# Matematica | COMPLEMENTARY MATHEMATICS II

## Matematica COMPLEMENTARY MATHEMATICS II

 0512300031 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2020/2021

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2018 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/04 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
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.
Prerequisites
KNOWLEDGE OF THE BASIC NOTIONS OF ALGEBRA, GEOMETRY AND ANALYSIS.
Contents
1) CONSTRUCTIONS WITH RULER AND COMPASS: RULES OF ELEMENTARY CONSTRUCTIONS. THE DUALISM POINT - COMPLEX NUMBER. THE ALGEBRAIC CONDITIONS FOR THE CONSTRUCTION OF A POINT. THE CLASSIC PROBLEMS OF ANTIQUITY. NOTES ON THE CONSTRUCTION OF REGULAR POLYGONS. NOTES ON CONSTRUCTIONS WITH PAPER FOLDING.
2) NATURAL NUMBERS AND INDUCTION. DIDACTIC AND THEORETICAL ASPECTS. LABORATORY. THE PYTHAGOREAN SCHOOL AND ARTMOGEOMETRY. PERFECT NUMBERS. PYTHAGOREAN NUMBERS
3) INCOMMENSURABILITY: INCOMMENSURABILITY OF THE DIAGONAL-SIDE IN THE PENTAGON AND SQUARE; BARBEAU'S THEOREM. OTHER RECENT DISCOVERIES ON IMMEASURABILITY: ANGULAR COMMENSURABILITY AND NIVEN'S THEOREM. PLATO, IRRATIONALS, AND INFINITY. ZENO'S PARADOX.
4) ITERATIVE PROCESSES: THE EXHAUSTION METHOD, ARCHIMEDES: THE STUDY OF THE CIRCUMFERENCE. LABORATORY ON THE APPROXIMATION OF \$ \ PI \$. RECURSIVE SEQUENCES. FIBONACCI AND THE METHOD FOR DETERMINING THE APPROXIMATIONS OF THE ROOTS.
5) ARISTOTLE AND SYLLOGISMS. THE PAGNAN DIAGRAMS.
6) MISCONCEPTIONS IN THE CRITERIA OF CONGRUENCE BETWEEN POLYGONS.

7) HILBERT'S PROGRAM.
8) LANGUAGES AND FIRST ORDER THEORIES. FORMAL SYSTEMS AND AXIOMATIC METHOD.
9) GÖDEL COMPLETENESS THEOREM. COHERENCE, COMPLETENESS, CATEGORICITY AND INDEPENDENCE. REMARKS ON GÖDEL'S INCOMPLETENESS THEOREMS.

10) EXAMPLES OF AXIOMATIC METHODS. NON-EUCLIDEAN GEOMETRY. ORDERED FIELDS AND ARCHIMEDEAN ORDERED FIELDS. COMPLETE ORDERED FIELDS.

11) THE LANGUAGE OF SET THEORY. THE THEORY OF ZERMELO-FRANEKEL. ORDINALS. INTRODUCTION TO ORDINAL ARITHMETICS. CARDINALS. INTRODUCTION TO CARDINAL ARITHMETICS.
12) THE AXIOM OF CHOICE AND SOME REMARKABLE CONSEQUENCES. THE CONTINUUM HYPOTHESIS. INDEPENDENCE FROM OTHER AXISOMS.
Teaching Methods
THE COURSE INCLUDES A PREVALENT PART OF THEORETICAL LECTURES ON BASIC CONCEPTS OF THE COURSE (40 HOURS / 5CFU). A MINOR PART OF LABORATORY LESSONS TO SHOW HOW THE THEORY KNOWS COULD BE USED IN EDUCATIONAL CONTEXTS (8 HOURS / 1 CREDITS ARE FORESEEN FOR THIS PART). FOR EXAMPLE: LABORATORY ON COMMENSURABILITY, ON CONGRUENCE CRITERIA OF POLYGONS AND ON SYLLOGISMS, ON READING AND COMMENTING OF SCIENTIFIC ARTICLES.
Verification of learning
THE FINAL EXAMINATION AIMS TO ASSESS KNOWLEDGE AND UNDERSTANDING CAPABILITIES OF THE CONCEPTS PRESENTED DURING THE COURSE, AS WELL AS THE ACQUIRED COMPETENCES.
THE ASSESSMENT WILL BE CARRIED OUT BY MEANS OF AN ORAL EXAMINATION.
IN THE ORAL EXAM THE FOLLOWING WILL BE ASSESSED: KNOWLEDGE OF THE COURSE TOPICS, ABILITY TO EXPOSE THEM IN A CRITICAL MANNER AND TO PLACE THEM IN A HISTORICAL CONTEXT.
IN BOTH PARTS THE ACQUIRED GENERAL CROSS COMPETENCIES WILL BE EVALUATED.
THE FINAL MARK IS UP TO THIRTY. THE "LAUDE" MAY BE ATTRIBUTED TO STUDENTS SHOWING ABILITY TO APPLY THE ACQUIRED KNOWLEDGE AND COMPETENCIES IN CONTEXTS DIFFERENT FROM THOSE PROPOSED DURING THE LECTURES.
Texts
G. GERLA, DAGLI INSIEMI ALLA LOGICA MATEMATICA. TENTATIVI DI FONDARE LA MATEMATICA, VOLUME I AND II

IT IS ALSO SUGGESTED TO READ

LEONESI S., TOFFALORI C. (2007). MATEMATICA, MIRACOLI E PARADOSSI. STORIE DI CARDINALI DA CANTOR A GÖDEL. MONDADORI.