Introduction to the Philosophy of Mathematics: History and Problems

Course Leader: Dr Paniel Reyes Cárdenas

Home Institution: UPAEP (Puebla, Mexico), The University of Sheffield (UK)

Course pre-requisites: Introduction to Philosophy, any basic logic course

Course Overview
The Philosophy of Mathematics is a relatively new discipline that comprises problems spanning from the history of mathematics and, especially for our concern, central philosophical questions. I would like to offer a module in which the students will be able to recognize and seriously consider these problems. This module, although presupposes some knowledge of logic and the history of philosophy, it is independent on its own from them, this new discipline is a whole different inquiry that sheds light in epistemological, metaphysical and methodological philosophical problems in general.

The module will be spinning out of two central concerns: students will be able to identify and think critically about the conceptions of mathematics in the classical philosophy and the history of mathematics and, more importantly, they will be introduced to the contemporary debates on the foundations of mathematics, the epistemology of mathematical knowledge and the relations of mathematics with philosophical inquiry. Those approaches will, although generally conceived, get the students ready to have a viewpoint on the topic.

Learning Outcomes
• To develop an ability to read, interpret, analyse and evaluate philosophical texts in its relation with other philosophical disciplines and with particular emphasis in Mathematical thought.
• To introduce students to a concrete application of Philosophical thought.
• To facilitate thoughtful and respectful dialogue on important philosophical, scientific and historical issues
• To foster writing skills by focusing on organization, thesis development and the use of secondary literature
• To synthesise and reflect upon the views of other scholars
• To challenge assumptions and unquestioned beliefs
• To think more critically and reflectively about the discipline of mathematics and the place of philosophy in inquiries about mathematical thought.

By the end of the course the students will have considered an important number of
arguments and topics in the philosophy of mathematics and will be open to truly interdisciplinary
research.

Course Content
Introduction to the Philosophy of Mathematics
Plato and Aristotle
Infinity and its problems
Kant and the reactions to Kant
Foundations of Mathematics: Geometry
Foundations of Mathematics: Arithmetic and Theory of Numbers
Logicism: Frege
Logicism: Set Theory and new approaches to Logicism
Formalism: Hilbert
Formalism: Deductivism, Platonistic formalism
Intuitionism: Brouwer
Intuitionism: intuitionistic logic
Nominalism: Reductive nominalism
Nominalism: Fictionalism
Structuralism: Benacerraf’s Puzzle
Structuralism: realist structuralism
Structuralism: eliminative structuralism
Synthetic Philosophy of Mathematics
Concluding Remarks: Philosophical arguments and new trends.

Instructional Method
The course will be comprised of 10 lectures and 10 seminars distributed in the available time.

Required Course Materials
Assistance and taking of notes during lectures, participation in Seminars. Reading materials shall be provided.

Assessment

The Standard assessment for the module will be by one essay between 2500 and 4000
words in length and one examination. The essay and examination will each count for
50% of the final grade.
Students may also request permission to be assessed entirely by a long essay (between
4500 and 6000 words in length).