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.