The Importance of Surprise in Mathematical Beauty

Abstract

Synopsis Mathematicians, mathematics education researchers, and philosophers have writ-ten about mathematical beauty and many of the qualities commonly associated with it, such as simplicity, brevity, enlightenment, etc. One key theme that under-lies many of these qualities is surprise or the unexpected. In this article, I discuss the integral role surprise plays in mathematical beauty. Through examples, I argue that simplicity alone is oftentimes not enough for a piece of mathematics to be considered beautiful, but rather it is unexpected simplicity that we seek. I propose, moreover, that surprise is necessary for enlightenment. The paper also reports results from an activity designed to elicit an appreciation of mathematical beauty from elementary preservice teachers; the majority reaction was a feeling of surprise. Understanding the relevance of surprise to mathematical beauty may offer us a feasible way to create opportunities for students to experience mathematical beauty.