The Princeton Companion to Mathematics by Timothy Gowers; associate editors June Barrow-Green & Imre Leader, Princeton University Press, 2008; 1008 pp, cloth, £60, ISBN 978-0-691-11880-2.
Once in a while a book comes along that should be on every mathematician’s bookshelf. This is such a book. Described as a ’companion’, this 1000-page tome is an authoritative and informative reference work that is also highly pleasurable to dip into. Much of it can be read with benefit by undergraduate mathematicians, while there is a great deal to engage professional mathematicians of all persuasions. The 200 entries were written by over one hundred contributors, selected for their expertise and expository skill.
The companion is organised in eight parts. Part I is a masterly introduction, presumably written by the editors, explaining the nature of modern mathematics, its language and grammar, and describing some of the main concepts and subject areas of mathematics (such as vector spaces, limits and hyperbolic geometry). This is followed, in Part II, by a historical overview, organised thematically in seven sections.
The core of the book is in the next two parts. In Part III about a hundred mathematical concepts are described in some detail, ranging from the axiom of choice, braid groups and Calabi–Yau manifolds, via elliptic curves, Jordan normal form and the Mandelbrot set, to matroids, Ricci flow and the Schrödinger equation. These feed into more substantial accounts in Part IV of twenty-six subject areas (such as analytic number theory, harmonic analysis, operator algebras and computational complexity); these take up about one-third of the book.
Part V presents descriptions of thirty-five important theorems and problems, old and new, ranging from the insolubility of the quintic and the prime number theorem to Gödel’s theorem, the Atiyah–Singer Index Theorem and the Birch–Swinnerton-Dyer conjecture.
Part VI presents mini-biographies, written by experts, of almost one hundred mathematicians, organised chronologically by birth date from Pythagoras to Bourbaki.
Part VII describes the influence of mathematics in such areas as biology, traffic flow, finance, cryptography, and the relationships between mathematics and music.
The book concludes in Part VIII with essays on general topics (such as problem solving, experimental mathematics, numeracy, and advice to a young mathematician) and a useful chronology of mathematical results and events.
The above outlines show that this is far from being a traditional encyclopedia of mathematics, but they can only hint at the enormous range of topics covered within its pages. As well as the usual mainstream topics, a welcome feature is the serious attention paid to such topics as the history of mathematics and combinatorics that are so often relegated to the sidelines or treated in an unscholarly manner. Although the line had to be drawn somewhere – for example, there is no discussion of mathematics education – the editors have done a fine job in embracing an impressively wide selection of important and interesting topics.
Finally, the publishers should also be congratulated on the high quality of the presentation. The print is easy to read, the paper is of high quality, the diagrams are clear, and the cost of the book, given its size, is remarkably low.
Strange Attractors: Poems of Love and Mathematics by Sarah Glaz and JoAnne Growney (editors); A.K. Peters, Wellesley, Massachusetts, 2008, 196 pp, £32.50, US$39.00, ISBN 978-1-56881-341-7.
It would seem a tall order to be asked to compile a substantial collection of ’poems of love and mathematics’. However this challenge has been successfully taken up by Sarah Glaz and JoAnne Growney: their anthology Strange Attractors contains 196 pages of poetry by about 150 poets. The oldest poems are by Solomon (c. 1000–928 BC) and Catullus (84–54 BC); but the bulk of the collection features writers from the eleventh to the twenty-first century AD – some with well-established poetic credentials (such as Dante, Emily Dickinson or Philip Larkin) and some rather surprising ’occasional’ poets like James Clerk Maxwell. Assembling such a diverse collection must have been a major task for the editors; and the extent of their literary research is also demonstrated by the inclusion of useful biographical notes on the poets and on many of the mathematicians who are mentioned. Some poems also carry explanatory footnotes.
The mathematical content of the poems is quite varied. Some involve little more than simple enumeration – as in the Catullus poem about counting a lover’s kisses. Some employ familiar mathematical words like ’equation’ or ’ratio’ (sometimes without much insight) in otherwise conventional love lyrics. There are more adventurous pieces like those by Jonathan Coulton or Ed Seykota which attempt to re-write Mandelbrot set mathematics in words arranged like a poem. One of the most unusual offerings is Carl Andre’s On the sadness which sets out a number of phrases in patterns governed by prime factors of the integers up to 47: the result is surprisingly endearing and wistful. The poems which seem to me to work best, however, are those which start with one (possibly quite simple) mathematical idea and work outward into emotional relationships. A good example is Yehuda Amichai’s addition of a human element to schoolbook problems about trains:
No one ever asked what happens
More movingly, Jonathan Holden describes his father’s advancing dementia in terms of the bit-by-bit loss of algebraic symbols:
...The letter A
This seems primarily a book for mathematicians with an interest in poetry. However it may not carry them much further into poetry than they have already travelled. Although there are some fine and serious poets represented, the arrangement of the collection (alphabetically by author within three rather arbitrary sections) makes its reading rather an uneven experience, with complex and thoughtful poems being sometimes followed by short humorous conceits. Poetry-lovers without much knowledge of mathematics may find this rather disconcerting and be inclined to doubt whether much light is shed on connections between mathematics and poetry.
The book is handsomely produced and will probably appeal to friends and families of mathematicians as a solution to a birthday present problem. It is very much a ’dipping-in’ volume – a bookshop browser may be delighted to stumble upon surprises such as a Venn diagram in a poem by the eminent New Zealand poet C.K. Stead or Henry Lok’s magic square poem intended for reading along diagonals as well as horizontally, like an Elizabethan form of Sudoku. In fact, a rewarding way to approach the book might be to generate pseudo-random integers betwee 3 and 198 and enjoy the pages as they come!