# 34740 - History of Mathematics

• Moduli: Silvia Benvenuti (Modulo 2) Maria Giulia Lugaresi (Modulo 3) Alessandro Gimigliano (Modulo 1)
• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
• from Feb 19, 2024 to May 21, 2024

• from Sep 18, 2023 to Nov 14, 2023

## Learning outcomes

At the end of the course, the students should:

- possess a hystoric  knowledge in some depth of the key-themes of mathematics and of mathematical thought; possess also a good panoramic view of the evolution of maths and of mathematical thought.

- be able to use these cultural tools in the professional activities when teaching maths.

-be able to use this knowlkedge in order to create efficient teaching materials to be used in the classroom.

## Course contents

The course is divided into 3 modules:

Module 1 (Alessandro Gimigliano) 4h I term, September-October.

Module 2 (Maria Giulia Lugaresi) 24h, II term.

Module 3 (Silvia Benvenuti) 20 ore, II term.

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The object of the course is the presentation of some of the principal results in several mathematical disciplinesin order to give an orientation of their hystorical development. We aim to give the students tools that may enable them to develope teaching strategies based also on the use of original hystorical sources.

Module 1

Chapter 1. Arithmetic. Sketches on ancients numerical systems (positional or not). Pythagora's Arithmetic.

Chapter 2. Ellenic and Ellenistic Geometry. Euclide, his Elements and the Optics. The first book . Archimedes and the exahustion method. Notes on Appollonius, Menelaus, Pappo.

Chapter 3: Resurgence of Geometry in the Reinassance, via perspective: Ghiberti, Piero della Francesca, Leon Battista Alberti, Dürer. Kepler.

Chapter 4.Medieval Geometry. Fibonacci, Oresme, Bradwardine, Buridano, Nicolas of Cusa.

Chapter 5. Application of algebra to geometry. Carthesian algebra. René Descartes: principal contents of la Géométrie.

Chapter 6. Lines of development od Proiettive Geometry. Pascal, Desargues, Monge and Descriptive Geometry .

Chapter 7. Projective Geometry in the XIX century. The synthetic approach (Poncelet, Chasles, Staudt) and the analytic-algebraic one (Gergonne, Moebius, Cayley).

Chapter 8. Non-Euclidean Geometries. Klein's program . The italian School ofAlgebraic Geometry and the problem of axiomatization.

Chapter 9 . Crisis of Foundations: Cantor, Hilbert, Russell, Goedel.

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Module 2:

Arithmetic in India and in the  arab world and its spreading in Europe.

Algebra. Algebra in the arab world its History and Geography. The work of Al-Khwarizmi. Arab mathbetween VIII and XV centuries (Al-Khwarizmi, Abu Kamil, Al- Karagi, Al-Kashi, Omar al-Khayyam, Al-Tusi). Examples from the works of Abu Kamil, Omar Al-Khayyam. Arab mathematics towards West.

Leonardo Pisano. The "Liber abaci". The schools of abacus, Luca Pacioli.

Italian algebrists in '500. The solution of third degree equations via radicals. Contributions of Scipione Del Ferro, Niccolò Tartaglia, Girolamo Cardano to the solution of third degree equations.The irrieducibile case in the works of Cardano and Bombelli. The solution of fourth degree equations by Ludovico Ferrari. The immaginary quantities, the Algebra by Rafael Bombelli.

François Viète and symbolism in algebra.

The birth of  infinitesimal calculus. Methods to determine the tangent to a curve: Descartes, De Baune, Hudde, Fermat.

The Nova Methodus by Leibniz: integral reading of the text. Exampes from Nova methodus: the law of refraction and the tangent to a curve.

Newton's opus. His memoires about differential calculus: Analysis of  equationis with an infinite number of terms, Methods of series and of fluxions, On the quadratura of curves. Application  of onvers and direct fluxions method. Method of first and last ratios. The dispute ol calculus.

The heirs of Leibniz's and Newton's traditions. The mathematics works of Jacob, Johann and Daniel Bernoulli. The Analyse des Infiniment petits by Hopital. The heirs of Newton's traditions: Brook Taylor, James Stirling, Abraham De Moivre, Colin Maclaurin.

The mathematics works of Euler. The treateses on infinitesimal calcuus: Introductio in analysin infinitorum, Institutiones calculi differentialis, Institutionum calculi integralis.

The diffusion of infinitesimal calculus in Italy.

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Module 3

In this module we will consider arguments abut history of ancient and modern math, not working with a chronological approach, but following several themes. Precisely, we will choose some results which are milestones for mathematics of all times, and for anyone of them we will analyze in some detail its origin, how it evolved and to what it leads.

So, an outline of the resulting prigram is (a subset of) the following:

1. The sum of the square on the omma dei quadrati costruiti on the catethuses: Pytagoras theorem, froi Babylon to Wiles.

See the program of A. Gimigliano's module, to which this module will add a lesson on Fermat Conjecture (statement, attempts tp prove it - among them the program of Sophie Germain) and Wiles's proof (contest, anecdots, mith of the isolated genius).

2. Geometries, form Euclid to Gauss - and beyond.

Starting from what has been seen on Euclid in the first module, we will reflect on the genesis of non-euclidean geometries, following first the suggestion of Imre Toth's phrase: “Mathematics is the sciece of liberty: non-euclidean geometry was not born for measures, but on the ground of the free human choice to deny in a non-destructive manner”, and then on the consequences on Math, Physics and Art of the introduction of the new geometries.

3. Classification of surfaces vs classification of n-varieties: topology, from Euler to Perelman.

The birth of topology is by convention associated to the one of graph teory, hence to the problem of Königsberg's bridges, attacked and solved by Euler. Starting from such problem, we will see how topology is the study of properties which are invariant for homeomorfisms. After clarifying what we mean with “classify” and the use of invariants, we will prove the theorem of classification of closed topological surfaces, and we will see how the generalization of this statement leads to Poincaré's Conjecture. THen we will see which has been, since the beginning the possible line of proof, and why the topological attempts failed (untill now, in order to explain what, in the end, has revealed to be the successful appoach and why, giving an idea of the proof (contributes of  Hamilton and Perelman).

4. The solutions to algebraic equations, from Babylon to Galois.

To what has been seen in the modulo of M.Giulia Lugaresi, we will add a lesson on Galois.

Module 1:

M. Kline, Storia del pensiero matematico, Einaudi.

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Module 2 and 3:

C. B. Boyer, Storia della matematica, Mondadori.

M. Kline, Storia del pensiero matematico, vol. 1, Einaudi.

B. D’Amore – S. Sbaragli, La matematica e la sua storia, voll. II-III, Edizioni Dedalo.

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Other material and notes will be made available by the teachers on the platform "Virtuale".

## Teaching methods

Lessons in classroom. Collaborative activities in the classroom, consulting original texts.

## Assessment methods

The final exam will be an oral test which starts with a subject chosen by the student.

There will be two separate exams, one for Module 1, the other for Modules 2 and 3. The final mark will be the avarage of the two results.

## Teaching tools

Notes and other materials given by the teachers.

## Office hours

See the website of Alessandro Gimigliano

See the website of Silvia Benvenuti

See the website of Maria Giulia Lugaresi