The Best Writing on Mathematics 2013
This collection, the fourth in a row, was brilliantly selected once more by the editor Mircea Pitici. Mind you, the papers were selected, not published in 2013. The current selection has papers published in the period 2009-2012. As usual, the foreword is provided by a mathematical celebrity. This time it's written by Roger Penrose. Reviews of the two previous collections can be found in this EMS database: see 2011 and 2012.
As in the previous collections we find here a diversity of generally accessible papers that report on issues from mathematics or about mathematics that not only professional mathematicians will (or should be) interested in, also politicians or anybody responsible for science and education are advised to read some of the 20 contributions.
There is an enlightening paper by Ph.J. Davis on the drastically changed landscape dominated by computers and multimedia in which mathematics has to operate nowadays. I. Stewart in a tribute to A. Turing explains how stripes on animals appear as a consequence of waves, while spots are the result of interfering waves. T. Tao illustrates how universal or global laws may result from unpredictable behaviour of a great many individuals. The small world phenomenon discussed by G. Goth is a counter-intuitive property of the many individuals populating e.g. social networks. This concept is by now fairly well known by the general public. To find the shortest path visiting several nodes in such a graph is known as a traveling salesman problem, and is hard to solve. It is NP-complete. The problem whether the class of NP problems equals its subclass of easier problems of class P is one of the big open problems in computer science and it shows up in a text by J. Pavlus. This is one of the Clay Mathematics Institute millennium prize problems. Solving it would gain you a prize of US$1,000,000.
Large numbers usually entail also randomness and probability and these topics are the subject of several of the papers in this collection. The very basics of randomness are bravely reviewed in a short paper by C. Seife. D. Knuth illustrates that small random perturbations of harmonics in music are more pleasing. This is a general phenomenon. Perfect symmetry is boring. Slight deviations make symmetry interesting. How come that gamblers become addicted? S. Johnson shows that this is because the gambler has the wrong intuitive interpretation of the probabilistic laws like "I've been loosing so long, hence the next time I should win". Notorious errors in the early days of probability theory are placed in an historical context by P. Gorroochurn. In financial mathematics E. Ayache uses metaphysical arguments to conclude that probability should not be applied to states of the market, but that the market itself should be a category of thought to substitute probability. Probability can analyse the past, but it cannot predict the future.
Geometry is another topic brought to the foreground in several of the texts. R. Gross uses the Jerusalem Chord Bridge to discuss Bézier curves, D. Silver shows that Dürer's "Painter's Manual" introduces the cuts of a cone. Although Dürer gives a construction method, he mistakingly thought that an ellipse was egg-shaped, i.e. thicker at the bottom than at the top. A somewhat understandable error because the cone is thicker there than at the top of the cut. More of a topological nature are the papers by K. Delp about how topological concepts like hyperbolic geometry that were used in design and fashion culture, while F. and W. Ross show graphical art that is produced by drawing just one very complicated Jordan curve. Mathematical history is reflected in the paper by D. Lloyd about the hoax that was caused by a picture of five neolithic carved stone balls from Scotland that presumably were models for the five Platonic solids. He gives good arguments against this conjecture. More history is represented by J. Bennett in his paper about the mathematical (mostly astronomical) instruments that were developed during the 16th to the 18th century.
The paper by F. Quinn is about the revolutionary ideas in the period 1890-1930 that have drastically influenced sciences. This revolution caused some bifurcation between core mathematics and the applied vision. The paper bridges the gap between history and education. The latter is another recurrent topic in these collections of "Best Writings on Mathematics". Quinn argues that the cultural attitude towards mathematics in general and to core mathematics in particular that we experience today is largely a matter of neglected public relations. Hence high time to revise our ideas about mathematics education, since that has not fundamentally changed since the Greeks. A. Sfard's contribution is another outspoken plea to drastically change our arguments of why we should teach mathematics and certainly how and what we should teach. Also E. Maloney and S. Beilock's text about `math anxiety', that is shown to exist at a much earlier age than generally accepted is relevant in this context.
Once more this is a marvelous selection of papers about mathematics written by the best. They do not drawn the reader into the mathematical jargon that is only of interest to the mathematical literate. In fact practically no mathematics is needed and formulas are almost completely absent. It is the best possible way of communicating mathematics to the non-mathematician and even the ones suffering from mathematical anxiety will enjoy reading the booklet. Of course this is only a relatively small selection but for the reader longing for more, Pitici gives in his introduction an even longer list of books, papers, websites and blogs that are equally worth reading. Pitici did once more an excellent job, and the result is highly recommended to all with a broad interest in science, history, art, education, philosophy,... which is almost anybody.