Essay on Mathematics and Typical Mathematics Department

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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 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.

Don’t get me wrong; I think conventional mathematical problems are one of the most important parts of learning mathematics. However, there could be other aspects to a solid mathematical education besides problems and reading theorems. Personally I love to read solid, austere, and elegant math, but I also enjoy the informal atmosphere of attempting to explain intuition.
Thinking about the meaning or intuition behind concepts is vitally important. The true meaning of things cannot be ascribed a rigorous mathematical definition, but mathematicians are constantly using their own mental “picture” or intuition about their research to advance their field.
Having such vague questions in assignments would give students a chance to really think about the line of thought emphasised through intuitive sentiment. Moreover, students would be encouraged to express their intuition and aesthetic appreciation, and might find that although the final product of mathematics is and should be proof, it is also worthwhile to examine one’s subconscious driving force behind creative output. Moreover, vague questions do not have to have vague answers; a good answer I think would be a combination of vague intuitive statements with precise examples and statements of theorems that would motivate the topic of discussion.
I also feel that mathematics is a very beautiful subject, filled with vast scenes of the elegant and sometimes bizarre objects that come out of the deceptively simple axioms and ideas; this aesthetic feeling is something that should
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