Corso di laurea magistrale in Filosofia

Familiarity with first-order logic.

This course will introduce students to some fundamental topics in the foundations and philosophy of mathematics, in line with the overall learning objectives of the MA program in Philosophy and the Philosophy International Curriculum to train the students' argumentative skills and their power of philosophical and logical analysis.

Familiarity with the fundamental problems and paradigms in the foundations of mathematics, and knowledge of the main limitative results in mathematical logic, and their philosophical significance. Increased ability of logical and methodological analysis of foundational and philosophical problems, thanks to the mathematical and conceptual tools acquired during the course. 

Lectures and discussions (36 hours overall). The course will be in person. Teaching materials (videos, texts, forums, and exercises) will be available on Moodle.

The exam will consist of a written and an oral component. The written component is a take-home problem set, that students will be given at the end of the course, and will have to deliver at least one week before the selected date for the oral examination. The oral examination will be a mixture of philosophical and technical questions. In order to prepare for the exam, students are required to do en-route assignments, i.e. problem sheets, every two weeks. Solutions to selected exercises will be discussed in class.  

The final grade will be given on a 30-points scale.

The oral interview will be in person. Students might ask the lecturers to do the interview online in case they cannot reach the exam venue due to exceptional circumstances.

The course is dived into two parts.

The first part will analyze the question of how we learn mathematical language, with focus on the language of arithmetic. We will review classical limitative results (Löwenheim-Skolem theorems and the compactness theorem, the halting problem, Gödel’s first incompleteness theorem, and Tarski’s undefinability theorem). A discussion of the philosophical significance of these results will follow. Finally, we will look at computational structuralism (Horsten) and abstractionism (Linnebo) as two strategies to answer the question of what fixes the reference of our arithmetical vocabulary.

The second part will deal with second-order solutions to the problem of referential indeterminacy in mathematics. After an introduction to second-order logic (SOL), with full and Henkin semantics, we will show that SOL does not share some of the meta-logical features of first-order logics (in particular, compactness and the Löwenheim-Skolem theorems), and can therefore single out unique models (up to isomorphism) of fundamental mathematical theories. More specifically, we will introduce the phenomenon of categoricity, and show that the second-order theories of the main number systems are categorical. If time permits, we will show that a quasi-categoricity result holds for second-order theories of sets. 

Required readings: 

Part I

1)McGee, V. (1997) How we learn mathematical language, Philosophical Review 106 (1):35-68.

2) Smullyan, R. (1992) Godel's Incompleteness Theorems, Oxford University Press, Oxford, Chapter 1.

3) Smith, P. Godel without (too many) tears,  freely available at this link, Chapters 1-7, 16.

4) Horsten, L. (2012)  Vom Zahlen zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism. Philosophia Mathematica 20 (3):275-288.

5) Field, Hartry (1994). Are Our Logical and Mathematical Concepts Highly Indeterminate? Midwest Studies in Philosophy 19 (1):391-429.

6) Linnebo, O. (2018) Thin objects, Oxford University Press, Oxford, Chapter 1 and chapter 10

Part II 

1) Shapiro, S. Foundations without Foundationalism, Oxford University Press, Oxford, 1991,  Chapters 1-6

2) Button, S. and S. Walsh, Philosophy and Model Theory, Oxford University Press, Oxford, 2018, Chapters 1, 2, 4, 6-8

3) Väänänen, J. "Second-order and Higher-order Logic", Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/logic-higher-order/

Optional readings:

Part I

Rayo, A., (2019) On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics, MIT Press.

Lectures are expected to start on April 13, 2023

The course will be in person. Some of the teaching materials will be available on Moodle. This course is addressed primarily to Philosophy and to Mathematics students, but it will be accessible to anyone with some background in first-order logic. 

Students willing to take the course are kindly invited to enroll on Campusnet.