‘Psychologists talk about things everybody knows, but in a language nobody understands’. About 50 years ago, I saw a remark like this in the preface of H. T. H. Piaggio’s introductory text on differential equations. Why he said such a thing in that context, I can’t now recall. But Piaggio, who also wrote with great style and clarity, would have been impressed by Ian Stewart’s ability to introduce difficult mathematical ideas in a language that everyone understands.

In *Visions of Infinity*, the difficult ideas are connected with ‘The Great Mathematical Problems’ (the book’s subtitle). Indeed, the book very much reminds me of Odifreddi’s book *The Mathematical Century: The 30 Greatest Problems of the Last 100 Years*. And although both authors refer to ‘the’ great problems, there is only partial overlap in their choice. Moreover, the contexts in which the problems are introduced (and summarised) are markedly different. What the books do have in common, however, is the provision of historical perspectives on (mainly) modern mathematics through an examination of many of the major problems with which it has been preoccupied.

Although no expected readership is specified for this book, the structure and nature of the chapters suggest that it is intended to include the general reader. For example, many elementary ideas from pre-high school mathematics are introduced or revised, and there is an abundance of everyday analogies to prepare the reader for the introduction of subsequent complexities. But many of the chapters (e.g. Goldbach Conjecture, Four Colour Problem, Kepler Conjecture etc) contain a minimum of technical detail — and Stewart makes this possible whilst simultaneously conveying the fascinating profundity of the problems in question.

Of the book’s 17 chapters, there are five that explore aspects of number theory. In addition to the Goldbach conjecture, there are those on the Mordell Conjecture, Fermat’s last theorem, the Riemann hypothesis and the Birch–Swinnerton-Dyer conjecture. Another three chapters concern topics from applied mathematics, such as Orbital Chaos, the P/NP problem and the Mass Gap Hypothesis.

However, it’s the chapter dealing with the Hodge conjecture that sees Ian Stewart’s expositional skills tested to the full. Within just eighteen pages of prose, he seeks to explain the following statement:

On any non-singular complex projective variety, any Hodge class is a rational linear combination of classes of algebraic cycles.

The basis of the explanation is an optimistically succinct introduction to the notions of algebraic varieties, complex projective space, homology, cohomology and the fundamental group (which was discussed in an earlier chapter). It then reveals the meaning of algebraic cycles and rational combinations thereof. For readers who are inured to such underlying abstractions, this is a great story. Neophytes may well be swept along by the sheer enthusiasm of the narrative.

There is more to this book than its specific mathematical content. It is rich in historical information, and readers gain much insight into the varied applications of mathematics and the lives of those who created it. Moreover, it is shown how the evolution of mathematical ideas is due to the collective input of a wide range of individuals. The chapter on the Goldbach conjecture (for example) has mention of around 50 mathematicians who have contributed to relevant aspects of number theory.

In fact, something approaching a ‘mathematician’s apology’ is contained within the later pages of the book. This is an attempt to justify the provision of public funds (taxes in the UK) for purpose of sustaining rarefied mathematical research. In this context, Ian Stewart refers to the beneficial effects of that may accrue in terms of applications. Fair enough; Odifreddi, however, says of certain university mathematicians:

They survive by producing research that too often has neither interest nor justification, and the university circles in which the majority of mathematicians work unwisely encourage them to “publish or perish”. As a result of this, there are now hundreds of specialized journals in which every year hundreds of thousands of theorems are published, the majority of which are irrelevant.

In spite of this sobering thought, Ian Stewart’s book will provide very stimulating reading for a readership that ranges from high school to postgraduate levels of mathematical competence (and beyond).

Peter Ruane** **was drawn into a career in mathematics by a book rather like this one. It was Sawyer’s *Mathematicians Delight*.