## Recent book reviews

Five recent book reviews

## Abraham De Moivre. Setting the Stage for Classical Probability and its Applications

Author(s):
David R. Bellhouse
Publisher:
CRC Press
Year:
2011
ISBN:
9781568813493
Short description:

In the first four chapters, the book deals about the De Moivre’s life in a Huguenot family at France, and then how he got established at London by political or religious concerns (both the same at that time). And the next ten chapters hinges on the three most important books written by De Moivre, which are “De Mensura Sortis”, “Doctrine of Chances” and “Miscellanea Analytica”, all about some probability problems based mainly on currently games of cards and dices. Also his short book “Annuities upon Lives” is extensively commented.

MSC main category:
01 History and biography
MSC category:
01A05, 01A45, 01A50
Other MSC categories:
01A85
Review:

This kind of books use to treat a twofold item as they tale the mathematician’s life and also the mathematical history through the developments made by the mathematician himself. But I would say this book even serves a fourfold purpose because the author gives a deeper insight into De Moivre’s reasons that inspired him during his life, so becoming an essay upon the history of mathematics, and he also covers some mathematicians' lives akin to De Moivre, showing us the typical environment of what is nowadays known as the “Republic of Letters”.

The book hinges on the three most important books written by De Moivre, which are “De Mensura Sortis”, “Doctrine of Chances” and “Miscellanea Analytica”, all about some probability problems based mainly on currently games of cards and dices. About the way they are explain let me use the author own words on pg. 74: “The details of the mathematics in De Mensura Sortis, some quite intricate, have been described fully elsewhere. [the author cites Schneider I. (1968) Der Mathematiker Abraham de Moivre. Archive for History of Exact Sciences 5: 177-317; and Hald A. (1990) A History of Probability and Statistics and Their Applications before 1750. New York: Wiley] Consequently, the focus of the remainder of this chapter, as well as the rest of the book, is to give the flavour of the mathematics and how it fits with the general message that De Moivre tried to convey”. With this aim, the author turn to analyze the division of stakes problem, the problem of the pool with its many variants, the gambler’s ruin problem, the duration of the play problem, some card games as “Basset”, “Pharaoh”, “Piquet”, etc, and also some dice games.

The author also uses the review of the De Moivre’s book “Annuities upon Lives” to show at least a moderate influence of the mathematician upon the price of life annuities in the marketplace during the first half of the eighteen century, contrary to some historian comments. And so, explains the current types of annuities upon land, the more usual, and upon one, two or up to three lives, all with several cases of payment to the heir.

Ultimately, the reader will discover the strong personality of De Moivre, who quarrelled to anyone who neglects him the merits of his work, as with Cheyne, Montmort, or Simpson among others, but who keeps his friendship tied during his long life, as with Newton, Halley, and others.

Reviewer:
José L. Guijarro
Affiliation:

