Numbers, Infinity, and Reality: An Interdisciplinary Undergraduate Philosophy of Mathematics Course – Course Design

The course Numbers, Infinity, and Reality aims to engage students with the following questions:

• What are numbers?
• What do we mean by infinity?
• How do we obtain mathematical knowledge?
• Does mathematics correspond with reality?
• Is mathematics invented or discovered?
• Who does (or gets to do) mathematics?

In addition to these specific questions, a more general aim of the course is to illuminate for students the many ways in which mathematics and philosophy have been, are, and should be intertwined.

Toward these ends, readings are selected with an eye toward accessibility for our different student constituencies while also being representative of the rigor and genre of each field. In addition, readings and topics are designed to provide balanced exposure to both disciplines. Readings have been grouped into the following units:

1. Historical foundations: Examples from classical Greece and China are adduced to showcase how linguistic and political contexts mitigate the ways in which mathematics is conceived in diverse societies.
2. Metaphysics and epistemology: Students are plunged into twentieth-century debates concerning the ontological status of mathematical entities and mathematical knowledge. Are such entities mind-independently real or socially constructed? What roles, if any, do deduction and intuition play in the acquisition and justification of mathematical beliefs?
3. Axiomatization: Students are exposed to the ways that starting assumptions and governing metaphors can shape number theory, geometry, and the mathematics of infinity.
4. Cross-disciplinary relationships: Students examine some of the ways that mathematics can be opposed to, subsumed by, or otherwise juxtaposed against discourses in the sciences, medicine, technology, and aesthetics.
5. Sociological implications: Students reflect on prevalent stereotypes about mathematics, romantic mythologies of mathematicians, typical encounters with mathematics in primary and secondary education, and ways in which gender and racial identities can intersect with or complicate the access to and standing of mathematical institutions.

In the next section, we provide three case studies from our assigned readings to illustrate how these themes are addressed in the course.