34740 - History of Mathematics

Learning outcomes

At the end of the course, the students should: - possess a hystoric knowledge in some depth of the key-themes of mathematics and of mathematical thought; possess also a good panoramic view of the evolution of maths and of mathematical thought. - be able to use these cultural tools in the professional activities when teaching maths. -be able to use this knowlkedge in order to create efficient teaching materials to be used in the classroom.

Course contents

The origins of mathematics in ancient civilisations, with particular attention to numeral systems, methods of calculation and the solution of practical problems in Egyptian and Mesopotamian mathematics.

The emergence of deductive mathematics in ancient Greece: Thales and Pythagoras, the discovery of incommensurable magnitudes, the theory of proportions and the gradual establishment of proof as a defining feature of mathematical knowledge.

Euclid’s Elements and the axiomatic organisation of mathematics; Archimedes’ contributions to the study of areas, volumes and infinitesimal methods, and Apollonius’ development of the theory of conic sections.

The development of Hellenistic and late ancient mathematics, with particular reference to Diophantus’ arithmetic, Pappus’ geometry and the transformation of mathematical methods during the transition from antiquity to the Middle Ages.

Mathematics in India, China and the Islamic world, including the development of positional numeral systems, algebra, trigonometry and computational techniques, as well as the role of Arabic scholarship in preserving and transmitting Greek mathematical texts.

Mathematics in medieval and Renaissance Europe: Fibonacci and the spread of Hindu-Arabic numerals, the development of abacus algebra, and the solution of cubic and quartic equations by Italian algebraists.

The emergence of symbolic algebra and modern mathematics through the work of Viète, Napier, Descartes and Fermat, with particular attention to logarithms, analytic geometry and the unification of algebraic and geometric methods.

The Scientific Revolution and the role of mathematics in the new science, with references to Galileo, Kepler and Pascal, and to developments in probability, mechanics and the study of curves.

The development of the calculus by Newton and Leibniz, considered both in relation to the problems that led to its creation and to the conceptual, methodological and notational differences between their approaches.

Eighteenth-century mathematics, with particular reference to the Bernoulli family, Euler, d’Alembert, Lagrange and Laplace, and to the development of analysis, differential equations, mechanics and probability theory.

Nineteenth-century mathematics: Gauss, Cauchy, Abel, Galois, Riemann and Cantor; the increasing rigour of analysis, the emergence of group theory, non-Euclidean geometries, function theory and set theory.

The foundational crisis and the main developments between the nineteenth and twentieth centuries, including Hilbert’s programme, mathematical logic, Gödel’s incompleteness theorems and the transformation of the language and structure of modern mathematics.

Readings/Bibliography

Boyer - History of mathematics

Teaching methods

Classes are structured as lectures and individual or small-group workshops,

Assessment methods

Oral exam

With regard to assessment, the use of AI is prohibited. Any use of AI constitutes a breach of academic integrity.

Teaching tools

An e-learning space is activated on the platform Virtuale

Office hours

See the website of Alessandro Gambini