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CODE 35288
ACADEMIC YEAR 2024/2025
CREDITS
SCIENTIFIC DISCIPLINARY SECTOR MAT/04
LANGUAGE Italian
TEACHING LOCATION
  • GENOVA
SEMESTER 2° Semester
TEACHING MATERIALS AULAWEB

OVERVIEW

Lectures are held in Italian or English, at the students' choice. The course has a strong multidisciplinary character, addressing, from a historical and critical perspective, topics at the intersection between mathematics and philosophy, from Antiquity to the 20th century.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to illustrate the historical-conceptual development of some central themes of mathematics, emphasizing the connections with other areas of knowledge and inviting students to a critical rethinking of various basic mathematical notions.

AIMS AND LEARNING OUTCOMES

The course will illustrate the development of the concept of mathematical infinite, from Euclid and Archimedes to Cantor, Dedekind, Hilbert, Russell, Brouwer and Gödel. In addition to purely mathematical aspects (e.g. the so-called exhaustion method, the emergence of differential calculus, the evolution of set theory and its 'paradoxes'), various issues of a more philosophical nature will also be discussed (e.g. Leibniz's, Hegel's and Bolzano's ideas).

TEACHING METHODS

The course follows a traditional approach.

SYLLABUS/CONTENT

Program

1.The problem of the infinite in mathematics: introduction

2. Aristotle’s distinction between actual and potential infinite

3. Euclides’ Elements: Book I, V, IX, XII

4. Archimedes: Dimensio circuli, Quadratura parabolae. Arenarius, Methodus.

5. The emergence of the concept of zero and Brahmagupta’s algebra.

6. Mathematics in Medieval Islam: al-Khwārizmī’s algebra; infinitesimal methods (al-Kindī, Banū Mūsā, Thābit ibn-Qurra, ibn al-Haytham).

7. Infinity and continuity in the western medieval thought

8. Galileo’s Discorsi e dimostrazioni matematiche .

9. Geometrical tradition and innovation in the Renaissance.  Kepler’s quadratures. Cavalieri’s and Torricelli’s geometry of indivisibles. The quadrature of the hyperbole and the logarithms.

10. The “problème des tangents”: Descartes, Fermat and Pascal.

12. Wallis and Barrow. Newton (series, fluxions, infinitesimals,  primae et ultimae rationes). Leibniz’s differential calculus and the  “labyrinth of continuum”.

13. Euler’s divergent series.  The question of the foundations of calculus from d’Alembert to Cauchy. Hegel’s remarks on the mathematicians’ infinite.

14. Bernard Bolzano; Wissenschaftslehre, Paradoxien des Unendlichen

14. Richard Dedekind: Stetigkeit und irrationale Zahlen; Was sind und was sollen die Zahlen?

15. Georg Cantor: the idea of “Mächtigkeit”; Über unendliche lineare Punktmannigfaltigkeiten (1879-1884) and the theory of ordinal numbers; Über eine elementare Frage der Mannigfaltigkeitslehre (1892);  Beiträge zur Begründung der transfiniten Mengenlehre, I-II (1895-1897); the continuum hypothesis.

16. Frege. Burali-Forti’s and Russell’s paradoxes. The well-ordering principle and the axiom of choice  (Zermelo, 1904 and1908).  Axiomatising set theory.

17. Hilbert’s “program”. The Löwenheim-Skolem theorem. Brouwer’s intuitionism. Gödel’s incompleteness  theorems.  Turing and the thorny issue of computability.

18. Some open questions.

RECOMMENDED READING/BIBLIOGRAPHY

Most of the texts discussed in class and various other materials will be made available to students on the Aulaweb site.

The following works are also recommended:

- F. Acerbi, Il silenzio delle sirene. La matematica greca antica, Carocci, Roma 2010

- C. Bartocci, Una piramide di problemi. Storie di geometria da Gauss a Hilbert, Raffaello Cortina, Milano 2012.

- C. Bartocci & P. Odifreddi (a cura di). La matematica, vol.  I. I luoghi e i tempi, Einaudi, Torino 2007.

- J. W. Dauben, Georg Cantor. His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton 1990.

- J. Ferreirós, Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics, second rev. ed., Birkhäuser, Boston-Basel-Boston 2007.

- I. Grattan-Guinness, The Search for Mathematical Roots, 1870-1940, Princeton University Press 2000.

- G. Lolli, La guerra dei trent’anni (1900-1930). Da Hilbert a Gödel, Edizioni ETS, Pisa 2011.

 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

EXAMS

EXAM DESCRIPTION

The examination consists of the preparation and presentation of a "seminar" on a specific topic chosen by the candidate in agreement with the teacher (the latter will provide all the necessary study materials).

ASSESSMENT METHODS

Oral exam; exam grades are on a 18-30 scale.

FURTHER INFORMATION

For any further information please write an email to me at the address bartocci@dima.unige.it

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Agenda 2030 - Sustainable Development Goals
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