Reflecting on beauty: the aesthetics of mathematical discovery

TL;DR

This paper explores the concept of mathematical beauty, highlighting its role in connecting diverse areas of mathematics and enhancing understanding, while examining the underlying aesthetic judgments involved.

Contribution

It provides an analysis of mathematical beauty as a central aesthetic concept, clarifying its connections to other notions and its influence on mathematical discovery.

Findings

Beauty reveals connections between unrelated problems.

Beauty correlates with depth, elegance, and simplicity.

Mathematical beauty guides discovery and insight.

Abstract

Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are based on remain unclear. In this article, we focus on the concept of mathematical beauty, as one of the central aesthetic concepts in mathematics. We argue that beauty in mathematics reveals connections between apparently non-related problems or areas and allows a better and wider insight into mathematical reality as a whole. We also explain the close relationship between beauty and other important notions such as depth, elegance, simplicity, fruitfulness, and others.