Integrating Mathematical Knowledge

What is MATHESIS ?

Mathesis is a system of higher mathematics, which is presented as a cumulative and modular body of knowledge in several volumes.

The content of Mathesis is that of a Bachelor's degree in mathematics at the highest level, augmented by essential and substantial supplements that are little or not studied at university.

The corpus is divided like a classical Bachelor's degree into three 'years', each consisting of two 'semesters'. These six semesters form six recurrent learning 'cycles' covering all subjects. Each semester or cycle consists of five or six compact courses.

The material is grouped into five main axes of mathematical knowledge: Logic and Combinatorics, Number Theory, Algebra, Geometry, Analysis and Topology.

The aim of Mathesis is to integrate the core of higher mathematics into a single, comprehensive, self-sufficient body of knowledge accessible to all.

Background and Purpose

My name is Jean Barbet, I am French, I hold a Master's degree in biology and environnemental sciences, and a PhD in pure mathematics (University of Lyon 1, 2010). I am a freelance mathematician and I have been teaching mathematics for more than ten years.

After a techno-scientific experience in ecophysiology research, I reoriented myself towards mathematics. I was then in search of a self-sufficient science where everything could be demonstrated, without theoretical or technological "black boxes".

Enrolled directly in the 3rd year of the Bachelor's degree, I experienced in my particular situation some of the material limitations of the university. Due to a lack of time, the volume of content taught is limited and due to a lack of resources, the teaching is concentrated on Algebra and Analysis. The multiplicity of teachers does not allow for a uniform approach either.

Having to do some personal work in the specialised books to fill in my gaps and complete my learning, this was the opportunity to start a continuous effort to integrate the knowledge and add to it elements that had been skimmed over or absent in the university teaching.

Number Theory and Geometry are ancient founding subjects and intrinsic motivations essential to Algebra and Analysis. Geometry is also essential to physics.

Mathematical Logic, the basis of theoretical computer science, is little taught, yet it is essential learning in relation to mathematical foundations and method.

What was a student exercise has become a continuing project: to gather the "essential" mathematical knowledge into a core, a complete and self-sufficient sum, accessible to all.

Mathesis is being developed as a 'system' of mathematics, and I wish to transmit this system as a written corpus and an online curriculum to enable anyone to learn higher mathematics from A to Z on their own.


I am progressively editing the volumes of the Mathesis written corpus in the form of electronic books ("e-books") containing the complete course, and accessible on Mathesis at the following address :


Each volume corresponds to one course, and the syllabus of the corpus/curriculum is available at the bottom of the page.

By purchasing the Mathesis e-books, you also support this project of integration of mathematical knowledge and you participate in its development. 

Epistemic and Pedagogical Approach

Mathesis is conceived as a programme for the natural construction of mathematical knowledge, in the form of a series of books individualised by courses.

While many textbooks or curricula introduce abstract concepts at the outset to reduce natural objects to simple examples, I propose to start from the natural intuition of elementary mathematical objects and to build abstraction in a progressive way.

The constraints of academic teaching and the mathematical content itself mean that the mathematical subject is often separated into 'domains'. However, these areas are not isolated, and I wanted to propose here a unique curriculum, which approaches them in a cyclical and transversal way, in contrast to the current practice.

The subject matter is divided into themes grouped into compact courses (5 or 6 per semester), and each cycle or semester forms a higher unit based on natural links between the different courses and themes.

These links can be relationships of logical necessity (certain notions are necessary to approach notions of comparable level) or relationships of thematic proximity (analogues of the same notion in different fields).

Each cycle proposes a return to the various subjects with successive additions, according to a cumulative progression comparable to elementary school pedagogy. Each field is thus also constructed for itself, the whole presenting a "helical" structure: one turn of the helix represents a cycle, and the contributions on a subject are accumulated vertically.

University courses and mathematics textbooks often separate lectures and exercises. In Mathesis, theory and practice form a unity, knowledge and skill are integrated, and exercises and problems naturally illustrate and complement the course.

Essential and substantial additions to the usual undergraduate curricula (in logic, number theory, geometry...), are added, to provide a comprehensive education, grounding all aspects of elementary mathematical knowledge and linking them together.

Because each course focuses on a particular theme, it is also possible to use Mathesis for modular learning by subject. By choosing a specific topic, the student can select the relevant courses to acquire that particular discipline.

Learning with MATHESIS

The five or six courses in each cycle each deal with a coherent and well identified topic. Each course is presented as a compact volume, corresponding to about twenty lessons or a month of teaching, and constitutes a valuable unit of learning in itself and a personal achievement.

The Mathesis approach is self-sufficient and complete: the single integrated corpus allows the acquisition of the necessary knowledge and skills without having to follow other textbooks or courses.

The individualisation of courses allows you to use Mathesis to complete your own curriculum or to satisfy your curiosity by choosing a particular course. It also allows you to work on a specific subject by selecting only certain courses, or to build a tailor-made learning programme (modular approach).

There are no substantial prerequisites for studying the corpus or following the curriculum: the elements are introduced as they are learned. Experience of high school mathematics is a comfortable advantage and a good starting point, but it is not essential.

Knowledge, and in particular mathematical knowledge, should in my view be an adventure. As a corpus, Mathesis is also a story, telling a journey through the modern mathematical universe. As a curriculum, Mathesis is also a safe learning path to mathematical science.


The Syllabus

The following syllabus is indicative and will be updated regularly. The curriculum is divided into three years of two semesters each. Each semester is a learning cycle, of which there are six. The courses are numbered within each semester or cycle.

First Year

Semester/Cycle I

The first cycle of Mathesis aims at setting up a complete foundation, at the level of the intuition of fundamental mathematical objects and their elementary conceptualisation via naive set theory and natural mathematical logic.

This first cycle is therefore first of all an introductory path, where we describe, axiomatise and prove the elementary properties of the fundamental arithmetico-geometric sets that are N (natural numbers), Z (integers), Q (rational numbers), R (real numbers), C (complex numbers) and H (quaternions).

This is an opportunity to acquire the modern mathematical method, which consists in particular in the conceptualisation of sets, descriptive and demonstrative rigour, and the systematic learning of demonstration rules.

A first elementary mathematical theory allows the integration of the first level of the system, in which many other subjects are covered, such as rational arithmetic, Euclidean geometry or curves in higher dimensional spaces.

All topics are approached at an elementary level by highlighting their natural connections, rather than waiting to be able to expound them from abstract theories. As an indication, the Mathesis student will discover mathematical infinity (Course n°2) and differential geometry (Course n°6) already in this first cycle.

List of courses for Semester I

Course/Volume 1 - Entering the Mathematical Universe: Natural sets, Mathematical logic, Set Theory and Mathematical Proofs

Discovery of the mathematical universe. Mathematical language and expression. Properties of natural sets. Basic set theory. Mathematical reasoning.

Course/Volume 2 - Sets, Applications and Numeration: from Finiteness to Mathematical Infinity

Products, relations and applications. Injections, surjections, bijections and the number of elements. Enumeration of finite sets. Mathematical infinity.

Course/Volume 3 - Elementary Arithmetic: from Natural Numbers to Rational Numbers

Axioms and arithmetic of the set N. Axiomatisation and description of the set Z, basic arithmetic. Axiomatisation and description of the set Q, arithmetic properties.

Course/Volume 4 - Euclidean Geometry: from Rational Numbers to Real Numbers

Course/Volume 5 - Basic Real Analysis: Sequences, Functions, Derivation, Integration

Course/Volume 6 - Higher Dimensions: Complex Numbers, Real Spaces, Curves, Quaternions

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