HISTORY OF MATHEMATICS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
- Contacts/Info
No specific prerequisites are required, other than a Bachelor's degree in a mathematical field or equivalent basic mathematical knowledge.
The examination consists of writing a paper on a topic agreed upon with one of the course instructors, chosen from those covered during the teaching period. The paper will be assessed considering its historical accuracy, the ability to correlate the topic with its developmental context and with other related areas of mathematics, and the quality of its explanation, particularly for a technical but non-specialist audience, and for an interested but non-technical audience. Alternatively, at the student's choice, three intermediate written tests will be conducted during the course period, aimed at verifying the learning and comprehension of the three parts into which the course is divided.
This course offers a comprehensive exploration of the evolution of mathematical thought and practice. The first part provides a concise overview of the development of Mathematics from prehistory to the close of the 20th century, highlighting major advancements and shifts in mathematical understanding. The second and third parts are monographic, allowing for a deeper investigation into specific areas of mathematical history. --- Upon successful completion of this course, students will have acquired the following skills: 1. Accurately identify and contextualize major mathematical results within their correct historical period and relevant historical-scientific framework. 2. Develop a comprehensive and unified understanding of mathematical disciplines and their interrelations, grounded in the foundational lines of thought they represent. 3. Explain the principal areas of Mathematics in non-technical terms, particularly those relevant to the curricula of upper secondary schools. 4. Communicate mathematical concepts to a diverse audience in general, non-technical terms, and effectively frame such communication within its appropriate historical context.
The first part of the course, taught by Prof. Benini, will be about the general history of mathematics from the ancient one (Mesopotamia, Egypt) till the beginning of 21st century. This part will take 32 hours. The second part of the course (16 hours), taught by Prof. Benini, will cover the history of computable functions, from their definition to computational complexity. The third part of the course (24 hours), delivered by Prof. Quadrelli, will deal with the history of the search for solutions of equations from ancient through modern times.
The general part of the course will cover the following historical periods: 1. Egyptian and Mesopotamian Mathematics (2 hours) 2. Greek Mathematics (4 hours) 3. Ancient Asian Mathematics (2 hours) 4. The Islamic Golden Age (2 hours) 5. Middle Ages and Renaissance (2 hours) 6. Mathematics in the 17th century: The dawn of modernity (2 hours) 7. Mathematics in the 18th century: The age of Euler (4 hours) 8. Mathematics in the 19th century: The age of revolution (5 hours) 9. The 20th century in Mathematics (7 hours) A) Prof. Benini: Computable Functions From Abacus to Computer: Tracing the Quest to Define Effective Calculability. - Hilbert, Post, Gödel, Schönfinkel: The foundational crisis and early formalisms of computation. - Church, Kleene: Lambda calculus and recursive functions. - Turing: The Turing machine and the limits of computation. - Von Neumann and the birth of Computer Science: Architectural paradigms and the practical realization of computation. - Automata, non-determinism, and computational complexity: Formal models of computation and resource analysis. - P vs NP: The central open problem in computational complexity. - Trickling incomputability: type theories and strong normalisation: Connections between computation and proof theory. Part B) Prof. Quadrelli: Finding Solutions to Equations Through the Centuries - Equations in ancient mathematics: Methods and solutions in Babylonian, Egyptian, Greek, Arab, and Indian mathematics. - Equations in the Middle Ages until Luca Pacioli: Algebraic developments and the transition to symbolic notation. - The Renaissance: Tartaglia, Cardano, Ferrari, and the mathematical duels: Discovery of cubic and quartic solutions and the rise of algebra. - Last developments before modern mathematics: Key advancements leading to modern algebraic theories (e.g., early ideas on polynomial roots, unsolvability).
The course will be delivered through lectures in English, utilizing slides and in-depth discussions at the blackboard. The slides are specifically structured to facilitate the learning objectives, particularly by providing concrete examples and guidance for the non-technical illustration of advanced areas of Mathematics. They also offer historical explanations of parts of Mathematics used in upper secondary schools and provide insights and references to original material within historical contexts.
All the information about the course will be available on the course website: https://marcobenini.me/lectures/history-of-mathematics/