Early one morning in the halls of a typical mathematics department, Katie, a graduate student in the field of higher category theory, walks into her final exam for grad algebra 1. She had enough sleep the previous night, and feels confident about her abilities. The first question is a routine application of Nakayama’s lemma, and the next an exercise in computing a $ \mathrm{Tor}$ group. After half an hour of deftly dealing out solutions, she comes to the last item:

**Explain the importance of module theory in ring theory using a few examples.**

*What kind of exam is this?* Katie thinks. *The question is not true, false, a computation, a proof, or undecidable in ZFC + V=L! Madness!*

### The Role that Essays Could Have in Math

I made this story up entirely. However, believe incorporating a small amount of such questions would be useful in emphasising intuition and the aesthetic side of mathematics, and this is something that could be used in upper undergraduate and all graduate courses.

More »