History of Mathematics
Program
The official syllabus.
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. The themes and instructors for these two parts may vary from
year to year, reflecting current research and scholarly interests.
In the academic year 2025/2026, the two monographic parts are:
(A) Prof. Marco Benini: Computable Functions From Abacus to Computer: Tracing the Quest
to Define Effective Calculability.
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.
- Parallel computing, artificial intelligence, and quantum computing: the new frontiers of contemporary computing.
(B) Prof. Claudio 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).
Text, slides and other material
The slides of the main part and about the monographic part A are available: please, select the right academic year
- First part: 2025/26 (printer-friendly version); 2025/26 (screen-friendly version)
- Monographic part A: 2025/26 (printer-friendly version); 2025/26 (screen-friendly version)
The textbooks used in this course are:
- Carl B. Boyer, A History of Mathematics, Wiley (2011) – also available in Italian
- Morris Kline, Mathematical Thought from Ancient to Modern Times, 3 vols, Oxford University Press (1972) – also available in Italian
Sources for the various sections of the first part of the course (ordered as cited in lessons):
- Egyptian and Mesopotamian Mathematics:
The Rhind papyrus, see also the British Museum
A History of Mathematical Notations vol I and vol II - Greek Mathematics:
Euclid Elements
Archimedes The Method
Archimedes Opera Omnia vol I and vol II
Apollonius Conics
Ptolemy Almagest
Diophantus Opera Omnia vol I and vol II - Ancient Asian Mathematics
Brahmagupta Brahmasphutasiddhanta
Aryabhata Aryabhatiya
The Nine Chapters on the Mathematical Art (preview), see also (complete but preliminary)
Fukagawa Hidetoshi, Tony Rothman Sacred Mathematics: Japanese Temple Geometry - The Islamic Golden Age
Muhammad ibn Musa al-Khwarizmi Algebra - Middle Ages and Renaissance
Leonardo Fibonacci Liber Abaci
Leon Battista Alberti De Pictura
Gerolamo Cardano Ars Magna
Rafael Bombelli L’Algebra
François Viète In Artem Analyticam Isagoge
Albrecht Dürer Underweysung der Messung
John Napier Mirifici Logarithmorum Canonis Descriptio
Luca Pacioli Summa de arithmetica geometria proportioni et proportionalità - Mathematics in the 17th century: The dawn of modernity
Isaac Newton Principia Mathematica, see also (English)
René Descartes Discourse on Method
Pierre de Fermat Varia opera mathematica
Bonaventura Cavalieri Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota
Gottfried Wilhelm Leibniz Nova Methodus pro Maximis et Minimis see also (English)
Christian Huygens De ratiociniis in ludo aleae - Mathematics in the 18th century: The age of Euler
Jacob Bernoulli Ars Conjectandi
Guillame de l’Hôpital Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes
Daniel Bernoulli Hydrodynamics
Leonhard Euler Institutiones calculi differentialis
Leonhard Euler Institutiones calculi integralis vol I, vol II, vol III, and vol IV
Joseph-Louis Lagrange Mécanique analytique vol I and vol II
Joseph-Louis Lagrange Œuvres complètes
Pierre-Simon Laplace Théorie analytique des probabilités
Pierre-Simon Laplace Philosophical Essay on Probabilities
Pierre-Simon Laplace Traité de mécanique céleste vol I, vol II, vol III, vol IV, and vol V (not translated)
Pierre-Simon Laplace Œuvres complètes
Brook Taylor Methodus Incrementorum Directa et Inversa
Gaspard Monge Géométrie descriptive
Jean D’Alembert Œuvres complètes
Carl Gustav Jacob Jacobi Œuvres complètes
August Ferdinand Möbius Œuvres complètes
Jean Baptiste Joseph Fourier Œuvres
NASA Space Flight Handbooks - Mathematics in the 19th century: The age of revolution
Augustin-Louis Cauchy Cours d’Analyse
Augustin-Louis Cauchy Œuvres complètes
Carl Friedrich Gauss Disquisitiones arithmeticae
Carl Friedrich Gauss Digitalised works in Gottingen
Johann Peter Gustav Lejeune Dirichlet Œuvres complètes
Nikolaj Ivanovič Lobačevskij Œuvres complètes (partial)
János Bolyai The science absolute of space
Bernhard Riemann Most works
Évariste Galois Oeuvres mathématiques
Karl Weierstrass Collected works vol I, vol II, vol III, vol IV, vol V, vol VI
Richard Dedekind Essays on the theory of numbers
William Rowan Hamilton Collected papers
Arthur Cayley Collected papers (too many to list the volumes)
Hermann Grassmann Die ausdehnugslehre
Leopold Kronecker Vorlesungen über Zahlentheorie
Georg Cantor Contributions to the founding of the theory of transfinite numbers
Cesare Burali-Forti Una questione sui numeri transfiniti - The 20th century in Mathematics
David Hilbert Collected works vol I, vol II, and vol III
David Hilbert Sur les problèmes futurs des mathématiques : les 23 problèmes
George Boole An Investigation of the Laws of Thought
Gottlob Frege The foundation of arithmetic
Gottlob Frege Begriffsschrift
Gottlob Frege Grundgesetze der Arithmetik
Giuseppe Peano Arithmetices principia: nova methodo
Bertrand Russell, Alfred North Whitehead Principia Mathematica vol I, vol II, vol III
Kurt Gödel On Formally Undecidable Propositions of Principia Mathematica and Related Systems
Henri Lebesgue Leçons sur l’intégration et la recherche des fonctions primitives
Henri Poincaré Analysis Situs
Ernst Zermelo Untersuchungen über die Grundlagen der Mengenlehre I
Nicolas Bourbaki Éléments de mathématique (list: the books are not in the public domain)
Felix Hausdorff Grundzüge der Mengenlehre
Hermann Minkowski Gesammelte Abhandlungen
Stefan Banach Théorie des opérations linéaires
Alexandre Grothendieck The Grothendieck Circle (a collection of most of his works)
Pierre Deligne La conjecture de Weil: I, La conjecture de Weil: II
Sophus Lie Theorie der Transformationsgruppen vol I, vol II, vol III
L. E. J. Brouwer Über Abbildungen von Mannigfaltigkeiten
Solomon Lefschetz Intersections and transformations of complexes and manifolds
Yuri Matiyasevich The Diophantineness of enumerable sets
Andrew Wiles Modular Elliptic Curves and Fermat’s Last Theorem
Albert Einstein Collected papers
Alan Turing On Computable Numbers, with an Application to the Entscheidungsproblem
Alonzo Church The Calculi of Lambda-Conversion
Stephen A. Cook The Complexity of Theorem-Proving Procedures
Edward Norton Lorenz Deterministic Nonperiodic Flow
Andrey Kolmogorov Foundations of the Theory of Probability
Thomas Bayes An Essay Towards Solving a Problem in the Doctrine of Chances
Clay Mathematics Institute The Millennium Prize Problems
Grisha Perelman The entropy formula for the Ricci flow and its geometric applications
Grisha Perelman Ricci flow with surgery on three-manifolds
Marco Benini 