Matematica | EPISTEMOLOGY OF MATHEMATICS

cod. 0512300046

EPISTEMOLOGY OF MATHEMATICS

	0512300046
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2021/2022

PERCORSO SHARED STUDY PATHWAY Cohort 2019/2020

	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/01	6	48	LESSONS	OPTIONAL SUBJECTS

	Objectives
	THE AIM OF THE COURSE IS TO PROVIDE KNOWLEDGE OF FOUNDATIONAL ASPECTS OF MATHEMATICS IN THEIR HISTORICAL DEVELOPMENT. KNOWLEDGE AND UNDERSTANDING: THE COURSE AIMS AT SUPPLYING STUDENTS WITH SOME BASIC MATHEMATICAL KNOWLEDGE, BY SETTING IT IN THE HISTORICAL CONTEXT OF ITS ORIGIN AND DEVELOPMENT. IN PARTICULAR, IT AIMS AT PROVIDING AN UNDERSTANDING OF THE FOUNDATIONAL ASPECTS OF MATHEMATICS, BY FOCUSING ON THE KEY MOMENTS OF THE MATHEMATICAL THINKING DEVELOPMENT. APPLYING KNOWLEDGE AND UNDERSTANDING: THE COURSE AIMS AT ILLUSTRATING IN A CRITICAL WAY THE ORIGIN AND THE DEVELOPMENT OF SOME BASIC MATHEMATICAL CONCEPTS, SUCH AS THE CONCEPTS OF SET, AXIOMATIC METHOD, INFINITE, ALLOWING THE STUDENTS TO UNDERSTAND THESE CONCEPTS FROM A HIGHER POINT OF VIEW AND MAKING THEM ABLE TO EXPRESS FORMALLY PROBLEMS WITH DIFFERENT LEVELS OF DIFFICULTY, IN ORDER TO BENEFIT FOR THE RESOLUTION.

THE AIM OF THE COURSE IS TO PROVIDE KNOWLEDGE OF FOUNDATIONAL ASPECTS OF MATHEMATICS IN THEIR HISTORICAL DEVELOPMENT.

KNOWLEDGE AND UNDERSTANDING:
THE COURSE AIMS AT SUPPLYING STUDENTS WITH SOME BASIC MATHEMATICAL KNOWLEDGE, BY SETTING IT IN THE HISTORICAL CONTEXT OF ITS ORIGIN AND DEVELOPMENT. IN PARTICULAR, IT AIMS AT PROVIDING AN UNDERSTANDING OF THE FOUNDATIONAL ASPECTS OF MATHEMATICS, BY FOCUSING ON THE KEY MOMENTS OF THE MATHEMATICAL THINKING DEVELOPMENT.

APPLYING KNOWLEDGE AND UNDERSTANDING: THE COURSE AIMS AT ILLUSTRATING IN A CRITICAL WAY THE ORIGIN AND THE DEVELOPMENT OF SOME BASIC MATHEMATICAL CONCEPTS, SUCH AS THE CONCEPTS OF SET, AXIOMATIC METHOD, INFINITE, ALLOWING THE STUDENTS TO UNDERSTAND THESE CONCEPTS FROM A HIGHER POINT OF VIEW AND MAKING THEM ABLE TO EXPRESS FORMALLY PROBLEMS WITH DIFFERENT LEVELS OF DIFFICULTY, IN ORDER TO BENEFIT FOR THE RESOLUTION.

	Prerequisites
	KNOWLEDGE OF THE BASIC NOTIONS OF ALGEBRA, GEOMETRY AND ANALYSIS.

	Contents
	FORMAL SYSTEMS AND THE AXIOMATIC METHOD. EXAMPLES OF AXIOMATIC METHODS. NON-EUCLIDICAL GEOMETRIES. COHERENCE, COMPLETENESS, CATEGORICITY AND INDEPENDENCE OF A FORMAL SYSTEM. THE LANGUAGE OF SET THEORY. THE THEORY OF ZERMELO-FRANEKEL. ORDINAL. ORDINAL ARITHMETICS. CARDINALS. CARDINAL ARITHMETICS. AXIOM OF CHOICE AND SOME OF ITS IMPORTANT CONSEQUENCES. HILBERT'S PROGRAM. GÖDEL'S INCOMPLETENESS THEOREMS. CONTINUOUS HYPOTHESIS. INDEPENDENCE FROM OTHER AXIOMS.

Contents

FORMAL SYSTEMS AND THE AXIOMATIC METHOD.
EXAMPLES OF AXIOMATIC METHODS. NON-EUCLIDICAL GEOMETRIES.
COHERENCE, COMPLETENESS, CATEGORICITY AND INDEPENDENCE OF A FORMAL SYSTEM.
THE LANGUAGE OF SET THEORY. THE THEORY OF ZERMELO-FRANEKEL.
ORDINAL. ORDINAL ARITHMETICS.
CARDINALS. CARDINAL ARITHMETICS.
AXIOM OF CHOICE AND SOME OF ITS IMPORTANT CONSEQUENCES.
HILBERT'S PROGRAM.
GÖDEL'S INCOMPLETENESS THEOREMS.
CONTINUOUS HYPOTHESIS. INDEPENDENCE FROM OTHER AXIOMS.

	Teaching Methods
	THE COURSE INCLUDES A PREVALENT PART OF THEORETICAL LECTURES ON BASIC CONCEPTS.

