"Mathesis - the Mathematical Universe" is a series of integrated courses covering the essentials of the university Mathematics undergraduate curriculum.
Mathesis offers to everyone the core of modern mathematical knowledge in a unique, cross-disciplinary programme of 3 years of two semesters each. The learning process is designed in a no-fail mode: the course focuses on the essential and is built from intuition to abstraction and not from abstraction in itself, natural exercises extend and complete it to integrate theory and practice. A simple working method is proposed in the preamble of each volume to guide the reader in his self-learning. Amateurs will find it a self-sufficient and accessible sum, and students of Mathematics, but also of Science and Philosophy, will find it a useful complement to their university course.
The "Mathematical Universe" series is presented in the form of successive integrated courses. Each one consists of a complete chapter or volume of learning built to progressively integrate all the necessary elements in a unique pedagogy that exposes all the topics in a circular fashion (see Mathesis - Integrating Mathematical Knowledge). The different courses follow each other naturally but can also be taken individually according to the student's interest or needs. These integrated courses are divided into three years of two semesters each, and numbered by their place in each semester.
This first 79-page volume of the "Mathesis - The Mathematical Universe" series is the first course of semester I of the first year of the MATHESIS curriculum. It lays all the necessary foundations, both theoretical and practical, for learning modern mathematics, and is the first step in this learning process, but it also stands alone as a general introduction to the subject. The course is divided into 23 sections and includes 45 figures, as well as numerous examples and exercises at an accessible level. It is designed with particular care for those who want to learn mathematics seriously but have been discouraged by their failures, or who do not know where to start but want to begin without further delay by following an authentic college level course in a simple, marked out and failure-free path. We cover natural sets of numbers, elements of set theory, natural and symbolic mathematical logic, the usual rules of proof, and we prove the first theorems.
This second 66-page volume of the "Mathesis - The Mathematical Universe" series is the second course of Semester I of the first year of the MATHESIS curriculum. It contains 18 lessons and 24 figures, and covers the essential set-theoretic notions of product, relation and application and all the concepts associated with functions, the mathematical conceptualisation of the "number of elements" of a set based on bijective applications, the numeration of finite sets and the enumeration associated with their elementary combinatorics, the rigorous definition of mathematical infinity and its elementary characterisations. The axiomatic description of the infinite sets N, Z and Q of the following course is also prepared, which will allow us to approach elementary arithmetic in a sound manner.
To Appear - Winter 2021-2022