# FOUNDATIONS OF MATHEMATICS

## Learning outcomes of the course unit

Knowledge and understanding. History of mathematics, epistemology and philosophy of mathematics together give important contributions to education and to student’s culture.

The course will contribute to training and education in epistemology and history of mathematics by the knowledge of the main problems 19th century mathematics and the foundations crisis through 20th century. The course, by the means of seminars for studying in depth, will prepare students to elaboration and application of their original ideas with a steady comparison with documents produced by research of the field.

Applying knowledge and understanding. Students will be required to find solution of problems involving different mathematical contexts, making also reference to the teaching. Moreover they will become able to choose the suitable frameworks in which to insert the topics in order to devise and to conduct personal argumentation regarding the course subjects.

Making judgement. Students are called to integrate their knowledge, to manage the complexity and to elaborate judgements considering adequately the historic, epistemological and contents parameters

Communication skills. Students will be aware of communicating their conclusion and their knowledge, explaining also the rationale of their choice to conversation partners with the same knowledge and also to non-qualified ones.

Learning skills. Students have to acquire the skill of learning advanced topics, by the means of an autonomous research of other texts making a deepening of the topics treated during the lectures

## Course contents summary

Concise history of Geometry up to Hellenism. The Syllogism and its evolution.

Non Euclidean Geometries

Concise history of Algebra up to the 19th century. The Analytical Society and its role. Logics before the 19th century. Boole, De Morgan and the algebraic logic; Peirce, Dedekind, Schröder.

From Algebra of Logics to algebraic Logic. Frege, Russell’s paradox and its outcomes. Introduction to first order Logic.

Reciprocal influences of Philosophy and Mathematics from Greek time to the modern times. The problem of foundations for Mathematics from 1875 to 1931. The problem of consistency and Gödel’s attainments.

Some aspects of mathematical language.

Epistemology of Mathematics after Gödel. Category Foundations, Alternative Mathematics. The philosophical meaning of Mathematics and its teaching. The reductionism.

## Recommended readings

Borga, M., Palladino, D. (1997). Oltre il mito della crisi –Fondamenti della Matematica nel XX secolo (1997) Brescia: Editrice La Scuola.

Mangione, C., Bozzi S. (1993). Storia della Logica – Da Boole ai nostri giorni. Milano: Garzanti.

Speranza, F. (1997). Scritti di Epistemologia della Matematica, Bologna: Pitagora Editrice.

Bagni, G.T. (2006). Linguaggio, Storia e Didattica della Matematica, Bologna: Pitagora Editrice.

Bagni, G.T. Elementi di Storia della Logica Formale. Bologna: Pitagora Editrice.

Marchini, C. Appunti delle Lezioni di Fondamenti di Matematica A.A. 2009/2010

## Teaching methods

Lectures will be mainly in transmissive style, but with a steady dialogue with students which can be called to the blackboard for discussing problems, or for showing their understanding of and taking part to the course. Student will be asked to take part to seminar for studying in depth some course topics.

## Assessment methods and criteria

Oral examination.

Assessment

Assessment will be made by a final oral, in which student must solve mathematical or interpretative problems.