Foundations of Mathematics I

The course introduces some key issues in the reflection on the foundations of mathematics, with particular attention to the relationship between intuition, the construction of mathematical entities, axiomatization and formalization, and to the crises in the foundations of mathematics. The topics will be addressed through historical examples and theoretical investigations, from Greek mathematics to the nineteenth-century crisis of foundations and to the theoretical systematizations proposed in the first half of the twentieth century. The transition from classical theories to modern theories and the relationship between models and axiomatic systems will be examined in detail, with particular attention to the emergence of non-Euclidean geometries and to the problem of the continuum. More specifically, in the second part of the course the relationship between the Euclidean line and the complete ordered field of real numbers will be studied in depth, through the analysis of different constructions of R and the study of the isomorphism theorem for complete ordered fields.
Introduction to the problem of foundations and analysis of the crisis of incommensurables
· Introduction to the foundations of mathematics.
· Numbers and magnitudes: the concepts of number and magnitude in Greek mathematics, with reference to Eudoxus' theory of ratios and Archimedes' postulate. The crisis of incommensurables: the discovery of the irrationality of √2. An approach to the study of the properties of numbers from a theoretical point of view and their connections with constructions in Euclidean geometry. Rational and irrational, algebraic and transcendental numbers. Some elements of modern algebraic proofs of the impossibility of solving the classical problems of Antiquity with straightedge and compass.
· Euclidean geometry: analysis of the foundational aspects of Euclidean geometry as an axiomatic theory. Common notions and postulates. The role of constructions in Euclidean geometry and the nature of geometric entities in Euclid's Elements. Analysis of prototypical propositions and proofs from Books I and II of Euclid's Elements. Proofs by contradiction and limitations of Euclidean theory. Analysis of the problem of the fifth postulate: formulation, independence from the other postulates, equivalent formulations, and issues concerning the constructibility of a line parallel to any given line through a point external to it. Propositions requiring the use of the fifth postulate.
The crisis of foundations in the nineteenth century
· Gerolamo Saccheri's work and the attempts to prove the fifth postulate. The "nature" of the straight line and the construction of the first models of non-Euclidean geometries. Models and theories, and the nature of geometric entities. Axiomatic theories and formal theories.
· Construction and critical analysis of models of elliptic and hyperbolic geometry.
· Towards a modern and formal characterization of geometric entities: Hilbert's Foundations of Geometry.
· The axioms of continuity and the problematization of the relationship between the "number line" and the Euclidean line.
Real numbers
· What real numbers are and why they are needed to construct a model of the axioms of Hilbert's geometry.
· Hilbert's axioms as a categorical system.
· Geometries over a field. Ordered fields. Archimedean fields. Continuous fields.
· Isomorphism theorem for complete ordered fields.
· Construction of the real numbers by Dedekind cuts.
· Construction of the real numbers by Cauchy sequences of rational numbers.
· Every model of Hilbert's axiomatic theory of plane geometry is isomorphic to the Cartesian plane over the real numbers.