# COMPLEMENTARY MATHEMATICS

## Learning outcomes of the course unit

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.

## Prerequisites

There are no compulsory prerequisites.

## Course contents summary

The course aims to provide students with the general criteria of analysis of a mathematical theory in its historical development and in its foundational aspects, with attention to the different criteria of rigor, to the methods and the role of intuition, of constructions, of formalism and contamination with other disciplines in some crucial phases of its historical development.

Therefore the contents proposed during the course of the lessons concern: in the first part of the course, Egyptian mathematics, Babylonian, Indian and pre-Hellenic Arabic, the Elements of Euclid, some works by Archimedes and Apollonius, the classical problems of Antiquity, Menelaus and pappus theorem in different geometries from the Euclidean one; in the second part of the course some basic passages are presented towards modern approaches to mathematics, with particular attention to non-Euclidean geometries, the passage to symbolic algebra and Cartesian geometry, the development of methods and the problem of the foundations of Analysis, the projective geometry, Klein's program, Hilbert's Grundlagen, Gödel's theorems.

## Course contents

CHAP 1: METHODS, FUNDAMENTALS AND THEORIES OF ANCIENT MATHEMATICS

The mathematics of the Egyptians and the Babylonians.

Greek mathematics: Thales, Pythagoras and his school, the crisis of the immeasurable. Zeno and the paradoxes of the infinite.

The three famous problems of Greek antiquity: quadrature of the circle, duplication of the cube, trisection of the angle and history of the solutions. Hippocrates and the quadrature of the lunulas.

Euclid: the "Elements", common notions, postulates and axioms, theory of parallels, theory of proportions, sizes, prime numbers, equivalence in the plane and in space. Euclid's work in the light of modern criticism. Archimedes: from the measurement of the circle to the volume of the sphere, the method of exhaustion. Apollonius: conic sections.

CHAP 2: METHODS, FUNDAMENTALS AND THEORIES OF MODERN MATHEMATICS

The birth of the concepts of limit, function, derivative, integral and the development of the Calculus. Numerical systems and properties in a historical perspective and axiomatizations: natural numbers, integers, rational numbers, reals.

Non-Euclidean geometries: historical and epistemological aspects, the Poincaré and Klein models.

The Erlangen program and the geometry of transformations: isometrics, similitudes, affinity, projectivity. Circular inversion.

The problem of the foundations of Geometry: Hilbert's axioms, independence, consistency, completeness.

Several concepts of completeness. Dialectic between intuition and formalism in the evolution of mathematical analysis and modern axiomatics. Hilbert's problems.

Cantor and the infinite: cardinal and ordinal, hypothesis of the continuous, ingenuous theory of the sets. Russell paradox. Peano arithmetic. Formalization and mechanization of mathematical reasoning: from Leibniz to formal logic, up to the first developments in information technology. Syntactic and semantic completeness. Incompleteness theorems.

## Recommended readings

The slides projected during the course in PDF format and all the material used during the lessons and exercises (translations of ancient works in digital format, extracts from thesis, notes with detailed demonstrations for further information) are made available to students and shared on the Elly platform at the end of each lesson.

In addition, students have access to a PDF online version of Euclid’s Elements, translated by Richard Fitzpatrick (2007) available online at: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

In addition to the material provided by the teacher, which comprehensively covers all the contents of the course, students with an interest in the historical aspects can deepen on the manuals:

Kline, M. (1972) Mathematical Thought from Ancient to Modern Times, Vol. 1-2-3

Available online at:

https://ia801301.us.archive.org/18/items/MathematicalThoughtFromAncientT...

C.B.Boyer, Uta C. Merzbach (2011) , A history of mathematics, ISBN: 978-0-470-52548-7

## Teaching methods

The course has a weight of 9 CFU, which corresponds to 72 hours of lessons. The teaching activities will be conducted by giving lectures in the classroom, alternating with exercises. During the lectures, the topics of the course are dealt with from a theoretical point of view and with detailed examples of proofs, constructions and problem solving, with some lecture lessons in which they also propose critical readings of historians and philosophers of mathematics or parts of original works. During the class exercises, students will be required to apply the theory to an exercise, a problem, a demonstration according to the methodological criteria illustrated in the lessons and in the bibliographic and teaching material. In addition to the teaching methods presented so far, in-depth seminars are organized on the topics of the course. The slides and documents used to support the lessons will be uploaded at the beginning of the course on the Elly platform; To download the slides it is necessary to register for the online course. All shared material is considered an integral part of the teaching material. Non-attending students are reminded to check the available teaching materials and the indications provided by the teacher through the Elly platform, the only communication tool used for direct teacher / student contact. On this platform, on a weekly basis, the topics discussed in class are indicated, which will then form the contents index in preparation for the final exam.

## Assessment methods and criteria

The assessment will take place on the basis of an oral test, with questions related to the contents of the course; some questions are asked concerning ancient mathematics and some related to modern mathematics. The questions are initially related to transversal issues to be treated longitudinally by the candidate; with this first type of questions we evaluate the communication skills and the ability to make use of the precise knowledge to build a broad and comprehensive overview on a macro-theme. Afterwards we go to deepen a single aspect asking to produce a demonstration, a resolution of a problem, a construction. Three general questions are asked to the candidate with relative details. Each question corresponds to 10 points. The test is passed if it reaches a score of at least 18 points. Praise is awarded only if the candidate shows autonomy of judgment and communication skills rather than good.