COMPLEMENTARY MATHEMATICS I

The Babylonian and Egyptian mathematics.
The Greek mathematics: Thales, Pythagoras and his school, the crisis of incommensurables. Zeno’s paradoxes.
The three famous problems of Greek antiquity: quadrature of the circle, duplication of the cube, trisection of angle. Hippocrates and the quadrature of lunula.
Plato: arithmetic and geometry, the platonics polyedra.
Aristotle: the structure of deductive science, syllogisms. Induction principle.
Euclide: the “Elements”, commons notions, postulates and axioms, theory of Paralleles, proportions theory, magnitudes, prime numbers, equivalence in plane and space. Euclide’s work in modern epistemology.
Archimedes: from the measure of circle to the volume of sphere, the method of exhaustion.
Apollonius: conic sections.
Numerics systems: natural, integer, rational, real, complex numbers. The fundamental theorem of Algebra.
Numbers e,
Non-Euclidean geometries: hystorical and epistemological aspects, Poincaré’s and Klein’s models.
The Erlangen program and the transformations geometry: congruence, similarity, affinity, projectivity.
The geometrical transformations in Escher’s works.
The geometrical transformations in the space.
The problem of foundations of Geometry: the Hilbert’s axioms, indipendence, coherence, completeness.