34734 - Elementary Geometry from an Advanced Standpoint

Academic Year 2017/2018

  • 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. 8208)

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. An ordered Archimedean field is a subfield of R; an ordered Archimedean complete field is R.

Desargues and Pappo-Pascal theorem.

Hilbert axioms are a categorical system and the unique model is the real Cartesian plane.

An example of a non-Archimedean field: R(t).

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.

Cantor set theory. The problem of the Foundations of arithmetics and set theory at the end of 1800. The most famous paradoxes.

The axiom of Choice, Zermelo's Theorem, Zorn's Lemma.

Logicism: G.Peano e G.Frege, Bertrand Russel and Alfred North Whitehead. Intuitionism: Kronecker, Henri Poincarè, Brouwer. Formalism: David Hilbert.

Gödel's incompleteness theorems.

Examination of some textbooks for the geometry taught in the first two years of high school, with particular attention to the initial choice of axioms.

 

Readings/Bibliography

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

For the first part of the course on plane curves:

E.Sernesi: "Geometria 1", Bollati Boringhieri, Torino 1989

http://www.dm.unibo.it/matematica/GeometriaProiettiva/hompg/hompg.htm

 

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.

Teaching tools

Files with exercises will be posted on the teacher's website.

At

[http://progettomatematica.dm.unibo.it/indiceGenerale5.html]

there are notes on argument of the course and interactive exercises.

Office hours

See the website of Monica Idà