Foundations of Mathematics I
A.Y. 2024/2025
Learning objectives
Understand some important "crises" of intuition in the development of mathematical thought and the corresponding theoretical resolutions: the discovery of incommensurable magnitudes; the concept of infinity; the crisis of foundations and the axiomatizations of the late 19th century (geometries, continuum, set theory).
Critically analyze the main developments and the mathematical and philosophical implications of these topics through key examples from the history of mathematics, examining the responses provided by mathematicians to these crises.
Critically analyze the main developments and the mathematical and philosophical implications of these topics through key examples from the history of mathematics, examining the responses provided by mathematicians to these crises.
Expected learning outcomes
Students must be able to comprehensively present their knowledge, demonstrating critical ability in analyzing foundational issues, both in concrete examples and at a transversal level. Additionally, they must acquire communication skills, arguing their choices and presenting their knowledge with a good balance between precision in language and clarity in exposition.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Introduction to the Problem of Foundations and Analysis of the Crisis of Incommensurables
● Introduction to the foundations of mathematics.
● Historical context: mathematics in ancient Greece.
● The concept of number and magnitude among the Greeks.
● The crisis of incommensurables: the discovery of the irrationality of √2
● Details on the discovery of incommensurables.
● Philosophical and mathematical consequences of the discovery.
● Ancient methods and approaches to dealing with incommensurable magnitudes.
Infinity in Mathematics
● The concept of infinity: from philosophy to mathematics.
● Potential infinity vs. actual infinity.
● Infinity in geometry, set theory, and analysis.
The Crisis of Foundations in the 19th Century
● Mathematics in the 19th century: a period of great changes.
● The problem of formalizing the intuition of the continuum and the "number line".
● The fifth postulate and non-Euclidean geometries.
● Cantor and naive set theory: introduction and early developments.
● The crisis of foundations: problems of rigor and consistency.
Responses to the Crisis of Foundations
● Hilbert and the formalization program.
● Klein and the Erlangen program.
● Set theories and axiomatizations.
Philosophical and Mathematical Implications
● Discussion of some philosophical debates regarding the foundations of mathematics related to intuition, axiomatization, and proof. Brouwer's intuitionism. Finitism and constructivism.
● The consequences of the crisis of foundations for modern mathematics.
● Introduction to the foundations of mathematics.
● Historical context: mathematics in ancient Greece.
● The concept of number and magnitude among the Greeks.
● The crisis of incommensurables: the discovery of the irrationality of √2
● Details on the discovery of incommensurables.
● Philosophical and mathematical consequences of the discovery.
● Ancient methods and approaches to dealing with incommensurable magnitudes.
Infinity in Mathematics
● The concept of infinity: from philosophy to mathematics.
● Potential infinity vs. actual infinity.
● Infinity in geometry, set theory, and analysis.
The Crisis of Foundations in the 19th Century
● Mathematics in the 19th century: a period of great changes.
● The problem of formalizing the intuition of the continuum and the "number line".
● The fifth postulate and non-Euclidean geometries.
● Cantor and naive set theory: introduction and early developments.
● The crisis of foundations: problems of rigor and consistency.
Responses to the Crisis of Foundations
● Hilbert and the formalization program.
● Klein and the Erlangen program.
● Set theories and axiomatizations.
Philosophical and Mathematical Implications
● Discussion of some philosophical debates regarding the foundations of mathematics related to intuition, axiomatization, and proof. Brouwer's intuitionism. Finitism and constructivism.
● The consequences of the crisis of foundations for modern mathematics.
Prerequisites for admission
The prerequisites for the course will be the knowledge acquired in the basic courses of the first year of the Bachelor's degree in Mathematics.
Teaching methods
● Lectures and interactive sessions
● Analysis of historical and modern texts and articles on the topics covered and discussions of significant case studies
● Possible group work
● Analysis of historical and modern texts and articles on the topics covered and discussions of significant case studies
● Possible group work
Teaching Resources
● Boyer, C. B. (1989). A History of Mathematics. John Wiley & Sons.
● Cantor, G. (1955). Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications.
● Dauben, J. W. (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press.
● Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.
● van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.
● Cantor, G. (1955). Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications.
● Dauben, J. W. (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press.
● Ewald, W. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.
● van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.
Assessment methods and Criteria
● Presentation of a seminar
or
● Traditional oral exam
Students must be able to comprehensively present their knowledge, demonstrating critical ability in analyzing foundational issues, both in the case of concrete examples and at a transversal level. Their knowledge of the problems will be assessed, as well as their ability to connect specific aspects of the individual case under examination with broader foundational issues. Students must show awareness of the implications of the problems encountered in concrete cases on a broader foundational perspective and the impact of a new theoretical arrangement on the formulation of problems within different theories. Additionally, students must demonstrate communication skills by arguing their choices and presenting their knowledge with a good balance between precision in language and clarity in exposition.
or
● Traditional oral exam
Students must be able to comprehensively present their knowledge, demonstrating critical ability in analyzing foundational issues, both in the case of concrete examples and at a transversal level. Their knowledge of the problems will be assessed, as well as their ability to connect specific aspects of the individual case under examination with broader foundational issues. Students must show awareness of the implications of the problems encountered in concrete cases on a broader foundational perspective and the impact of a new theoretical arrangement on the formulation of problems within different theories. Additionally, students must demonstrate communication skills by arguing their choices and presenting their knowledge with a good balance between precision in language and clarity in exposition.
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 6
Lessons: 42 hours
Professors:
Branchetti Laura, Mantovani Sandra
Shifts:
Professor(s)
Reception:
By appointment
Online, Microsoft Teams
Reception:
Thursday 12.45-14.15, by appointment
Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50