Exploring the infinite. An introduction to proof and analysis (Q2832132)

Anybody who wants to become an experienced mathematician needs a distinguishing characteristic called routine, and to achieve a sufficient amount of routine requires what in German is usually called ``Sitzfleisch''. However, mathematics is much more than just knowing definitions and results, or acquiring technical skills to carry out difficult calculations: what really matters is to understand a new problem, to jump from one unfounded conjecture to another, and finally to solve the problem with imagination and daring. This is meant when some people speak of mathematics as fine arts, rather than science.NEWLINENEWLINE The aim of this book is to encourage students to chose this imaginative path, to develop a certain variety of skills, and to plunge into the adventure of mathematical exploration. The outcome is an interesting book somewhat beyond the scope of other books for freshmen. It consists of two parts. The first part serves as an introduction to techniques of proof and abstraction which aims to prepare the reader to get used to mathematical reasoning. Since this is at the heart of any mathematical field, the specific nature of the topics treated here it is not really important. In the second part, this process is applied to some parts of a first year calculus course covering real numbers, sequences and series, continuous functions, and differentiation.NEWLINENEWLINE The book is very well written. The reviewer particularly liked the detailed problem sections which are scattered over the whole book, because they are not the usual collections of more or less boring calculations, but are often organized in the ``true or false?'' style of correct assertions (to be verified by a rigorous argument) or incorrect assertions (to be falsified by a counterexample). This is the best way to teach mathematics, and it contains elements of an oral examination.NEWLINENEWLINE As the author herself writes in the Preface, a proof should be better presented as ``a clear, logical explanation of the solution to an interesting non-obvious question'', rather than ``the construction of formal arguments using only axioms, definitions, and previously-established theorems''. The reviewer fully agrees on this point of view and therefore highly recommends the book.