0.3: What this book is not
There is an approach to teaching a transition course which many instructors favor. It is to have a problem-solving course, in which students learn to write proofs in a context where their intuition can help, such as in combinatorics or number theory. This helps to make the course interesting, and can keep students from getting totally lost.
We have not adopted this approach. Our reason is that in addition to teaching the skill of writing a logical proof, we also want to teach the skill of carefully analyzing definitions. Much of the instructor’s labor in an upper-division algebra or analysis course consists of forcing the students to carefully read the definitions of new and unfamiliar objects, to decide which mathematical objects satisfy the definition and which do not, and to understand what follows "immediately" from the definitions. Indeed, the major reason that the epsilon-delta definition of limit has disappeared from most introductory calculus courses is the difficulty of explaining how the quantifiers \(\forall \varepsilon \exists \delta\) , in precisely this order, give the exact notion of limit for which we are striving. Thus, while students must work harder in this course to learn more abstract mathematics, they will be better prepared for advanced courses.
Nor is this a text in applied logic. The early chapters of the book introduce the student to the basic mathematical structures through formal definitions. Although we provide a rather formal treatment of first order logic and mathematical induction, our objective is to move to more advanced classical mathematical structures and arguments as soon as the student has an adequate understanding of the logic underlying mathematical proofs.