Module description
What is the subject matter of mathematics? Is it abstract mathematical objects, or can apparent facts about mathematical objects be reduced to facts about something else? Assuming we have knowledge of mathematical facts, how is this knowledge acquired? Despite being essential to the sciences (and often thought of as one of the sciences), the non-empirical nature of mathematics raises epistemological and metaphysical questions quite distinct from those that arise in, say, physics. This course will examine approaches to answering these questions, including varieties of Platonism, and various forms of nominalism. We’ll also take a close look at the role of mathematics in the sciences, with the aim of evaluating one of the key arguments in the debate between the Platonist and the nominalist: the indispensability argument.
Assessment details
Summative assessment: 1 x 3,000-word essay (100%)
Formative assessment: 1 x 2,500-word essay
Educational aims & objectives
The students will be introduced to and receive training in certain key ideas from the Philosophy of Mathematics. In particular, students will gain some or all of the following:
- An understanding of early twentieth century schools in philosophy of mathematics, including logicism, formalism, and intuitionism.
- An understanding of the arguments against these positions, including arguments based in Gödel’s incompleteness theorems.
- Familiarity with the Quine-Putnam indispensability argument and the more recent ‘enhanced’ indispensability argument.
- An understanding of contemporary nominalist positions, including fictionalism.
Learning outcomes
By the end of the module, the students will be able to demonstrate intellectual, transferable and practicable skills appropriate to a level-6 module and in particular will be able to demonstrate:
- Knowledge of the main positions in the philosophy of mathematics.
- The ability to assess and develop these positions.
- To exercise their powers of intellectual criticism by critically commenting upon the views discussed.
Teaching pattern
One one-hour weekly lecture and one one-hour weekly seminar over ten weeks
Suggested reading list
- Colyvan, Mark. (2001), The Indispensability of Mathematics, Oxford: Oxford University Press
- Leng, Mary. (2010), Mathematics and Reality, Oxford: Oxford University Press
- Maddy, Penelope (2011), Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford: Oxford University Press
- Benacerraf, Paul (1973), ‘Mathematical Truth’, Journal of Philosophy, Vol. 70, No. 19
- Baker, Alan (2005), ‘Are there genuine mathematical explanations of physical phenomena?’, Mind, Vol. 114, No. 454