Elegance or Alchemy? An International Cross-Case Analysis of Faculty and Graduate Student Perceptions of Mathematical Proofs

Artist Marcel Duchamp once said, ``The painter is a medium who doesn’t realize what he is doing. No translation can express the mystery of sensibility, a word, still unreliable, which is nonetheless the basis of painting or poetry, like a kind of alchemy" (Moffitt, 2012). Just as there is a puzzling aspect of creating art or writing poetry, the aesthetic quality of mathematical proofs is a mysterious and ill-defined concept. Like many other subjective terms, it can be difficult to reach a consensus on what elegance means in a mathematical context. In this thesis, I try to better understand faculty and graduate students’ perceptions of elegance in mathematical proofs. To do this, I conducted an international cross-case analysis that involved participants from three groups: graduate students studying mathematics in the United States, graduate students studying mathematics in Ghana, Africa, and research faculty of mathematics in the United States. My goals in this thesis were to learn how participants perceive elegance in proofs, better understand how participants’ perceptions of elegance compare to their perception of other constructs, such as surprise, creativity, and rigor, and determine which proof constructs our participants seem to value most. I gathered data from each group and compared these three goals amongst all three groups.