- Docente: Monica Idà
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)
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from Sep 21, 2022 to Dec 23, 2022
Learning outcomes
Notions about foundations of mathematics with particular regard to geometry in their classical and modern development. Specific knowledge for math teaching.
Course contents
The first part of the course consists in the first notions of projective geometry and the study of plane affine and projective algebraic curves, with particular regard to their singularities.
The second part of the course concerns the axiomatic methods in mathematics and in particular the Foundations of Geometry.
We start with the Definitions, the Postulates and the Common Notions in Euclid's Elements, pointing out some problems in Euclid axiomatization: intersection of two circles, of a line and a circle, the superposition method, points lying between two points in a line, lines lying between two lines in a pencil. Playfair axiom and the fifth postulate.
Hilbert plane geometry: the 5 groups of axioms.
Independence, consistency, relative consistency, categoricity and completeness of a axiom system.
The cartesian plane P_F over a field F. Ordered, Pythagorean, Archimedean, Euclidean fields and corresponding properties of P_F. The constructible field and the Hilbert field.
Hilbert axioms are a categorical system and the unique model is the real Cartesian plane.The consistency of Hilbert geometry follows by the consistency of the real numbers, which in turns follows by the consistency of the natural numbers: Hilbert's second Problem.
Logicism: G.Peano e G.Frege, Bertrand Russel and Alfred North Whitehead. Intuitionism: Kronecker, Henri Poincarè, Brouwer. Formalism: David Hilbert. Gödel's incompleteness theorems.Readings/Bibliography
E.Sernesi: "Geometria 1", Bollati Boringhieri, Torino 1989
http://www.dm.unibo.it/matematica/GeometriaProiettiva/hompg/hompg.htm
Gli Elementi di Euclide. A cura di Attilio Fraiese e Lamberto Maccioni. Unione Tipografico - Editrice Torinese 1970.
David Hilbert, Fondamenti della Geometria. Feltrinelli 1970
Robin Hartshorne, Geometry: Euclid and Beyond. Springer 2000
Teaching methods
The course consists of 6 CFU, 5 of front lectures and 1CFU, corresponding to 12 hours, of exercise sessions.
Assessment methods
Exam consisting of a written test and an oral test.
Students are admitted to the oral exam only if the written exam is sufficient (i.e.18/30). Both tests must be held in the same exam session.
The written exam consists of some exercises on the first part of the course.
The final mark for the course Elementi di Algebra e Geometria da un punto di vista superiore is the average of the marks in Elementi di Algebra and in Elementi di Geometria.
Teaching tools
Files with exercises will be posted on Virtuale
Office hours
See the website of Monica Idà