Corso di laurea magistrale in Filosofia

The course aims to familiarize students with the philosophical background of modern axiomatics, with particular reference to the early twentieth-century debate on implicit definitions and abstraction principles. It will allow students to engage with texts by Frege, Hilbert, and other classical authors, as well as with contemporary developments of their positions. Based on the background provided in the course, the students will be especially challenged to form their own judgment on these philosophical theses.

After attending the course, students will be expected: a) to have acquired knowledge and demonstrate understanding of the philosophical texts presented during the course; b) to be able to apply the notions learned during the course to central issues in the philosophical reflection on mathematics; c) to have improved their critical skills in engaging with the course materials; d) to have improved their communicative skills by participating actively in the discussions; e) to demonstrate their learning skills by elaborating on the course materials.

Lectures and seminar-style discussions.

The oral examination will test the students' knowledge and understanding of the course materials, their capacity to situate the primary sources in their historical context and to analyze their philosophical contents, the ability to engage with the philosophical problems discussed during the course in a critical way, the acquisition of argumentative skills.
Students who cannot attend the course are required to prepare for the oral examination with the help of additional literature. They are kindly asked to contact the instructor in advance for advice on the additional materials.

The course addresses different issues that arose at the intersection between philosophy and mathematics in the late nineteenth century: How does modern axiomatics differ from Euclidean axiomatics? Are there true statements about abstract objects such as numbers, points, or functions? Or do axiomatic systems refer to higher-order concepts or structures that can have infinitely many interpretations in terms of objects? What is the role of axioms in mathematical proofs? What is the correct way to demonstrate consistency and independence results for axiomatic systems? To what extent can these techniques provide a foundation of mathematics?
These questions will be addressed on the basis of the textual analysis of primary sources and works by Gottlob Frege, Giuseppe Peano, and David Hilbert. The selected texts will provide a general introduction to a classical debate in the twentieth-century philosophy of mathematics over implicit definitions and definitions by abstraction. The course materials will offer a starting point also for a discussion of contemporary understandings of mathematical abstraction, in the neo-logicist discussion on abstraction principles, in relation to the practice of mathematics, in connection with mathematical explanations within mathematics and in the sciences.

Exam programm

– Frege, Gottlob. Philosophical and mathematical correspondence, edited by Gottfried Gabriel et al.; abridged for the English edition by Brian McGuinness and translated by Hans Kaal. Oxford: Blackwell, 1980.
– Frege, Gottlob. Collected papers on mathematics, logic, and philosophy, edited by Brian McGuinness. Oxford: Blackwell, 1984.
– Hilbert, David. Foundations of geometry, translated by Leo Unger from the German 10th edition (1968). La Salle, Ill: Open Court, 1971.
– Mancosu, Paolo. Abstraction and infinity. Oxford: Oxford University Press, 2016.

Students who cannot attend the course are required to prepare for the oral examination with the help of additional literature. They are kindly asked to contact the instructor in advance for advice on the additional materials.