the Ruler and the Compass

Exploring Infinity


The Ruler and the Compass presents a philosophical, that is, a conceptual and integrated, approach to modern mathematics.

I offer articles at various levels on topics ranging from elementary to more advanced mathematics, focusing on the natural construction of concepts and their genealogical relationships.

Mathematics is the science of the infinite.

Hermann Weyl

Your author

My name is Jean Barbet. Holder of a PhD in pure mathematics (mathematical logic), I am an independent mathematician and online teacher. Through my research and my original teaching, I want to promote the idea of a conceptual mathematics, with its interest and meaning in itself, and not enslaved to its applications, or limited to a set of techniques.

Once a student in Bachelor’s degree after a reconversion from studies in Life and Environmental Sciences, I found myself temporarily confronted with failure, because of my shortcomings and an overly technical approach to the teaching of certain courses. I had to reconstitute for myself the core of academic mathematical knowledge that I needed, not in the form of disparate knowledge and meaningless practices, but in the form of a system showing the connections between the various elementary branches of mathematical science. I drew from it the motivation and structure that enabled me to complete my training as a mathematician, and I continue to nurture this construction that guides and orients me today in my multiple mathematical activities.

Numbers govern the world.

Pythagoras of Samos

Technology, science and philosophy

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

The most ancient civilisations, such as the Egyptians and Babylonians, possessed advanced mathematics, but in the form of empirical and intuitive knowledge. It was the illustrious Pythagoras of Samos who is said to have introduced the logic of philosophy into mathematics, turning it into a genuine science in which, according to the injunction of Thales of Miletus, every statement must be demonstrated.

Mathematics participates in 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. I believe that its perfect clarity and precision should serve as a model for every technology, every science and every philosophy.

There is no royal road to geometry.

Euclid of Alexandria

Why “The Ruler and the Compass” ?

The ancient Greek mathematicians considered that the ‘ideal’ geometric constructions were those that could be made using a ruler and a compass: starting from a segment taken as a unit and its extremities, they only allowed themselves to construct points, circles and straight lines that could be obtained in a finite number of steps by the sole use of the ruler and the compass, and only from objects constructed in the previous steps. This method symbolises for me the break with a purely technical approach, which allowed the emergence of a genuine mathematical science that found its questions, its interest and its meaning in itself.

In ancient times, arithmetic, or the theory of numbers, and geometry, or the theory of figures, were the two pillars of mathematical science. The modern viewpoint is based on two methodological approaches: algebra (born out of the theory of equation solving) and analysis (born out of the study of infinitesimal phenomena and limit processes), which complement the association between arithmetic and geometry. Set theory made it possible to define infinity, to provide a rigorous natural foundation for mathematical science, and to give rise to modern mathematical logic. In short, logic, arithmetic, geometry, algebra, analysis… are all dimensions of this science of infinity that mathematics is, and which I want to integrate and combine in ‘The Ruler and the Compass’, to present this integration of modern mathematics in the same spirit as ancient Greek mathematics.

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