Elementary Mathematics from an Advanced Standpoint 1

The course aims to provide a critical re-reading of selected elementary mathematical topics, analyzed through methods typical of the Elementary Mathematics from a Higher Standpoint tradition. The objective is to promote a deeper understanding of fundamental mathematical concepts by examining their historical development, the comparison between different languages and theories, and the role of formal tools in the construction of mathematical theories.
The topics covered in the course will include:
1. Sets and foundations of mathematical language. General introduction to first-order languages and theories. The language of set theory. Abstract sets and the membership relation. Axiom of extensionality. Axioms of the empty set, pair, union and power set. Axiom of separation. Operations on sets and non-existence of a universal set. Relations and functions; families and products of families of sets. Cardinality, Cantor's theorem and the Cantor-Bernstein-Schröder theorem. Reflexive sets, successor, hereditary sets, axiom of infinity and construction of the set ω of natural numbers. Principle of induction, strong induction and principle of the minimum. Properties of ω, Peano axioms and operations on natural numbers. Finite sets and axiom of choice.
2. Development of algebraic symbolism and evolution of algebra as a discipline. Historical evolution of the ways of representing and treating algebraic relations. From rhetorical and syncopated language to modern symbolism. The role of notation in the transformation of algebra into an autonomous discipline and in the generalization of mathematical reasoning.
3. Conics from a higher standpoint. Study of conics through the comparison of different approaches: sections of the cone, loci in the plane and curves representable by equations. Analysis of the transition from a synthetic and geometric treatment to an analytic and algebraic description. Elements of the projective interpretation of conics and of the role of a change of standpoint in the classification of geometric objects.
4. From tables to modern formulas and the concept of function. Analysis of the transition from tabular forms to symbolic and functional representations. The examples of Ptolemy's theorem and logarithms. Discussion of the role of tables, formulas and curves in the construction of the modern concept of function. Analysis of Descartes' Géométrie and development of analytic geometry. Reading and analysis of selected significant passages from Descartes' Géométrie, with attention to the role of algebraic symbolism in the treatment of geometric problems. Development of analytic geometry and the coordinate method. Equations of curves and functions.
5. Evolution and formalization of logic: from Aristotle to Boole. Elements of the historical development of logic, from Aristotelian logic to the progressive formalization of reasoning. Introduction to the evolution of modern logic from the seventeenth century to Boolean algebra.