You are here

Mathematics Old and New

Saul Stahl and Paul E. Johnson
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

This is a Dover republication of the first edition of a book called Understanding Modern Mathematics (Jones and Bartlett, 2007). The contents of the original book are now contained in the first five chapters of this new version. Those chapters deal with probability, statistics, voting systems, game theory and linear programming. This new edition adds chapters on planar and spatial symmetries, the bell-shaped curve and map colouring.

With memories of over forty years of teaching in UK schools and universities, I can’t recall a course that would have conformed to the diverse range of contents in this book. And yet, considering the needs of mathematical education in the USA, the publisher’s blurb suggests that it is suitable for use with ‘advanced high school students and college undergraduates in all fields — as well as readers with an interest in mathematics and its history’.

The authors are rather more specific by saying that their book is aimed at the popular ‘Topics in Mathematics’ course that is a requirement for the B.A. degree in many universities. Such a course would have no prerequisites above and beyond high school algebra. Indeed, there is no use here for calculus in any shape or form — neither is there a required knowledge of geometry or trigonometry etc. What is required is a willingness to embark upon the development of the transferrable skills of open-ended mathematical enquiry and not to regard mathematics as a set of immutable facts devoid of historical or cultural context.

I guess that the title refers to ‘mathematics old and new’ because three of its nine chapters cover more recently formalized topics (voting systems, linear programming and game theory). But the chapters on more standard topics include material not usually covered elsewhere. For example, the chapter ‘Map Colorings’ commences with an historical survey of the 4-colour problem; it extends to colouring problems on other surfaces and concludes with discussion on the Ringel-Youngs theorem. Chapter 7 (Spatial Symmetry) ends with an account of the story of Monstrous Moonshine — an outcome of the classification of finite simple groups developing from to the work of J. Conway and R. Griess.

Three quarters of the material covered in this book would seem to meet the needs of students in business studies or the social sciences — but it should be of interest to a wide readerhip, since the treatment is intellectually engaging and shows how mathematics can be applied to a myriad of (non-simplistic) real-world applications. There are hundreds of exercises, ranging from the routine to the open-ended and for which answers, hints or solutions to all exercises are provided in Appendix C.

Highly recommended – and not just for the readership defined by the authors.

Peter Ruane has taught many aspects of this material (except game theory and voting systems) at school and university levels. He recalls most clearly a multitude of naïve misconceptions regarding the simplest aspects of probability among mature students.

The table of contents is not available.