Reading, Writing and Proving. A Closer Look at Mathematics
The book is intended for undergraduate students beginning their mathematical career or attending their first course in calculus. The book starts from the very beginning (quantifiers, basic set theory) and, little by little, a reader is led to more difficult parts of analysis (ordering of real numbers, the concept of function, sequences, finite and infinite sets, countable and uncountable sets, and metric spaces). The book is concluded by eleven projects to be worked out as a result of independent though guided study. These include among others the problem of the irrationality of e and π, the structure of the Cantor set, the Cauchy-Schwarz Inequality, and the properties of algebraic numbers. Throughout the book, the so called Polya’s fourth-step process is applied: students are led to 1) learn to understand the problem, 2) devise a plan to solve the problem, 3) carry out that plan, and 4) look back and check what the results told them. This concept is very valuable, since the book written in this way not only presents the facts but also tries to show what mathematics really is: the very concepts of definition, theorem, example, and comment are introduced, and also, the fact that mathematics consists of a careful reading and writing is emphasized, with special stress on the concept of proving facts in a rigorous way. An additional material to the book, corrections, and other documentation can be found, as the authors suggest, on the web page http://www.facstaff.bucknell.edu/udaepp/readwriteprove/. The book is written in an informal way, which could please the beginners and not offend the more experienced reader. A reader can find a lot of problems for independent study as well as a lot of illustrations encouraging him/her to draw pictures as an important part of the process of mathematical thinking.