M A T H E S I S

About…

**MATHESIS **presents a **conceptual** approach to modern mathematics.

We offer **articles** at different levels on elementary and advanced mathematical topics, with an emphasis on the natural construction of concepts and their genealogical relationships. We will soon propose courses in the format of e-books, so that you can learn the essentials of mathematics on your own in a perfectly rigorous way.

**Mathematics is the science of the infinite.**

Hermann Weyl

**Your author**

My name is **Jean Barbet**, I am French and in live in Strasbourg. I hold a PhD in pure mathematics and I am an independent mathematician and online teacher. Through my research and original teaching, I promote the idea of a conceptual mathematics, having its interest and meaning in itself, and not reduced to its applications, or limited to a set of techniques.

Once an undergraduate student in mathematics after a conversion from life and environmental sciences, I was temporarily faced with failure, due to my shortcomings and an overly technical approach to teaching in certain courses. I had to reconstruct for myself **the core of higher mathematics** I needed, not in the form of disparate knowledge and meaningless practices, but in the form of a system manifesting the connections between the various elementary branches of mathematical science. It provided me with the motivation and structure to complete my training as a mathematician, and I continue to feed this construction that guides and directs me in my multiple mathematical activities today.

**Numbers rule the universe.**

Pythagoras of Samos

**Technique, science and philosophy**

**Mathematical science** was born from practical considerations related to counting, calculation and measurement… and from theoretical considerations related to the nature of numbers, continuum and figures… .

The most ancient civilisations, such as the Egyptians and Babylonians, had advanced mathematics, but in the form of empirical and intuitive knowledge. It was **Pythagoras** who introduced the logic of philosophy into mathematics to make it a true science, where according to the injunction of **Thales**, one must prove every statement.

Mathematics is part of **technique** (through calculation, reasoning, geometric representation, etc.), **science** (as a body of knowledge established by a rational method) and **philosophy** (as a discourse based on precise concepts and propositions established by logical arguments).

Through this unique triple participation, mathematics, which was **the first of the sciences** and the only one whose results are timeless, is a central element of human civilisation. Its perfect clarity and precision should serve as a model for every technique, every science and every philosophy.

**There is no royal road to geometry.**

Euclid of Alexandria

**From a Ruler and a Compass**

The ancient **Greek mathematicians** considered that the ‘ideal’ geometric constructions were those that could be made with a **ruler and compass**: starting from a segment taken as a unit and its extremities, one was allowed to construct only points, circles and lines that could be obtained in **a finite number of steps** by the sole use of the ruler and compass, and only from objects constructed in the previous steps. For me, this method symbolises the break with a purely technical approach, having allowed the emergence of a true **mathematical science** that finds its questioning, its interest and its meaning in itself.

In antiquity, arithmetic or number theory and geometry or figure theory were the two **pillars of mathematical science**. The modern viewpoint is based on two methodological approaches: algebra (born of the theory of equation solving) and analysis (born of the study of infinitesimal phenomena and limit processes), which complete the association between arithmetic and geometry. **Set theory** made it possible to define infinity, to give a rigorous foundation to mathematical science, and to bring about modern mathematical logic. In short, logic, arithmetic, geometry, algebra and analysis are all dimensions of the **science of infinity** that I want to integrate and cross-reference in MATHESIS, in order to present this integration of modern mathematics in the same spirit as the ancient Greek mathematics.