## Learning objectives

Knowledge and ability to understand: through the lectures held during the course, the student will acquire the methods and knowledge necessary to analyze a mathematical theory from a historical and foundational point of view, to contextualize it, to compare it with other theories of other historical periods or with different objectives and problems. In particular, he will learn some methods to deal with problems of a practical and theoretical nature, ancient and modern axiomatic constructions, different roles of mathematical formalization both aimed at generalizing the results to other classes of problems, or other areas of mathematics, and aimed at transforming demonstration processes in increasingly stringent and codified procedures. The student will learn the structure of some great works of organic arrangement of knowledge in a unitary corpus and founded on principles and methods (e.g. Elements of Euclid, as well as heuristic methods that anticipate theorization, the dynamics of crisis and revolution that have made mathematics evolve over the centuries, the relationship between ancient and modern mathematics.

Ability to apply knowledge and understanding:

Through the classroom exercises related to some topics of the program, students learn how to apply the knowledge acquired in problems, constructions and proofs of ancient and modern works. In particular, the student will be asked to: apply the methods presented to solve problems with ancient methods, even though they know faster solutions based on modern techniques; carry out rigorous constructions and proofs, specifying the criteria and principles adopted as valid; build and validate models for a simple axiomatic system; apply new concepts in problems similar to those known but in a different environment (projective geometry, non-Euclidean geometry, finite geometry).

Autonomy of judgment:

The student must be able to understand and critically evaluate the essential points of a method or theory that characterize it and guarantee its transparency, consistency and rigor case by case. It will also have to use the acquired knowledge to critically analyze textbooks, documents and manuals assessing whether the proposed reconstruction is consistent and compatible with a historical-epistemological approach between those studied and whether the reconstruction is faithful or rigorous or approximate, also comparing different sources.

Communication skills

Through the lectures and the dialogue with the teacher, the student acquires the specific vocabulary of several ancient and modern mathematical theories and the ability to make connections and comparisons, both by building longitudinal and long-term panoramas on the same theme (eg. infinite from the Greeks to modern mathematics), and by comparing two different approaches to the same problem (eg axiomatic approach or for transformations and invariants in the arrangement of Geometry). It is expected that, at the end of the course, the student will be able to transmit, in oral form, the main contents of the course and transversal themes. The student must communicate his / her knowledge with appropriate strategies, knowing both to build large frameworks and to go into the details of proofs, methods and procedures.

Learning ability

The student who has attended the course will be able to deepen their knowledge on the history and foundations of mathematics regarding the macro-themes and the theories examined, through the independent consultation of specialized texts and scientific or popular magazines, in order to carry out in-depth studies in a historical or didactic perspective and to undertake subsequent training courses in the area of the foundations, history and teaching of Mathematics.