Five recent book reviews
This interesting book (originally published in German as "Der Zahlen gigantische Schatten" in 2005) is a fascinating reading on the history and use of numbers. Using a description of lives and works of great mathematicians, musicians, physicists and philosophers (Pythagoras, Bach, Hofmannsthal, Descartes, Leibniz, Laplace, Bohr and Pascal), the author shows connections between number theory and other fields of science (e.g. astronomy, physics and geometry) as well as the role of numbers in painting, architecture, music, religion, politics, economics, thinking, etc. Each of the eight chapters contains many photographs and pictures of high quality (the book contains 163 pictures), as well as many historical and mathematical notes and explanations and a list of references to material that is both popular and professional. A deeper background in mathematics is not necessary in order to read, enjoy and learn from this book. The book can be recommended to all readers interested in the world around us and wanting to understand the importance of numbers in our daily lives.
The name of the author, Martin Gardner, is well-known throughout the world. He has produced more than 60 books and many of them are still in print. The first edition of the present book was published in 1959. Since it is long time ago, the chapters in the new edition are complemented with afterwords, addendums, postscripts and numerous bibliographies. Of course, the solutions of the posed problems are also included. The contents of the book can be characterised as “recreational mathematics”. The author explains that he is not a creative mathematician but a journalist who loves math and who enjoys writing about what the real mathematicians discover. And we see that his writing is very successful. The book is divided into short chapters with very different topics. A few of them are mentioned in the title of the book. For example, hexaflexagons are a special class of flexagons which are objects created by folding strips of paper in various ways. Probability paradoxes contain, among others, the birthday paradox and some paradoxes arising from the (mis)use of conditional probability. Further problems concern, for example, the Moebius band, mathematical card tricks, memorising numbers and fallacies. The book demonstrates principles of logic, probability, geometry and other fields of mathematics. I believe that many readers will enjoy the book with great pleasure.
This book is a unique reference for anyone with a serious interest in mathematics. It is edited by Timothy Gowers, a recipient of the Fields Medal, and it includes entries written by several of the world's leading mathematicians. The entries have many goals, such as to introduce basic mathematical tools and vocabulary, to trace the development of modern mathematics, to explain essential terms and concepts, to describe the achievements of scores of famous mathematicians and to explore the impact of mathematics on other disciplines such as biology, finance and music. The book is divided into eight chapters. The introduction presents basic mathematical language and some fundamental mathematical definitions. The origins of modern mathematics are described in the second chapter. It starts with numbers, geometry and the development of abstract algebra and through algorithms and proofs in mathematics gets to the crisis in the foundations of mathematics. The next chapter is devoted to important mathematical concepts presented in alphabetical order. Thus, one can begin reading with an axiom of choice and end with the Zermelo-Fraenkel axioms. Then the book turns its attention to mathematical branches. From algebraic geometry to stochastic processes, brief descriptions of the main areas of mathematics are provided. Crucial theorems and fundamental open problems occupy the fifth chapter. Then the book presents some achievements of famous mathematicians, starting with Pythagoras and ending with Nicolas Bourbaki. After this, several examples of the influence of mathematics on other branches of science are presented. The companion ends with general thoughts on mathematics and mathematicians. The book contains some valuable surveys of the main branches of mathematics that are written in an accessible style. Hence, it is recommended both to students of mathematics and researchers seeking to understand areas outside their specialties.
This volume contains fourteen survey lectures: three on algebraic geometry, two on differential geometry, one on the Poincaré conjecture, one on dynamic systems, one on number theory, one on the fundamentals lemma, one on the André-Oort conjecture, one on quadratic forms, one on algebraic topology, one on mathematical physics, and one on probabilities. As tradition dictates, all of them contain a discussion of important achievements (in this case in the period 2004-2005). For instance, the article devoted to number theory reports on the Green-Tao result on arithmetic progressions in the sequence of primes. The book also contains contributions on the Mumford conjecture (following the work of Madsen and Weiss), a proof of the Parisi formula by Guerra and Talagrand, on ‘Formes quadratiques et cycles algébriques’ (following Rost, Voevodsky, Vishik, Karpenko and others), and so on. As one of the most important sources of reports on major achievements in contemporary mathematics, the volume will find a prominent place on every shelf.
This book is the product of the 2004 MSRI (Mathematical Sciences Research Institute, Berkeley) conference “Assessing Students´ Mathematics Learning: Issues, Costs and Benefits”. The conference articulated different purposes of assessment of student performance in mathematics. The book focuses on ethical issues related to assessment, including how assessment interacts with concerns for equity, sensitivity to culture and the severe pressures on urban and high-poverty schools. The book introduces different frameworks, tools and methods for assessment, comparing the kinds of information they offer about a student’s mathematical proficiency. The book describes complexities of assessment when English is not a student’s native language. If a student is not fluent in English, is their failure to solve a problem a result of not understanding the problem or of not understanding the mathematics? The book highlights the kinds of information that different assessments can offer, including many examples of some of the best mathematics assessments worldwide.