THE MATHEMATICAL UNIVERSE

"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.

6 courses/volumes

Entering the Mathematical Universe

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**.

Sets, Applications and Numeration

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.

Elementary Arithmetic

To Appear - Winter 2021-2022