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.