Five recent book reviews
This is the 19th printing in the Canto series of a classic. Written in 1940, it was extended with a foreword by C.P. Snow since the 1967 edition. The chemist and novelist Snow gives a human biography of his friend Hardy, and Hardy himself contemplates at the age of 62 on his waning mathematical abilities and on the usefulness of dull elementary mathematics as opposed to the harmless beauty of pure mathematics, which he considered to be a body of creative art presumably without any application.
In 1940, at the age of 62, and at the verge of WW II, G.H. Hardy wrote down his apology, in an attempt to justify his life as a mathematician: is the work that a man does during his life worth doing, and why does he do it?
The second question has two possible answers: either he does it because he is good at it or because he is not good at anything and it just so happened. The first reason applied to himself: There is no doubt that I was right to be a mathematician, if the criterion is to be what is commonly called success. Note the past tense. Hardy is convinced that mathematics, more than any other art or science is a young man's game. All great mathematicians achieved their main results at a young age, and they either died young or did something else later in life. So he is implicitly regretting his waning mathematical skill.
The other question about why mathematics is worth doing takes more pages. He claims that mathematics is an unprofitable, perfectly harmless and innocent occupation. With unprofitable he means not directly of use like medicine or physiology is, and harmless refers to not applicable in e.g. warfare.
As for the usefulness, it should be clear that he refers to what he calls 'real' mathematics, i.e. what we should call pure mathematics, which is quite different from the repulsively ugly and intolerably dull school mathematics which indeed have applications in daily life and are applied by e.g. engineers. If it is useless, then it is harmless. If it can not be used, then it can not be used for good purposes, but neither can it be misused for evil ones.
Pure mathematics are universal, deep and beautiful. A chess problem is mathematics but unimportant and not serious. Moreover it is general and of all times: Babylonian and Assyrian civilisation have perished; Hammurabi, Sargon, and Nebuchadnezzar are empty names; yet Babylonian mathematics are still interesting. Pythagoras, Newton and Einstein are people who influenced science considerably. As examples of serious mathematics he gives Euclid's proof of the infinity of prime numbers and the proof of Pythagoras on the irrationality of √2.
A proof gets some aesthetics from unexpected twists, which is not found in a proof by enumeration or using standard techniques. The deeper the result, the less straightforward the proof will probably be.
Hardy also contemplates the difference between pure and applied mathematics. For example the applied geometry of our surroundings will change when a massive gravitational object will be brought into the room. However, the theorems of pure geometry that have been proved in that same room do not change. In a sense the mathematician is in more direct contact with reality than e.g. a physicist. For example a chair can be a collection of whirling electrons or anything else, depending on the model used to describe the physics, but a mathematical object is exactly what it is, no interpretation possible. There are many possible models for the physical world, yet 317 is a prime not because we think it is, [...] but because it is so.
The foreword by C.P. Snow takes about one third of the booklet. It has been added since the 1967 edition of the Apology. Snow is a chemist and novelist. When Hardy returned from Oxford to Cambridge in 1931. Snow was invited by Hardy because he wanted to speak about cricket, a passion they both shared. They became good friends thereafter. The foreword is basically a short, yet very human biography of Hardy, not about his mathematical achievements but rather about his opinions and his character as a human being.
Hardy's text is now more than 70 years old, and it is clearly a plea for pure mathematics, but obviously Hardy is also formulating a justification of his own life. Some of his viewpoints are controversial, and some arguments may no longer hold (his beloved field was number theory which became a very important element in decoding the German Enigma machine during WW II, although he may have considered that as dull and applied, and not of the pure and creative sort). Yet his ideas are so clearly formulated, that it is still advisable for any mathematician to read it, discuss it with colleagues, and think about an apology for his or her own life as a mathematician. So we are very lucky that the booklet is still available in print.
The apology by Hardy is publically available courtesy of the University of Alberta Mathematical Science Society. The foreword by Snow is not.
In 10 chapters, some historical approaches to elements from calculus are explained. This involves summation of infinite series, computation of surfaces and volumes, limits, continuity and differentiability. Proofs are by figures and intuition. Each chapter ends with problems additional information and suggestions for further investigation.
This book is an historical approach to calculus. In ten chapters, a rough outline of calculus is given, approaching the topics discussed such as they were originally addressed by their "inventors", but using familiar modern concepts, terminology, and notation. To give an example: chapter 1 discusses infinite sums and how they also show up in the ideas of Archimedes on computing the area of a parabolic segments, filling up the area with smaller and smaller triangles. In a concluding section called "Furthermore" it is again Archimedes who uses successive polygons to approximate the area of a circle and hence the value of π. It is a further challenge/exercise to find a formula that gives the sum of the squares and cubes of the first n integers.