## Calculating Catastrophe

Author(s):
Gordon Woo
Publisher:
Imperial College Press
Year:
2011
ISBN:
978-1-84816-739-1
Price (tentative):
US$29 / £19 (paperback) Short description: This is an update of the author's previous book The Mathematics of Natural Catastrophes (1999). It is an abundant source of data of natural and man-made hazards that can occur or have occurred. Rather than giving detailed mathematics it explains the underlying principles to a broad audience. MSC main category: 60 Probability theory and stochastic processes MSC category: 60G70 Other MSC categories: 60E99, 91F10, 91B30, 91B74, 86A17, 86A10 Review: Let me start by a warning. Just as his earlier The Mathematics of Natural Catastrophes (World Scientific, 1999) which was not a book about mathematics; this one is not about calculating' either. This one is a timely update of the predecessor, broadening it and including data of the last decade. The reader will not find the precise models, or computational methods to rigorously simulate or predict catastrophes. There are however a lot of data and many underlying principles are explained. The first two chapters form a phenomenal collection of data about all kinds of hazards both natural (extra terrestrial, meteorological, geomorphic, or hydrological) or man-made (political violence, infectious diseases, industrial accidents, or financial crises) that have happened or could happen in the future. Notable (but essential in chaotic systems) are the almost philosophical reflections about whether some event is a cause or a consequence of another one. The problem being posed, the subsequent chapters will point to some possible answers. Chapter 3 discusses the different scales and units in which the strength of all these phenomena are measured. Richter's scale for earthquakes is well known, but how to measure e.g. a volcanic eruption, and how far in time and space will cataclysmic effects propagate? That brings us to uncertainty and evidence. Historical and philosophical issues of probability theory and related notions are contemplated in chapter 4. This in turn forms the basis to explain some statistics and stochastic processes (chap. 5). Although precise prediction of the start of a war or a financial crash is very difficult, it is possible to recognize the conditions for instability and indicators for an imminent outbreak (chap. 6). Threats of terrorist attacks follow different mechanisms (chap. 7). The next chapter on forecasting is somewhat more precise on earthquakes, tsunamis, tornadoes and floods. Once a disaster is predicted, one should decide what precautions to take, what scenarios to follow and when and how to inform the public (chap. 8-9). After the event is over, insurance companies have to deal with the consequences. What should they cover? How to calculate the risk? What and how to reinsure? (chap. 10-12). The final chapter deals with long-term planning (global warming, global war). As it is stated in the conclusion: the majority of catastrophes will not be controlled by force or science. The only thing one can do is to try and understand the principles. That is exactly what the author has achieved for a very broad audience. Mathematical knowledge from secondary school suffices. After the reader-mathematician has accepted that this is not a book about the mathematics, but a book for a broad audience about facts and principles, he will definitely enjoy the reading. The erudition and literacy of the author are amazing and an intellectual pleasure to read. Besides the names of scientists, you can find a lot of names of artists and citations from their work: from Henry Longfellow and Graham Green to Fyodor Dostoevsky and from Paul Cézanne to Franz Kafka, Samuel Becket and Hokusai. Even phenomena such as Harry Potter and Star Wars are part of the game. Read and enjoy all of that! Reviewer: A. Bultheel Affiliation: KU Leuven ## Journey through Mathematics. Creative Episodes in Its History Author(s): Enrique A. González-Velasco Publisher: Springer Year: 2011 ISBN: 978-0-387-92153-2 Price (tentative): hardcover 59,95 € (net) Short description: This book describes the history of mathematics that gave rise to our modern concepts in calculus: trigonometry, logarithms, complex numbers, infinite series, differentiation and integration, and convergence (limits). MSC main category: 01 History and biography MSC category: 01-02 Other MSC categories: 01A05 Review: The book grew out of a mathematical history course given by the author. It has a list of 39 pages with references to historical publications which are amply cited and from which many parts are worked out in detail. This is organized in 6 chapters describing the evolution of concepts from ancient times till the 18th century to what is now generally used in calculus courses. González-Velasco has done a marvelous job by sketching this very readable historical tale. He stays as close as possible to the original way of thinking and the way of proving results. He is even using the notation and phrasing and explains how it would be experienced by scientists of those days. However at the same time he makes it quite understandable for us, readers, used to modern concepts and notation. A remarkable achievement that keeps you reading on and on. In my opinion, this is not only compulsory reading for a course on the history of mathematics, but everyone teaching a calculus course should be aware of the roots and the wonderful achievements of the mathematical giants of the past centuries. They boldly went where nobody had gone before and paved the road for what we take for granted today. What follows is a brief summary of the subjects treated in ech chapter. The first chapter on trigonometry starts with the Greek, the Indian, and the Islamic roots (mostly geometric) of trigonometric concepts. One has to wait till the XVIth century when trigonometric tables were produced before the term sinus was used and 2 more centuries before the notation sin, cos,... was used. The second chapter on the logarithm is a natural consequence of the trigonometric tables as an aid for computation.$\sin A \cdot \sin B=\frac{1}{2}[\cos(A−B)−\cos(A+B)]$could be used to multiply numbers$x\approx \sin A$and$y\approx\sin B$. Napier (1550-1617) and Briggs (1561-1630) worked out the concepts of the logarithm in base e and base 10. Later de St. Vincent (1584-1667) connected this to areas below (integrals of)$1/x$, and Newton (1642-1727) with infinite series, while Euler (1707-1783) generalized it to a logarithms in an arbitrary basis. Complex numbers are introduced in chapter 3. This is tied up with the solution of a cubic equation (Cardano, 1501-1576) as square roots of negative numbers. Bombelli (1526-1572) described complex arithmetic and Euler even studied the logarithm of complex numbers, but it was only Wallis (1616-1703) and Wessel (1745-1818) who gave the geometric interpretation and made complex numbers accepted (if you can draw them, they must exist). To Hamilton (1805-1865) they were a couple of real numbers and Gauss (1777-1855) introduced the letter i for$\sqrt{-1}$. Next chapter treats infinite series. Summation of numerical sequences was known to the Egyptians and the Greek, but it was Leibniz (1646-1716) who first summed the inverse of the triangular numbers$1/k(k−1)$and Euler computed$\sum 1/k^=π^2/6$. As for function expansions, the Indians knew a series for$\sin(x)$in the XIVth century, but in Europe one had to wait till the XVII-XVIIIth for Newton and Euler. However it was Gregory (1638-1675) with his polynomial interpolation formulas who later inspired Taylor (1685-1731) and Maclaurin (1698-1746) to develop their well known series. Chapter 5 about calculus is the major part (about a quarter) of this book. Fermat (1601-1665), Gregory and Barrow (1630-1677) contributed but of course Newton and Leibniz are the main players here with the well known dispute of plagiarism as a consequence. Here the author gives a careful and detailed analysis of their contributions and concludes that they worked independently, but that Leibniz's publications lacked clarity, which made him difficult to understand for his contemporaries. The last chapter is about convergence. Leibniz and Newton's ideas were still rather geometric: derivatives were tangents and integrals were quadratures, thus essentially finite. The notion of limit was lacking, which was only developed later in contributions by Fourier (1768–1830), Bolzano (1781-1848), Cauchy (1789-1857), Dirichlet (1805-1859), and others. This chapter has also a remakable original section on the less known Portugese mathematician da Cunha (1744-1787) whose much earlier contribution went largely unnoticed. Reviewer: A. Bultheel Affiliation: KU Leuven ## Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice Author(s): Alexandre V. Borovik Publisher: Providence, R.I. : American Mathematical Society Year: 2010 ISBN: 978-0-8218-4761-9 Price (tentative):$58
Short description:

