Knowledge and understanding: the students will be expected to acquire essential knowledge about the main philosophical problems raised by observation of the practice of mathematics.
Use of knowledge and understanding: while grappling with theoretical questions, the course will provide the students with tools to apply rigorous reasoning models in diverse circumstances.
Ability to draw conclusions: the course will emphasize the importance of rigorous critical argumentation. In particular, the use of logical and formal tools will be encouraged.
Communication skills: in both classroom discussion and individual writing, importance will be given to the ability to present the issues clearly and systematically, as well as to argue rigorously in presenting one's personal considerations.
Ability to learn: the course will stimulate the student’s independent reflection through the progressive verification of the knowledge acquired, while also providing didactic and bibliographical indications for further analysis of the course's topics.
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Lectures accompanied by group discussions. The student’s active participation will be encouraged.
Written exam in (take-home) paper form and class presentation. Eligible topics and criteria for the papers will be made available during the class.
The course "Philosophy of Mathematics: Epistemology and Practice" will address central problems in the philosophy of mathematics. It will do so by integrating complementary perspectives.
This second Module (Module B – S. De Toffoli) will provide a general overview of different approaches in the epistemology of mathematics. It will then zoom in on practice-based approaches that consider the subjects of knowledge to be embodied, fallible, and operating within a social network. This is in stark contrast with traditional approaches that consider highly idealized subjects operating in isolation.
Afterward, the problem of mathematical rigor will be considered. In particular, the relationship between informal proofs (which are common currency among mathematicians), and formal proofs (in a specific formal system) will be discussed. To deal with this topic in detail, we will analyze the role of diagrams and visualization both in the practice of ancient mathematics as well as in contemporary mathematics.
We will also talk about the possibility or impossibility of obtaining certainty in mathematics and how mathematics can be conceived as a fallible human practice. We will thus explore the complex relationship between idealized analyzes of mathematics and mathematical reasoning in practice.
The topics discussed in this module will be integrated by the discussion regarding the foundations of mathematics. These are addressed in the first module of the course. Further topics in epistemology and the philosophy of mathematics will be covered in the course "EPISTEMOLOGY AND MATHEMATICS".