The other chapters follow a similar pattern, each time ending with the "Furthermore" section giving additional information of historical mathematicians, and some (usually rather elementary) exercises. In this style we meet Ibn al-Haytham an Jyesthadeva again on infinite sums. Among others, the reader meets Fermat and Descartes and later Cavalieri and Roberval and their study of curves and the computation of areas and volumes, which bring the reader to the 16th and 17th century. Although the exposition still relies largely in graphics, there is gradually more algebra sneaking in. Still looking at areas under curves like the hyperbola leads to the concept of logarithms and the exponential function as developed by de Sarasa, Brouncker and Wallis. Enter Newton and Leibniz. They lay the foundations of differential calculus, the interpretation of differentials and introduce a notation that looks familiar to a reader of the 21st century. The final chapter is then about continuity as discussed by Bolzano, Weierstrass, Dirichlet, and others. They introduce more rigor and are gradually leaving intuition behind.
The few names of mathematicians that I mentioned above are only some examples. Many more that have contributed are mentioned in due course.
The topics are treated in a rather intuitive way, using extensively the many figures of the text. These are manually produced with ruler and pencil, which in time of computers is somewhat surprising, but it blends well with the general approach of the book. There are of course formulas, but no formal proofs are included. So it is not surprising that the book stops with the chapter where the mathematics take off to more abstraction and less intuition.
The book is a collection of topics that more or less follows the evolution of calculus throughout the centuries. It is neither a full history of mathematics or even calculus, nor is it a collection of biographies of prominent mathematicians. It is a mixture of these, with emphasis on the mathematics itself. If used to teach calculus, then it is certainly an unusual approach. Many topics are not discussed and there is no classical set of drilling exercises for derivatives and integrals as is usual in a calculus course. It may be a good source of inspiration to formulate some assignments for homework if used as a textbook besides a more traditional course.
This is a new edition of the book written by Sara Turing, Alan's mother, that was first published in 1959. This centenary edition has been extended with a foreword by Martin Davis and an afterword by John Turing, Alan's brother. The latter places the loving account of his mother into a somewhat different perspective.
The year 2012 has been declared the Turing year because Alan Turing was born in 1912. He died in 1954 at the age of 41 by cyanide poisoning. In his short life Alan Turing has contributed a lot to different areas of science. He got an OBE (Order of the British Empire) for his contribution in breaking the code of the Enigma machine that was used by the Germans during WW II. With other code-breakers at Bletchly Park he designed the bombe an electromechanical machine to discover the settings of the rotors in the German encoders to decrypt their messages. Later he was involved in the Manchester Computers project and he designed his Universal Turing Machine, an abstract computer device. His incentive for this was that he wanted to solve the Entscheidungsproblem as formulated by Hilbert in 1928. His Turing test was designed to define "intelligence": can one distinguish a machine from a human, by just asking questions? This is the start of artificial intelligence. Turing's efforts have certainly contributed to the impetus leading to the development of computers and computer science. All in all, enough reason to celebrate his 100th birthday.
Shortly after Alan's death, his mother wrote the first version of this book. It is a loving account that a mother can give about the much too short life of her famous son. Alan's father was stationed in India and Alan and his older brother John were regularly left in the good care of a family in Scotland while his mother went to accompany his father. So apart from her own memories, Sara also draws on letters from Alan and from other people who have known Alan in his childhood or at a later stage of his life. For example, about his childhood, she mentions that Alan impressed his teachers with his sharp mind. She however mentions also his bad handwriting and his untidiness. She further describes as she had experienced his studies and his career. Obviously for her account of his scientific career, she has to rely even more on letters and testimonies by others. Like in the first edition, also this one contains two unfinished/unpublished texts and correspondence by Alan Turing: one on computing machines and one on morphogenesis, a topic he became interested in lately.
This edition has an introduction by Martin Davis who gives a short survey of Alan's main scientific achievements, and also mentions Alan's homosexuality. In fact Alan's untimely death has been conjectured to be a suicide. Alan's dead body was found with a poisonous half eaten apple nearby (hence the apple on the cover on this edition). Homosexuality still being illegal in those days, explains the intimidation by the police, and this could have been a possible motive for his suicide. Sara never mentions Alan's sexual tendency and considers Alan's death an unfortunate accident. In the afterword, his brother John does confirm Alan's sexual preferences and claims that also his mother knew about it. This is one example where he disagrees with the account of his mother. Sara died in 1976, which allows John in this edition to give his own account of his relationship with Alan, and at several points he places some of the impression given by his mother in a different perspective.