This book concerns cognitive aspects of the mathematical activity but it is written from a practicing mathematician’s point of view. Thus, topics on mathematical psychology, philosophy and education naturally coexist with numerous examples on a wide range of disciplines related to mathematics. The book covers a big amount of material along the three big parts in which it is divided:
Part I) Simple Things: How Structures of Human Cognition Reveal Themselves in Mathematics
Part II) Mathematical Reasoning
Part III) History and Philosophy

URL for publisher, author, or book:
http://www.ams.org/publications/authors/books/postpub/mbk-71
MSC main category:
00 General
MSC category:
00A30 Philosophy of mathematics
Other MSC categories:
00A35 Methodology of mathematics, didactics; 97C30 Cognitive processes, learning theories; 97C50 Language and verbal communities
Review:

The big aim of this book is to reflect on what moves and happens in our brains when doing mathematics, as it is pointed out in the Preface. In order to deal with this challenging task, mathematics is placed “under the microscope”. This means that the author concentrates in simplest –but not necessarily trivial – activities of the mathematical practice such as order, symmetry and parsing (these are recurrent examples in the book, but there are many more). However, in doing this zoom-in, the author tries to keep in mind the big picture, that is, the essential vertical unity that mathematics possesses. Therefore, the following question properly summarizes the aim of the book: what are the “atoms” that build up the mathematics universe? Moreover, a motivation for writing this book is the crescent interest of mathematical research community in mathematics education. In particular, this book addresses the following questions: what kind of mathematics teaching allows for the production of future professional mathematicians? What is it that makes a mathematician? What are the specific traits which need to be encouraged in a student if we want him or her to be capable or rewarding career in mathematics?

This journey made between the microscopic and the universal vision of the mathematics, with an educational rationale as background, is also visible in the titles and contents of the three parts in which the book is divided:

Part I) Simple Things: How Structures of Human Cognition Reveal Themselves in Mathematics- The first chapter, “A taste of things to come”, serves to open the book and to set the tone of its narrative. In tune with the fourth chapter, “Simple things”, these chapters start from simple mathematics observations (such as “subitizing”) but come across some less trivial ones (such as tropical mathematics). The second chapter, “What you see is what you get”, emphasizes the role of visualization in mathematics and introduces a recurrent example in the book: the theory of finite reflection groups. In the third chapter, “The wing of the hummingbird”, an attempt to unite the visual and symbolic aspects of mathematics is done and some limitations of visualization are pointed out. The question of “parsing” leads to talk about Catalan numbers. The two last chapters, of this first part, address the issue of infinity. The fifth chapter, “Infinity and beyond” discusses how we interiorize infinity (visual images, potential infinity, Achilles and the Tortoise, geometric intuition). The sixth chapter, “Encapsulation of actual infinity”, has a more educational taste.

Part II) Mathematical Reasoning- In this part, there is a shift from subconscious and semiconscious activities of the mathematical practice to more conscious ones. The seventh chapter, “What is it that makes a mathematician?”, introduces the difficulties of the mathematician’s work and it ends with an interesting list of twelve archetypal mathematical problems. In the eight chapter, “''Kolmogorov's logic'' and heuristic reasoning”, two examples of the application of some basic heuristic principles of invention of mathematics: 1) Hedy Lamarr and her contributions to spread-spectrum communication; 2) turbulence in the motion of a fluid. The ninth chapter, which has a more technical character, deals with the issue of “Recovery vs. discovery”. In the tenth and last chapter of this part, “The line of sight”, the author gives a first-hand account of the life of cycle of a mathematical problem, in which the key concept is convexity.

