Elementary Mathematics from an Advanced Standpoint 1

A.Y. 2025/2026
6
Max ECTS
42
Overall hours
SSD
MAT/04
Language
Italian
Learning objectives
The aim of this course is to provide an introduction to the axiomatic Zermelo-Fraenkel Set Theory. The notions of finite and infinite sets, natural numbers, ordinals and cardinals will be given and studied, together with the related arithmetics. Furthermore, various equivalent forms of The Axiom of Choice will be given, highlighting the importance of such an axiom from both a foundational and a practical point of view.
Expected learning outcomes
Acquisition of awareness of the need for a formal, rigorous and axiomatic theory of sets, in contrast to the naive set theory, usually taken as a basis for mathematics. Critical capacity to use axioms and comprehension of the role of paradoxes.
Single course

This course can be attended as a single course.

Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
Course syllabus
The course aims to provide a critical re-reading of elementary mathematical topics, analyzed with the methods typical of the Elementary Mathematics from an Higher Standpoint tradition.
The objective is to promote the understanding of some fundamental concepts by examining their historical development, fostering the ability to connect different languages and theories, and providing critical tools.

The topics covered in the course will include:

1. Sets and structural algebra: operations, order relations, equivalence classes.
2. Algebra and language: from al-Khwarizmi's Treatise to the geometric approach; notions of proof and generalization.
3. The Pythagorean theorem: generalizations from the Pythagoreans to Euclid and to the Arabic world.
4. From tables to modern formulas and the concept of function: examples of Ptolemy's theorem and logarithms.
5. From computation to theory: the Indian tradition and later contributions.
6. A multicultural mathematics: interactions between traditions (Greek, Arabic, Indian, modern).
7. Contemporary perspectives: Villani (synthetic/analytic, curvature); Grothendieck (construction of objects, intersection between theories).
Prerequisites for admission
Basic knowledge of Mathematical Analysis 1, Geometry 1 e Algebra 1.
Teaching methods
Lectures with historical and applied examples; guided reading and discussion of classical texts; exercises; individual in-depth study.
Teaching Resources
Carruccio, E. Matematiche elementari da un punto di vista superiore, Bollati Boringhieri.
D'Amore B. et al, Numeri, La Scuola.

Ulteriori testi e articoli segnalati a lezione.
Assessment methods and Criteria
Oral exam aimed at assessing:

mastery of theoretical contents;
ability to compare different versions of mathematical objects and theories from a higher standpoint;
clarity of exposition and correct use of mathematical language.
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 6
Lessons: 42 hours
Professor(s)
Reception:
By appointment
Online, Microsoft Teams