell in the esoteric world of mathematics!’ Having organized my thoughts for the follow-up appointment, I asked him if he ever used PET scanners, and whether he knew that this technology depends on the discovery of the positron, whose existence was predicted by Paul Dirac’s study of the equation \[ (\beta m c^2 + c(\alpha_1p_1+\alpha_2p_2+\alpha_3p_3))\psi(x,t) = i\hbar\frac{\partial\psi(x,t)}{\partial t}.\] He took one look at this equation, and said ‘I leave the use of those scanners to my colleagues — thank goodness!’

This anecdote pertains to one of the main themes of this book, the general perception that mathematicians are unworldly and mathematics is of little practical relevance. One reason for such misconceptions is that, unlike electronics or civil engineering, mathematics seems not to lead to the production of tangible objects — such as mobile phones, tablets, tunnels or bridges. And alienation from the world of mathematics is compounded by unpleasant memories of mathematics in primary and secondary school.

Hence, this book is almost an apologia for the fact that mathematicians appear to be denizens in the arcane world of university departments or hi-tech industries, and that mathematics seems to be the preserve of the privileged few. It consists of 25 short chapters, each one by a different author, and organized into three parts.

- Who are mathematicians?
- On becoming a mathematician.
- Why I became a mathematician.

The contributors are well known in the world of mathematics, and nearly all work in mathematics at the university level. Moreover, they constitute a wide range of personalities and don’t conform to what may be thought of as popular stereotypes (‘beautiful minds’). Inevitably, they’ve all had different home backgrounds and varied early life experience, and their mathematical specialisms are correspondingly diverse.

The themes covered here are many and varied, and they include the following

- Social class and mathematical values in the USA.
- Dynamics of mathematical groups.
- Psychology of being a mathematician.
- Mathematics, teaching and lecturing.

Overall, the main consensus among the 25 contributors is that they all really love their chosen profession, and they feel privileged to be paid for the work they do. There is, of course, no clearly defined viewpoint on the nature of the mathematical personality; but that very question is explored in an informative and entertaining manner. In particular, I thoroughly enjoyed the articles in part three, which consists of many poignant descriptions of the how an interest in mathematics emerged from a variety of childhood backgrounds.

However, an unwitting emphasis in some of the articles is that mathematics becomes a creative enterprise only at the PhD level. But, as Hyman Bass indicates in his chapter ‘Mathematics and Teaching’, much creative output can be seen among students from the age of 5 years and upwards, and I specify the following instances from my own experience.

- Years ago, when teaching a class of five year olds, I had them making geometrical shapes with equipment called ‘geostrips’, which they pinned end-to-end to make triangles, quadrilaterals and any other shapes they wished. One kid made regular polygons, beginning with an equilateral triangle, but he didn’t have enough strips to build anything beyond the decagon. When he showed me this collection, this five year old said ‘I can’t make a circle, ‘cos that’s a billagon’.
- On another occasion, I was asked by a teacher to do some diagnostic work with an eight year old, who was disruptive during maths lessons. After a short while, I noticed that she had devised many original (but not always efficient) arithmetic algorithms. For example, when I asked her to find half of 5738 she immediately wrote down\[2 \frac12 3\frac12 1\frac12 4 =2869\] and she could do this at high speed for any number. This child’s ‘problem’ was that she was more inventive than the teacher, who viewed mathematics only through textbook standard methods.
- During a lesson investigating properties of a square, a nine-year old showed me his perfectly good method for deriving an infinite number of ways of dissecting a square into two congruent parts, and his explanation of this method was tantamount to formal proof.

Appropriately, another of the book’s themes is that creativity should not be solely equated with Kantian originality, but also regarded as a manifestation of the Platonic viewpoint. So, buy this book, and read all about it!

Peter Ruane has taught mathematics to people in the age range from 5 to 55 — that is, from school arithmetic to transfinite arithmetic, and from use of geostrips to projective geometry.