	Verification of learning
	THE EXAM AIMS TO EVALUATE THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS INTRODUCED DURING THE LECTURES, AS WELL AS THE ACCURACY AND INDEPENDENCE IN USING SUCH TOOLS. THE EXAMINATION CONSISTS OF AN ORAL INTERVIEW (ABOUT 45 MINUTES) WHERE IT WILL BE EVALUATED THE KNOWLEDGE ACQUIRED ON BASIC AND MOST ADVANCED CONCEPTS IN PROPOSITIONAL LOGIC, FIRST-ORDER LOGIC AND BOOLEAN ALGEBRAS. STUDENTS MUST FIRST DEMONSTRATE THAT THEY KNOW THE CONCEPTS (DEFINITIONS) COVERED DURING THE COURSE AND THAT THEY UNDERSTOOD THEM, SHOWING THAT THEY CAN INDEPENDENTLY BUILD EXAMPLES. LATER THE QUESTIONS WILL BE AIMED AT UNDERSTANDING IF STUDENTS KNOW HOW TO USE THOSE CONCEPTS AND DEFINITIONS AND KNOW THE FUNDAMENTAL PROPERTIES SEEN DURING THE COURSE (THEOREMS). ONLY IF BOTH THE PREVIOUS PARTS ARE SUCCESSFULLY OVERCOME THE REASONS WHY THESE PROPERTIES HOLDS WILL BE DISCUSSED (DEMONSTRATIONS). THE FINAL MARK IS UP TO THIRTY. LAUDE WILL BE ATTRIBUTED TO STUDENTS THAT PROVE THEMSELVES TO BE ABLE TO INDEPENDENTLY USE THE KNOWLEDGE AND SKILLS ACQUIRED ON THE MOST ADVANCED TOPICS OF THE COURSE, AND ARE ABLE TO FIND CONNECTIONS WITH CONTEXTS DIFFERENT THAN THOSE PRESENTED IN THE LECTURES.

Verification of learning

THE EXAM AIMS TO EVALUATE THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS INTRODUCED DURING THE LECTURES, AS WELL AS THE ACCURACY AND INDEPENDENCE IN USING SUCH TOOLS.

THE EXAMINATION CONSISTS OF AN ORAL INTERVIEW (ABOUT 45 MINUTES) WHERE IT WILL BE EVALUATED THE KNOWLEDGE ACQUIRED ON BASIC AND MOST ADVANCED CONCEPTS IN PROPOSITIONAL LOGIC, FIRST-ORDER LOGIC AND BOOLEAN ALGEBRAS. STUDENTS MUST FIRST DEMONSTRATE THAT THEY KNOW THE CONCEPTS (DEFINITIONS) COVERED DURING THE COURSE AND THAT THEY UNDERSTOOD THEM, SHOWING THAT THEY CAN INDEPENDENTLY BUILD EXAMPLES. LATER THE QUESTIONS WILL BE AIMED AT UNDERSTANDING IF STUDENTS KNOW HOW TO USE THOSE CONCEPTS AND DEFINITIONS AND KNOW THE FUNDAMENTAL PROPERTIES SEEN DURING THE COURSE (THEOREMS). ONLY IF BOTH THE PREVIOUS PARTS ARE SUCCESSFULLY OVERCOME THE REASONS WHY THESE PROPERTIES HOLDS WILL BE DISCUSSED (DEMONSTRATIONS).

THE FINAL MARK IS UP TO THIRTY. LAUDE WILL BE ATTRIBUTED TO STUDENTS THAT PROVE THEMSELVES TO BE ABLE TO INDEPENDENTLY USE THE KNOWLEDGE AND SKILLS ACQUIRED ON THE MOST ADVANCED TOPICS OF THE COURSE, AND ARE ABLE TO FIND CONNECTIONS WITH CONTEXTS DIFFERENT THAN THOSE PRESENTED IN THE LECTURES.

	Texts
	M. BORGA, D. PALLADINO, OLTRE IL MITO DELLA CRISI: FONDAMENTI E FILOSOFIA DELLA MATEMATICA DEL XX SECOLO. EDITRICE LA SCUOLA. G. LOLLI, TAVOLI, SEDIE E BOCCALI DI BIRRA: DAVID HILBERT E LA MATEMATICA DEL NOVECENTO. RAFFAELLO CORTINA EDITORE. S. LEONESI, C. TOFFALORI. MATEMATICA, MIRACOLI E PARADOSSI. STORIE DI CARDINALI DA CANTOR A GÖDEL (2007) MONDADORI. G. GERLA, DAGLI INSIEMI ALLA LOGICA MATEMATICA. TENTATIVI DI FONDARE LA MATEMATICA, VOLUME I E II. ILMIOLIBRO.IT.

Texts

M. BORGA, D. PALLADINO, OLTRE IL MITO DELLA CRISI: FONDAMENTI E FILOSOFIA DELLA MATEMATICA DEL XX SECOLO. EDITRICE LA SCUOLA.
G. LOLLI, TAVOLI, SEDIE E BOCCALI DI BIRRA: DAVID HILBERT E LA MATEMATICA DEL NOVECENTO. RAFFAELLO CORTINA EDITORE.
S. LEONESI, C. TOFFALORI. MATEMATICA, MIRACOLI E PARADOSSI. STORIE DI CARDINALI DA CANTOR A GÖDEL (2007) MONDADORI.
G. GERLA, DAGLI INSIEMI ALLA LOGICA MATEMATICA. TENTATIVI DI FONDARE LA MATEMATICA, VOLUME I E II. ILMIOLIBRO.IT.