# Matematica | COMPLEMENTARY MATHEMATICS II

## Matematica COMPLEMENTARY MATHEMATICS II

 0512300031 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2013/2014

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/04 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 GIANGIACOMO GERLA T
Objectives
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.
MAKING JUDGEMENTS: THE COURSE INTENDS TO MAKE STUDENTS BECOME CRITICAL AND INDEPENDENT ABOUT BASIC ISSUES ON WHICH MATHEMATICS IS GROUNDED.
IT AIMS AT MAKING STUDENTS TO ACQUIRE THE ABILITY TO PERCEIVE MATHEMATICS NOT AS A SEPARATE AND DEFINITIVELY ESTABLISHED BODY, BUT AS ONE OF THE FUNDAMENTAL ELEMENTS OF THE CULTURE OF DIFFERENT PERIODS, AND CONSEQUENTLY SUBJECT TO GROWTH AND TO INTERACTION WITH OTHER CULTURAL AREAS. IN THIS WAY STUDENTS WILL BE ABLE TO UNDERSTAND MATHEMATICAL MODELS RELATED TO CONCRETE SITUATIONS ARISING ALSO FROM OTHER DISCIPLINES AND THEY WILL BE ABLE TO USE THESE MODELS ALSO IN OTHER SITUATIONS.
COMMUNICATION SKILLS: THE COURSE AIMS AT PROVIDING STUDENTS WITH BOTH MATHEMATICAL AND LINGUISTIC TOOLS USEFUL TO MAKE THEM ABLE TO COMMUNICATE, BOTH TO EXPERTS AND NON-EXPERTS, PROBLEMS, IDEAS AND SOLUTIONS REGARDING MATHEMATICS AND ABLE TO CLEARLY AND RIGOROUSLY EXPLAIN THE ACQUIRED KNOWLEDGE.
LEARNING SKILLS: DURING THE COURSE, THE REASONS THAT LED TO THE BIRTH AND TO THE DEVELOPMENT OF SOME FUNDAMENTAL MATHEMATICAL CONCEPTS ARE HIGHLIGHTED. THIS FOSTERS IN STUDENTS THE DEVELOPMENT OF A FLEXIBLE AND ANALYTICAL MINDSET ALLOWING THEM TO IDENTIFY AUTONOMOUSLY WHICH KIND OF KNOWLEDGE HAS TO BE EXAMINED IN DEPTH AND HAS TO BE ACQUIRED FOR THE PROBLEM MANAGEMENT BOTH IN MATHEMATICAL CONTEXT AND IN OTHER CONTEXTS SUCH AS THE BUSINESS ONES.
Prerequisites
KNOWLEDGE OF THE BASIC NOTIONS IN ALGEBRA, GEOMETRY, ANALYSIS.
Contents
COMPARISON BETWEEN INFINITE SETS. TRANSFINITE ARITHMETIC. CRISIS OF THE SET THEORY. ANTINOMIES. THE AXIOMATIC METHOD. THE FOUNDATIONAL POINT OF VIEW AND THE STRUCTURALIST ONE. HILBERT’S PROGRAMM. THE PROPOSITIONAL CALCULUS. THE PREDICATE CALCULUS. THE HILBERT’S PROGRAM FAILURE. GÖDEL THEOREMS. INTUITIONISM. CATEGORIES’ THEORY. PROBABILTY THEORY AND ITS PARADOXES.
Teaching Methods
FRONT LECTURES AND LABORATORY ACTIVITIES.
Verification of learning
THE ASSESSMENT WILL BE CARRIED OUT BY MEANS OF AN ORAL EXHAMINATION. DURING THE EXAMINATION, KNOWLEDGE OF THE MATHEMATICAL CONTENT OF THE ARGUMENTS, CAPABILITY TO EXPOSE THEM IN A CRITICAL MANNER AND TO CONTEXTUALIZE THEM IN THE HISTORICAL DEVELOPMENT WILL BE EVALUATED.
Texts