A New Perspective of Entropy

Abstract

This article describes a new connection between two seemingly disparate topics in science, namely entropy and higher mathematics. It does not assume prior knowledge of either subject and begins with a brief introduction to information theory and a concept known as Shannon entropy, which we simply refer to as entropy. We then survey the vast landscape of higher mathematics, giving special attention to advanced analogues of high-school algebra and geometry known as abstract algebra and topology, respectively. Our goal is then to show that entropy, abstract algebra, and topology are inextricably linked through a version of a well-known formula from calculus known as the Leibniz rule. This result is given in the author’s recent work in [Bra21], and this present article is intended to give an overview of the ideas by gently introducing them from the ground up.