Mynard F. Introduction to the language of mathematics

New York: Springer, 2019. — 191 p.This text is meant to be used as a textbook for a transition course from mathematics centered on calculation techniques to proof-based mathematics. Most US universities offer such a class after two semesters of Calculus, more for the sake of ensuring a minimum of mathematical maturity than because Calculus itself is of any help. Such a course often focuses on proof writing and is foundational for upper level mathematics courses. Proofs, however, are only a part of the foundations that need to be laid out to prepare for upper level courses. Hence, this text is about foundations and discusses not only proofs but the kind of system and conventions you need to build your proofs in. The radical change of perspective is often disorienting to students, and such courses often have high failure rates and leave students frustrated.
For students who have come to think of mathematics as Calculus and its applications, it comes as somewhat of a shock that mathematics is about something else entirely. Here is what I want to tell them before engaging on that road: This is a course unlike any other mathematics course you have taken so far. Where you have been focusing on basic techniques and calculations, you will focus on arguments from now on. Consider that you have been taking pre-mathematics courses until now and that you are taking a first mathematics (as what mathematicians do) course, where you will learn the language and way of thinking of mathematicians—mathematicians rather than mathematics—for mathematics is a human activity, driven by human impulses to understand and to model esthetically and efficiently.
This is a formal course, focused on formalism, though the goal of reaching fluency in reading proofs and proof making goes way beyond formalism. As such, the material is often disorienting for students at first and appears very dry and abstract. It is not unlike learning a foreign language from scratch: before you can see the beauty of poetry in that language, you have a long way to go in the sometimes tedious task of learning vocabulary and grammar, before developing the necessary intuition. Learning this material will probably be more difficult than any other mathematics class before, but it is worth the effort: you will sweat and puff pushing a very heavy door, but the land on the other side of the door is the secret garden of mathematicians, a land of beauty and harmony that you will surely enjoy exploring.