34740 - History of Mathematics

Learning outcomes

At the end of the course, the student: - will have an in-depth historical and epistemological knowledge of the main key themes of mathematics and mathematical thought; he also has a good general overview of the evolution of mathematics and mathematical thought; - will be able to use these cultural tools from a professional point of view, applying them to the 'theory of obstacles' and therefore to the evaluation and concrete and effective intervention relating to some objective difficulties of students in learning mathematics; - will be able to use this knowledge to develop effective teaching materials to be tested in the classroom.

Course contents

The History of Mathematics course for the a.y. 2022/23 will be held in two modules, one by Silvia Benvenuti and the other by Maria Giulia Lugaresi, with the integration of two lessons held by Rita Fioresi. Silvia Benvenuti's module will have a thematic approach, while Maria Giulia Lugaresi's will follow a more chronological approach: the two modules will however proceed in parallel, taking care to deal with logically connected topics in a chronologically coherent way.
We report here the draft programs of both modules, with the caveat that the topics mentioned are certainly overabundant to be able to be treated in the 9 credits at our disposal: a selection of the same will therefore be implemented, according also to the requests of the students.


SILVIA BENVENUTI
Draft program


In this module we will address some topics of the history of mathematics, ancient and modern, working not with a chronological approach, but by themes. Precisely, we will choose some results that constitute milestones of mathematics of all time, and for each one we will go into detail where it comes from, how it has evolved and where it leads. The module is completed by two separate sections, one dedicated to the relationship between history and teaching and one dedicated to the history of women in mathematics.


The resulting program is therefore broadly the following:


History and didactics: in what sense can the knowledge of history themes be a formidable arrow in the bow of a mathematics teacher in schools of all levels?


Women in Maths: from Teano to Maryna Viazovska, vicissitudes and successes of female mathematics. Among the protagonists of this story, in addition to the aforementioned: Hypatia, Hildegard of Bingen, Émilie du Châtelet, Maria Gaetana Agnesi, Sophie Germain, Sonja Kovalevskaja, Emmy Noether, the girls of ENIAC, Margareth Hamilton.


Formulas that changed the world: where did they come from, how did they develop and where will they take us?

The Pythagorean theorem, from Babylonian to Wiles.
Geometries, from Euclid to Gauss - and beyond.
Topology, from Euler to Perelman.
The resolution of algebraic equations, from the Babylonians to Galois - see the program of the module by M.Giulia Lugaresi, to which only the lesson on Galois will be added in this module.

Integrated into this module are the two lessons by Rita Fioresi on Archimedes.


MARIA GIULIA LUGARESI 
Draft program


Module 2 of the History of Mathematics Course has as its object the presentation of the main results of the various mathematical disciplines to guide students on their historical development. The course aims to provide students with tools for designing and developing mathematics teaching methodologies starting from the use of original historical sources.


The topics that will be addressed will be divided as follows:
Chapter 1. Arithmetic. Outline of the ancient numbering systems (positional and non-positional). Pythagorean arithmetic. Indian and Arabic arithmetic and its transmission to the West.
Chapter 2. Algebra. The birth of algebra in Arab civilization. Historical and geographical context. The mathematical work of Al-Khwarizmi. Arabic mathematics between the eighth and fifteenth centuries (Al-Khwarizmi, Abu Kamil, Al-Karagi, Al-Kashi, Omar al-Khayyam, Al-Tusi). Examination of some examples from the works of Abu Kamil, Omar Al-Khayyam). The transmission of Arabic mathematics to the West.
The figure of Leonardo Pisano. The Liber abaci. The abacus schools, Luca Pacioli.
Italian algebraists of the sixteenth century. The solution by radicals of third degree equations. The contributions of Scipione Del Ferro, Niccolò Tartaglia, Girolamo Cardano for the solution of third degree equations. The irreducible case in the works of Cardano and Bombelli. Ludovico Ferrari's solution for quadratic equations. Imaginary quantities, Rafael Bombelli's Algebra.
Chapter 3. Geometry. Euclid and his Elements. The first book and the question of parallels.
The return of geometry in the Renaissance: the work of Federico Commandino. Galileo and the Galilean school. The mathematical work of Bonaventura Cavalieri: the geometry of indivisibles. The indivisible curvilinear in the work of Evangelista Torricelli.
Chapter 4. Applications of algebra to geometry. Notes on the mathematical work of François Viète, Isagoge in Artem Analyticem. Cartesian algebra. Life and Philosophical Works of René Descartes. The Discourse on Method and the three essays in the appendix (Dioptrics; Meteore; Géométrie). Examination of the main contents of the three books of the Géométrie.
Chapter 5. The birth of infinitesimal calculus. Methods for finding the tangent to a curve: Descartes, De Baune, Hudde, Fermat.
Leibniz's Nova Methodus: full reading of the text. Examples taken from the Nova methodus: law of refraction and tangent to a curve.
Newton's mathematical work. The memoirs on differential calculus: Analysis of equations with an infinite number of terms, Methods of series and fluxions, On the squaring of curves. Application of the direct and inverse fluxion method. The method of the first and last reasons.
The dispute over the infinitesimal calculus.

Teaching methods

Lectures, collaborative activities in class, consultation of original texts.

Assessment methods

Oral exam on the whole program

Teaching tools

Slides and other materials provided by the teachers

Office hours

See the website of Silvia Benvenuti

See the website of Maria Giulia Lugaresi