This book is not one in which you will find much details about the scientific work of Alan Turing. This is a somewhat biased account by primary sources, yet with many citations from third persons, about the life of Turing as a person. It is mainly told by a 70 year old loving mother, as a tribute to her son who died too young, but somewhat annotated by the afterword of Alan's brother.
A marvelous book, abundantly illustrated. It contains many applications in art science and technology, all with an underlying geometric flavour. There are some theorems and proofs, but these can be just skipped without harming the overall reading in case the reader is not interested in the technicalities.
What is immediately striking at first glance is the luxury of this publication: thick paper with hundreds of colorful glossy pictures and graphs. If you love books and geometry, this is one to fall in love with. Heavy stuff though: about 1.4 kg, it is a treasure that you would not like to take in your hand lugage on a plane.
In fact, the book has been available in German as Geometrie und ihre Anwendungen in Kunst, Natur und Technik (Elsevier, Spektrum Akademischer Verlag, 2005/2007). This is the English translation, extended with 60 pages and extra illustrations (there are about 900 of them). The author is professor at the Universität für angewandte Kunst in Vienna, and this might explain that this book, with geometry as the binding factor, has so many and very diverse applications. Moreover, this is not his first book on this kind of topic and he has also books on software for computer geometry (OpenGL®). Additional information about this book and links to other publications can be found at the book's website www.uni-ak.ac.at/geometrie.
This picture book is more than just a coffee-table book unless it is a table in the coffee room of a math department, because it contains not only many pictures, but also gives theorems and proofs(!). Although the proofs are not so very technical and are more descriptive geometrical than analytical with a minimum of formulas. If the reader is not interested in these proofs, there is no harm done or discontinuity in the appreciation of the global story told when they are just skipped.
The emphasis is clearly on the applications of geometry. In 13 chapters of increasing complexity, the reader is confronted with many expected but also with many unexpected applications. Sometimes, the application is more physics than geometry, but if it has an important geometrical component, it is reason enough to include it.
The author starts with points, lines, and elementary curves in the plane, to move on to projections. Already there the reader finds applications such as what can be learned from the shadows of objects or about the retro-reflector in a bicycle wheel. Entering the 3D world starts with polyhedra, then moves on to curves in 2D and 3D, to arrive at cones and cylinders as the simplest examples of what is further elaborated: developable surfaces, conic sections and surfaces of revolution. On a more advances level we find helical, spiral and minimal surfaces and an introduction to splines and NURBS for modeling general curved surfaces. All of this is amply illustrated with many applications from industrial design, architecture, cartography, connecting pipes, gear wheels, animal horns, DNA, and many more.
After that, the chapters start dealing with the more applied sciences. Chapter 9 is about optics: the human eye and photography and reflections and refraction. The next two chapters deal with the geometry of motion: curves generated by all sorts of mechanical devises, and orbits in astronomy. The last two chapters are about tilings of the plane and symmetry and other remarkable patterns appearing in nature. The latter two are are promoted from an appendix in the German edition to proper chapters in this one.
There are also two appendices left in the form of short courses. One is about free hand drawing. As the author rightfully claims, in this computer age where pictures and graphs are rendered digitally by computer software, generating a result that is unnaturally close to perfection, free hand drawing becomes a rare skill while it should be a basic one for communication. The second course is about photography: the rules of perspective brought in practice. Both of these can be read independent from the rest of the text.
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, these Dirichlet series are expected to have analytic continuation and to satisfy functional equations, having applications to analytic number theory. The group of symmetries of functional equations of the series are arbitrary finite Weyl groups. This book is centered in the case of Weyl groups with Dynkin diagrams of type A, giving a complete proof of the analytic continuation and functional equations in this case. It is a technical piece of work addressed to experts in the area.
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, these Dirichlet series are expected to have analytic continuation and to satisfy functional equations. By contrast, these Weyl group multiple Dirichlet series differ from usual L-functions in that they are usually functions of several complex variables, and in that their coefficients are multiplicative only up to roots of unity.
The group of symmetries of functional equations the series are arbitrary finite Weyl groups. This book is centered in the case of Weyl groups with Dynkin diagrams of type A. The coefficients of the Weyl group multiple Dirichlet may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and once it is proven that they are equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. This is accomplished through a series of surprising combinatorial reductions. The main bulk of the book (chapters 6 to 17, out of the 20 chapters in total) consists on the proof of this result.
The book also includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation. It is a technical piece of work addressed to experts in the area.