In both versions of the course Numbers, Infinity, and Reality, grades are determined primarily by essay assignments, with a small percentage of overall grades dedicated to participation. Although mathematical problem sets are used during class sessions, they are not collected for grades but do constitute part of class participation. The honors version of the course features longer, more comparative essays, while the non-honors assignments were brief reflection-style essays. In the honors iteration of the course we also administer a cumulative final examination with the aim of assessing accurate use of terminology, broad comprehension of main debates and authors, and sufficient grasp of basic mathematical concepts pertaining to the course themes.
In addition to simple reading responses (one-page written reactions to readings), the non-honors iteration of the course focuses on five 2–3 page “reflection" essays. For these assignments we ask students to respond creatively and analytically to prompts distributed ahead of time. Some examples of prompts are given below.
Figure 6. David Hilbert (left) and Bertrand Russell (right). Convergence Portrait Gallery.
“Illustrative” essays assigned to students in the honors version of the course are designed to showcase student understanding of the debates, theories, and texts from the course syllabus. Rather than providing a prompt for the students, we task them with identifying one of the central debates or issues that have been examined in class and composing a five-page essay in which they explicate that topic. We emphasize that students should showcase two of the relevant readings in order to inform their illustration of the topic. We are clear that students should not attempt to evaluate which theory or viewpoint is more correct or incorrect and thus should avoid defending their own perspective regarding the debates or the viewpoints.
In the final weeks of the honors course we ask students to compose an eight-page argumentative essay in which they defend their own conclusions about some particular question or debate within the interdisciplinary space between mathematics and philosophy. We warn students that they should not attempt to connect their thesis to every single reading or viewpoint; rather, we encourage them to home in on some specific aspect. Frameworks for successful essays include, but are not limited to: (1) defending one view from an objection that a rival view makes against it; (2) revealing some point of constructive similarity between two otherwise different views; or (3) showcasing a potential problem or limitation internal to a view.
The honors course final exam is cumulative and assesses comprehension of basic terminology, historical context, logical techniques, and the relative pros and cons of key theories within the philosophy of mathematics. The purpose of this exam is to ensure sufficient breadth of engagement with the syllabus texts, whereas the essay assignments described above aim more at encouraging depth of student engagement. In Spring 2021 our final exam consisted of twenty questions in multiple choice, matching, or true/false formats. Roughly half of the questions were focused on texts that were more strictly mathematical, while the other half addressed strictly philosophical texts. Of course, most of the questions were ultimately in the intersection between mathematical and philosophical content. Sample questions include asking students: (1) to identify items that Euclid assumed without proof, featuring options such as the claim that ellipses always intersect (incorrect) or that circles are completely determined by a radius and a center point (correct); or (2) to select from a list the proposition that best summarizes Gödel's attitude toward Cantor's Continuum Hypothesis, featuring options such as that it can neither be proven true nor proven false (correct) or that it can be proven both true and false, rendering it incoherent (incorrect).
Figure 7. Georg Cantor (left) and Kurt Gödel (right). Convergence Portrait Gallery.