Part III) History and Philosophy- This last part of the book turns into more general and philosophical facets of the mathematical activity. The eleventh chapter, “The ultimate replicating machines” deals with “memetics” and the “reproducible” aspect of mathematics. In the last chapter, “The vivisection of the Cheshire Cat”, the author concludes the book by explaining his position with respect to the principal issues of the philosophy of mathematics. He finishes by emphasizing the importance of the dialogue of mathematicians with cognitive scientists and neurophysiologists, for the future of mathematics as discipline.

As this brief description of the contents highlights, as well as the vast and varied bibliography, this book covers a big diversity of issues (and some highly topical). Moreover, this is done in a very fresh and familiar style (figures and photographs included are worth mentioning) but serious and non-trivial at the same time. In fact, there are some technical parts non-affordable with only school mathematics. These parts are conveniently indicated in the text, facilitating the reading to those who are not expert on advanced mathematics. Thus, this book will be interesting –perhaps for different reasons – to math majors at universities, to graduate students in mathematics and computer science, to research mathematicians, to computer scientists, but also to practioners and theorist of general mathematical education, to philosophers and historians of mathematics, and to psychologists and neurophysiologists. Either if you are interested in enriching your mathematical culture beyond mathematics itself or you are looking for interesting and illuminating examples as an extra ingredient for your classes, this is a highly recommendable book.

Reviewer:
Blanca Souto-Rubio
Affiliation:

## Imagine Math. Between culture and mathematics

Author(s):
Michele Emmer (ed.)
Publisher:
Springer
Year:
2012
ISBN:
978-88-470-2426-7
Price (tentative):
64,15 € (net)
Short description:

This book is similar to the volumes Mathematics and Culture also published by Springer and edited by M. Emmer.
It illustrates the diversity in applications and interaction that mathematics has in our current society, in particular when it does not concern science, technology and engineering.

URL for publisher, author, or book:
http://www.springer.com/mathematics/applications/book/978-88-470-2426-7
MSC main category:
00 General
MSC category:
00A99
Other MSC categories:
00A65, 00A66, 00A67
Review:

The many diverse texts that make up this book, link mathematics to
history, different kinds of art (music, literature, architecture, film,...),
science, medicine, economics etc. Let me illustrate this with a number of examples.
A first chapter is a tribute to Benoît Mandelbrot.
The next is about the restoration of the Teatro Le Fenice in Venice
(destroyed by fire in 1996 - what's in a name).
This brings us to an homage to Andrea Pozzo, a Baroque artist famous for
his use of geometry and perspective in his frescos.
Near the end of the book there
is another homage to Lucia Pacioli who was also a pioneer in the use
of perspective.
There are several contributions devoted to women in mathematics.
Hypatia lived in Alexandria in the 4th century and was the first
known woman to contribute to mathematics, but brutally murdered for some
obscure reasons. There have been many other women in the course of
history that contributed to the progress of mathematics:
Emily du Châtelet, Maria Gaetana Agnesi, Sophie Germaine, Mary Fairfax Sommerville,
Sonya Kovalevsky, and Emmy Noether are all briefly placed in the spotlight.
The part on mathematics and art considers different aspects like
the use of geometrical curves and surfaces in architecture or the
role of mathematics in art throughout the past and how these mathematical rules also
appear in nature. Also literature can be analysed using mathematical
structures and statistical analysis,
but conversely, a set of generating rules may also construct an artificial language.
An analysis is given of the inescapable labyrinth' that Jorge Luis Borges created with
mathematical precision in his story The library of Babel'.
Under the part "applications" we find essays about Lorentz knots,
the statistics of words and leading numbers, a numerologic analysis of the
Seal of the US (the number 13 of the original 13 states is omnipresent),
and aperiodic tilings (the birth of quasi-crystals).
A contribution about a numerical model used for connecting artificial
parts to the aorta falls under medical application. Other links are
found between differential equations and origami.
A study if the number system of Mesopotamia is of historical nature.
There is not only a stage play about Hypatia, but there are also
films about mathematics (like the Moebius strip by Edouard Blondeau), but it is also interesting
to see how in the course of history the casting of characters playing
the role of a mathematician in a movie has evolved.
A final chapter is about the graphical representation of hidden rules
that are used in music: visual art produced from music.

The book is about mathematics, but the formulas are only sporadically used.
Theorems and proofs are fully absent.
The texts are often written by non-mathematicians. Hence it is easily
accessible for anyone having a general interested in mathematics and
the interaction with many other aspects of science, society, and knowledge
which are not the obvious engineering applications. The nice thing
about it is that the interaction often goes both ways.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven