Learning objectives
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
Prerequisites
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Course unit content
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.
Full programme
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 to paradoxes
The suggestions of Logicism
Types theory
Russell and Peano
The prodroms of Principia Mathematica
Short analysis of Principia Mathematica
Other types
Non-logical axioms
Other foundational systems based on logicism
The theory NF
The theory ML
General aspects of constructivism
Constructive definitions and proofs
Problem relative to constructive real numbers
Intuitionism
Brouwer
Intuitionistic logic
Intuitionistic arithmetic
Some ideas about intuitionistic Analysis
Hilbert and Brouwer
Formalism
From the consistency of geometry towards consistency of analysis
The restarting of the programme of late Hilbert
Hilbert’s logical system
Propositional calculus
First order predicates calculus
Hilbert’s programme
The confirmation to Hilbert’s programme
Gödel and his work
The problem and the consequences of consistency
Completeness theorem
Incompleteness theorem
The process of ‘gödelisation’
Primitive recursive functions
Formal arithmetic
The incompleteness phenomenon
The second incompleteness theorem
The communication of incompleteness
Incompleteness consequences
Some other Gödel results
Continuum hypothesis and axiom of choice
Relativization
Construibilitiy
Beyond Gödel’s model
A contribution to proof theory
After Gödel’s theorems
A panorama of Foundation after 1931
The logical development of mathematics
Formal semantics
Model theory
Limitative theorems
General recursive functions
Church’s thesis
Turing’s machine
Nicolas Bourbaki
Direct proof of consistency
Gentzen
Proofs of Analysis consistency
Constructivisms
Markov’s algorithms
Bishop’s constructivism
Ultrafinitism
Effective computability
The ultrafinitism theses
Vopěnka and alternative mathematic
Sets in alternative theory
Classes and semi-sets
The finite and the infinity in alternative meaning
The category approach to foundations
From sets to morphisms
The ‘birth’ of categories
A first order axioms
The first categorical foundation of mathematics
The second categorical foundation of mathematics
Some topics of the contemporary Philosophy of mathematics
Empiricism in mathematics
Lakatos
Euler’s formula and Cauchy’s proof
Empiricism and Didactics
The role of proofs
Other philosophical proposal for mathematics
Frege once again
Neo-positivism and Wittgenstein
Quine’s epistemology
Problems of Platonism
Conclusion
Bibliography
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 information
Lecture notes available at web-site http://www.unipr.it/arpa/urdidmat/Fond09_10
