# 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.

## Course contents

Concise history of Geometry

Foundations begin from Geometry

The Mediterranean Antiquity

Egyptian Geometry

Mesopotamian Geometry

Geometry in non-Mediterranean culture

Chinese mathematics

Indian mathematics

Greek geometry before Euclid

The geometers’ roll

Theorems

Euclid and his works

Euclid

The Elements

Relationship between Euclid and Aristotle

Aristotle’s Deductive Science

The success of the Elements and of the Deductive Science

Geometry as a set of statements

Evidence postulate for terms

Evidence postulate for statements

Equality in Aristotle and in Euclid

The word ‘equal’ in the Elements

Equality as superposition

Homogeneity

Equality of ratios

The role of definitions in Aristotle and Euclid

Thence, Euclid

Greek geometry after Euclid

Archimedes

Apollonius

Late Hellenism

Conclusion

Deductive tools

The Greek legacy

A historical survey of Greek logic

Euclid’s logical tools

Aristotle’s contribution to logic

Terms

Figures

Quantity and quality

The square of propositions and their truth table

Rules for syllogisms

Terms universally present

Rules about terms and propositions

The modes of syllogism

First figure modes

Second figure modes

Third figure modes

Fourth figure modes

Transformations of syllogisms

The presentation of Summulae logicales

Syllogisms which conclusion is particular

Castillon

Extensional features of syllogism

Leibnizs ’mathematical version- Euler’s diagrams

Extensional analysis of some syllogism

Barbara – Barbari

Cesare – Cesaro

Bocardo

Fresison

Propositional calculus

Greek ancient propositional calculus

From Middle-Ages on

Truth, validity and provability – The case of Geometry

The Euclid’s flaws

The syntax presentation of non-Euclidean Geometries

True, Valid, Theorem

Boole’s contribution to logic

Before Boole

Algebra in United Kingdom

English logic before Boole

The logic works of De Morgan

The Boole’s work

The influence of De Morgan and William Hamilton debate

General aspects of Boole’s logic

Boole’s calculus

A short comparison of Boole’s and De Morgan’s work

Boole’s logic after Boole

The Boole’s flaws

Jevons

Venn and the others

Peirce and Schröder

Set introduction and logicism approach

Cantor and the ‘birth’ of sets

How is changed the scenario

Cantor’s contribution

The infinity

An abstract notion of set

Cardinal numbers and their properties

Three problems with cardinal numbers

Ordinal numbers and their properties

Implicit axioms of set theory

Frege

Frege’s work

Frege’s aims

The controversy against Empiricism

The controversy against Psychologism

Frege and Kant

The controversy against formalism

Other controversies

Frege’s logical calculus

The language

Axioms and rules

Extension and intension

The problems of Foundations

The axiomatic method

Hilbert and his Grundlagen

The main ideas of the Grundlagen

The axiomatic setting of the Grundlagen

Some remarks about the axioms

Development of the Grundlagen

Consistency

Axioms as definitions

A period of crisis

Antinomies

Russell’s paradox

Russell’s letter

The role of membership relation

The ‘reflexive’ cases of membership relation

Russell and the barbers

Frege’s reaction

Other antinomies

Berry’s paradox

Richard’s paradox

Zermelo-König’s paradox

Paradoxes analyses

Vicious circle and the self-reference

“Exemplo de Richard non pertine” – Ramsey

The neo-Cantorian solution

From Cantor to Zermelo

(A-posteriori) justifications of the Zermelo’s axiomatic system

The naïve set theory

The set-theoretical operations

How to choose?

Axiomatic of operations

Isolation axiom schema

Relationship between isolation axiom and other axioms

Infinity axiom

The original Zermelo’s proposition

The general inspiring principles

Fundamental definitions and the first two axioms

The isolation axiom

Other two axioms

Product and choice axiom

Infinity axiom

Zermelo’s axiomatic system development

The system ZFS

Criticism of isolation axiom

Replacement axiom

Foundation axiom

Theories with classes

The theory NBG

Comparison of ZF and NBG

The finite number of axioms for NBG

The theory MKM

Other solution

## Recommended readings

Borga, M., Paladino, 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

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

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

## Assessment methods and criteria

Oral examination

## Other informations

Lecture notes available at web-site http://www.unipr.it/arpa/urdidmat/Fond09_10