Do I count? Stories from Mathematics

This is a book about the  doing of mathematics, the making of mathematics. After the reading of the first pages I believed that it had been written by a very good mathematician to be read by non-mathematicians. But this was a wrong perception. I think now that although a lot of people with a good cultural level could enjoy with the reading of the book, mathematicians will enjoy more. The first chapters mainly deal with numbers, from some naive questions about the numbers the bees could count, to more subtle results as the equality $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$. As it should be, some pages are devoted to prime numbers: Euclid's proof of its infinitude, Mersenne's prime numbers, and some much more difficult problems: Goldbach's conjecture, prime's distribution, the $3x+1$ problem, and many others. This first part finishes with a question about which this reviewer has spent some time: how many entries of the grid must be filled in by a Sudoku maker in order to construct a puzzle with a unique solution? I have learnt from Ziegler that this is an unsolved problem, but the answer is not greater than $17$. The author finishes the part of the book mainly devoted to mathematics with some science-fiction wishful thinking about answers to some of the most important open problems: according with the (optimistic) author's wishes, the $P\neq NP$ problem and Riemann Conjecture will be solved in 2015, and Navier-Stokes equations will be solved two years later. To finish the description of this first part of the book it is worthwhile mentioning that a section is devoted to explain in detail several situations of misuse of mathematics in the real life. In the second part of the book Ziegler pays special attention to where and how maths are done, and he presents very interesting aspects of the life of some brilliant mathematicians. Concerning the first item it is explained how Paul Erdös obtained many important results by traveling from one city to another and, preferably, with a cup of coffee. There is also a long list of famous cafés all around Europe were many brilliant ideas have grown-up and, of course, in the last decades many significant results have been obtained in the computer. All this is explained by the author in a very pleasant way. Some stories are well known; how Poncelet, Leray and André Weil among others were able to create high level mathematics in captivity, the seven years spent by Wiles in an attic room in Princeton before presenting the (noncomplete) solution of Fermat's Last problem, or Smale's solution of Poincaré's Conjecture for $n\neq3$ in the beaches of Rio de Janeiro. But some other ones were unknown for this reviewer. In particular I cannot resist myself to write here part of the content of the section devoted to how maths can be done in church. After some comments of general character the author transcribes the following acknowledgment in the paper by M.I. Hartley and D. Leemans entittled Quotients of a universal locally projective polytope of type $\{5, 5, 5\}$, Mathematische Zeitschrift, 247, (2004): `The first author would like to acknowledge and thank Jesus Christ, through whom all things were made, for the encouragement, inspiration, and occasional hint that were necessary to complete this article. The second author, however, specifically disclaims this acknowledgement". The final chapters concentrate on some aspects of the life of some brilliant mathematicians; the connection between chess and maths, with Wilhelm Steinitz and Emanuel Lasker as protagonists, the relationship between Sofia Kovalevskaya and Gösta Mittag-Leffler as a possible explanation of why there is no Nobel Prize in Mathematics, the mysterious disappearance of Alexander Grothendieck, some not well known aspects in the life of Paul Erdös or Gian-Carlo Rota, and some others. I would like to finish this report by copying some words of Ziegler about our discipline. ``Mathematics is difficult. One could, of course, try to base instruction in mathematics on the message that it's all quite simple; indeed, that approach has been tried repeatedly. But it's not true. Mathematics is not simple. Mathematics is difficult. My colleagues and I would not find doing mathematics nearly as attractive if it were easy. Sadly for the textbook writers, mathematics is and remains difficult. But we should treat that fact as a challenge and emphasize it almost like a marketing slogan". Finally, of course, I encourage you to read this book. I have enjoyed a lot doing it and I have learnt many interesting things about mathematics, about the process of learning and doing mathematics and about the life of some outstanding mathematicians.

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