Math Without Numbers - Milo Beckman *****

In some ways, this is the best book about pure mathematics for the general reader that I've ever seen.  At first sight, Milo Beckman's assertion that 'the only numbers in this book are the page numbers' seems like one of those testing limits some authors place on themselves, such as Roberto Trotter's interesting attempt to explain cosmology using only the 1,000 most common words in the English language, The Edge of the Sky. But in practice, Beckman's conceit is truly liberating. Dropping numbers enables him to present maths (I can't help but wince a bit at the 'math' in the title) in a far more comprehensible way. Counting and geometry may have been the historical origin of mathematics, but it has moved on.

The book is divided into three primary sections - topology, analysis and algebra, plus a rather earnest dialogue on foundations of mathematics exploring the implications of Gödel's incompleteness theorems, and a closing section on modelling (including automata and 'science'). What this approach enables Beckman to do brilliantly is to move the image of mathematics away from school maths and onto what professional mathematicians spend their time on. What's more, and perhaps more impressively for a reader who has only ever been interested in applications, it gives the best appreciation I've seen of what the point of pure mathematics is - why some find it so delightful and interesting.

Along the way in those summary headings we come across shapes, manifolds, dimensions, infinity, maps, abstraction, structures and inference. We do eventually meet, for example, sets - though they come surprisingly late when taking a conventional view. Of course not everything can be covered in detail. Groups for example, crop up with brief coverage of both symmetry groups and wallpaper groups - but we are never told what a group is. Of course, most topics have to be handled distinctly briefly. This isn't a long book (I'd say it's just the right length to be enjoyable without being either trivial or getting bogged down), but Beckman fits a lot in.

I do have a couple of small issues. As mentioned, we're told from the start the only numbers in the book are the page numbers. This isn't strictly true - numbers as words crop up reasonably regularly. And though it does provide the freedom I mentioned, in one case - Cantor's diagonal argument for the infinity of the continuum - I found the non-numeric explanation far harder to get your head around than the traditional approach using numbers. It was also, perhaps, a little unfair to include (presumably as a diversion - they aren't given any context) a pair of logic puzzles without providing the solutions: one was straightforward, but the other had some issues. In terms of content, things went ever so slightly astray when Beckman strayed into science, telling us that Newton's gravitational relationship depended on the weights of the two bodies.

No book is perfect, though. The fact remains that Math Without Numbers is a brilliant introduction to pure mathematics and a delight from end to end.

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Review by Brian Clegg