Corso di laurea in matematica

The teaching revisits basic mathematics and science topics, which could or could not be faced at upper secondary school, providing a different point of view based on an axiomatic approach and a broad perspective, which allow on the one side to strengthen and deepen knowledge of concepts, methods and theories already encountered and, on the other, to understand its significance and development, even by means of the relationships between mathematics and other sciences in the historical development of disciplines.

- Knowledge and understanding of the hypothetical-deductive method as a method of mathematical inquiry and discovery;

- knowledge and understanding of direct and indirect proof methods, as well as reductio ad absurdum and induction proofs;

- analysis of euclidean and non-euclidean geometries and arithmetic with natural numbers as axiomatic systems;

- application of the axiomatic method for the resolution of new problems and proof, especially in arithmetic and geometry;

- knowledge of scientific practices (genesis and development of concepts, methods and theories), from ancient civilizations to the 19th century;

- periodization and geographic location of contributions and results;

- critical ability of thinking about strengths and weaknesses of past scientific procedures, in comparison with current ones.

Proof in mathematics: Deductive reasoning; Direct proof and induction proof; RAA indirect proof and counter-noun proof. Euclid's axiomatic method: Euclid's axiomatic method; Modern axiomatic method. Hilbert's axioms for geometry (I): Axioms of Incidence; Theories and models; Axioms of betweenness; Axioms of congruence. Hilbert's axioms for geometry (II): Axioms of continuity; Axioms of parallelism; Geometric systems. Consequences of the axioms for geometry: Geometry of the circle; Geometry of the quadrilateral; Geometry of the triangle. Natural numbers according to Peano: Peano's axioms; Various formulation of the axiom of induction; Recursive definitions.

From abacus to the computer: The words of science, origins and evolution; Abachi and numerical systems; Rulers and tables; Calculators. Precision and Approximation Mathematics: Probability theory and calculus of variations. Measuring space and time in history: Astronomy and cosmology; Calendars.

Online interactive teaching through the use of given presentations and videos, as well as questions to simulate the final examination.

Three different possibilities exist, which are not equivalent to each other.

First possibility: those who choose the course for the Mathematics degree will be examined for a total of 6 credits (modules of geometry and arithmetic: 1 to 6), not for the 3 credits of Hystory of Sciences. The course will be valid for 6 credits (6 CFU).

First possibility: those who choose the course for the Strategic Sciences degree will be examined for a total of 6 credits (modules arithmetic and history of sciences: 1, 2, 6, 7, 8 and 9). The course will be valid for 6 credits (6 CFU).

Third possibility: the course could also be chosen for a total of 6 credits (3 + 3 at your choosing) by students from other degrees. 

It is possible to choose all the modules (1 to 9): in this case, the course will be valid for 9 credits (9 CFU). 

For each chosen possibility, the examination is only written and consists of a computer-based test. The examination takes place at the same time for all the students (independently of their choice).

Slides and didactic material of the modules that make the course, accessible on the start@unito platform subject to registration and authentication (link: http://start.unito.it/enrol/index.php?id=27). 

Greenberg, M.J. (1993). Euclidean and Non-Euclidean Geometries: Development and History (3rd Ed.). New York: W.H. Freeman and Company.