# 851-0196-00L Philosophy of Pure and Applied Mathematics: From Foundations to Practice

Semester | Autumn Semester 2023 |

Lecturers | Y. P.‑H. Hamami |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | This course is a general introduction to the philosophy of mathematics for science, mathematics and engineering students. It will introduce the main views and debates on the nature of mathematics present in contemporary philosophy. A special focus will be put on questions pertaining to the foundations of mathematics as well as on philosophical issues emerging from actual mathematical practice. |

Objective | The objective of this course is to help students develop a reflective stance on what mathematics is and on its special place in the landscape of human knowledge. We expect students to be able to report the main philosophical conceptions of what mathematics is. We also expect them to be familiar with key debates in the philosophy of mathematics. |

Content | This course is a general introduction to the philosophy of mathematics for science, mathematics and engineering students. It will introduce the main views and debates on the nature of mathematics present in contemporary philosophy. A special focus will be put on questions pertaining to the foundations of mathematics as well as on philosophical issues emerging from the actual practice of mathematics. The course is composed of four parts. Part I: Foundations of Mathematics. In this first part of the course, we will present the debates concerning the foundations of mathematics at the turn of the twentieth century. We will review the three main philosophical conceptions of mathematics developed during this period: logicism, formalism and intuitionism. Part II: Ontology and Epistemology of Mathematical Objects What is the nature of mathematical objects? And how can we acquire knowledge about them? Here we will present several ways of approaching these questions. We will discuss philosophical views that conceive mathematical objects as similar to physical objects, as creations of the human mind, as fictional characters, and as places in larger structures. We will see the strengths and weaknesses of these different views. Part III: Philosophy of Mathematical Practice In this part of the course, we will be concerned with a recent movement in the philosophy of mathematics dealing with the actual practice of mathematics. We will see two trends of research developed within this tradition. The first one aims to explain how we can think and reason mathematically with non-linguistic representations such as diagrams and symbolic notations. The second one asks whether there could be such things as explanations in mathematics and if yes what they are. The paradigmatic examples we will discuss here are mathematical proofs that not only establish that a theorem is true but also explain why it is true. Part IV: The Applicability of Mathematics to the Natural World It is a truism that mathematics is used everywhere in the natural and social sciences. But how come that mathematics applies so well to the natural world? If mathematics is just a pure game with symbols, or a pure invention of the human mind, it seems difficult to explain why it is so useful when formulating scientific theories about the world. In this part of the course, we will discuss this problem known as the applicability of mathematics, and we will see different philosophical solutions that have been developed to address it. |