Five recent book reviews
This interesting book collects the best poetry written by mathematicians and poets including Nobel and Pulitzer Prizes winners and Poet Laureates (e.g. R. Alberti, J. Bernoulli, L. Carroll, J. V. Cunningham, R. Dove, R. Forest, J. C. Maxwell and K. O'Brien). This international collection contains about 150 poems with a common theme: love - romantic love, spiritual love, humorous love, love between parents and children, love of mathematics, etc. The editors have gathered poems from the last 3000 years; they include a fragment of The Song of Song by King Solomon (about 1000 BC), as well as contemporary American poetry. The poems are divided into three parts: romantic love, encircling love and unbounded love. Each poem has a strong link to mathematics (in content, form or imagery) and engages a variety of mathematical topics (for example counting, rings, infinity, Fibonacci numbers, zero, Venn diagrams, functions, distance, calculus, geometry and puzzles). The authors add a “contributions' note” containing biographical notes about contributors and mathematicians appearing in the poems and bibliographical references of sources. These sections give an important impulse for those wishing to find various mathematical concepts appearing in the poems. The book can be recommended to every reader who loves poetry and mathematics and who wants to explore the way that mathematical ideas inhabit poetry.
This book, written by a professor emeritus of mathematical physics, is a bold attempt to describe specific features of mathematical thinking. It presents and elaborates many methodological and philosophical problems connected with this frequently posed and until now unsolved problem. It aroused considerable attention immediately after its publication and critical reviews and discussions are still running. There was a criticism concerning the choice of the word brain instead of mind in the title but Ruelle’s Johann Bernoulli lecture presented in Groningen in April 1999 explains his position, namely the comparison of PC and brain activities. The scope of the book is “to present a view of mathematics and mathematicians that will interest those without training as well as the many who are mathematically literate”. An investigation of the activities of a human brain can be found in the Greek and Renaissance periods and the names and results of Leonardo da Vinci, Descartes, Newton and Galileo Galilei are mentioned and discussed in the book. However, Ruelle’s attention is primarily focused on personalities that directly influenced mathematics of the 20th century and, consequently, B. Riemann (in particular his conjecture concerning primes), F. Klein (Erlangen program), G. Cantor, D. Hilbert and K. Gödel are frequently recalled throughout the book, and beside the life works also the life stories of A. Turing and A. Grothendieck are described at length.
Ruelle also tries to answer many philosophical problems connected with mathematics and human thinking. However, a serious critique of his conclusions was heard just from philosophical circles and some of the reproaches are worth mentioning. The thesis that “the structure of human science is dependent on the special nature and organization of human brain” is questionable because of our limited knowledge of the brain’s activities. Ruelle’s conclusion that “mathematics is the unique endeavour where the use of a human language is, in principle, not necessary” is hardly correct. Also his link to previous investigators of the same theme, in particular to H. Weyl (Philosophy of Mathematics and Natural Science, 1949) and Saunders Mac Lane (Mathematics: Form and Function, 1986) is insufficient. Perhaps also Ruelle’s inclination to physics and the way of thinking in physics was an obstacle to attaining more general and correct conclusions. However, all referees praise the book for its ability to present the exceptionally creative ways of mathematical thinking as well as their impact on mathematicians’ lives and they recommend it as a daring attempt to elucidate the uniqueness of mathematical thinking in comparison to other scientific activities as well as the similarities in the behaviour of all scientists.
The book under review is the third of four parts of the second edition of the book ‘Winning Ways for Your Mathematical Plays’, which was successfully published a quarter of century ago. In the second edition, some new parts have been added. The third volume is devoted to games that are played in Clubs, these games usually use coins or paper and pencil. In the book, a lot of games are discussed, usually with their winning strategies. More famous games like chess and go are also considered, the authors describe their rules and some facts concerning their history, however, they do not discuss their strategy. The book is full of pictures and diagrams, which makes the reading of the book quite comfortable. Although the topic belongs to recreational mathematics, all studied problems are treated very precisely.
There are many books dealing with origami, i.e. the art of folding realistic objects (animals) and geometric structures (polygons and polyhedrons) from a single sheet of paper without cutting or pasting. Origami seems to belong to recreational mathematics although it does not need any mathematics. The book under review differs from others on this topic. There are interesting calculations and results like the golden ratio and values of some goniometric functions. The book will appeal to the geometric intuition of the reader, containing a lot of figures and coloured photos of folded structures. It can be strongly recommended to everybody interested in recreational mathematics and elementary geometry.
Martin Gardner is a popular American writer specialising in “recreational mathematics”. The basis for this book comes from the “Mathematical Games and Recreations” column published in Scientific American magazine from 1957 to 1986. The book presents plenty of well-known and some less well-known mathematical problems. Some of the subjects discussed in the book are: the five platonic solids, tetraflexagons, digital roots, the Soma cube, the golden ratio, mazes, magic squares, eleusis, origami and mechanical puzzles. The presentation of problems is completed with many illustrative pictures and photos in every chapter. It is possible to find traditional solutions of every problem and also some newer views, explanations and proofs of solved problems. Every chapter also includes an extensive bibliography. Martin Gardner has already published more than 60 books encouraging readers to play with mathematics and Origami, Eleusis, and the Soma Cube is a well prepared addition to his bibliography.