\"Uber Nichts

TL;DR

This paper explores the concept of Nothing through philosophical and mathematical lenses, analyzing zero and the empty set's roles in formal systems and their philosophical implications for understanding absence and creation.

Contribution

It offers a comprehensive analysis of zero and the empty set, highlighting their historical evolution and foundational roles in mathematics and philosophy.

Findings

Zero as an algebraic and structural element

Empty set as a foundational axiomatic object

Active nothing's role in algebra and physics

Abstract

This paper investigates the concept of Nothing from both philosophical and mathematical perspectives, distinguishing between absolute non-being (nihil) and relational negation as a principle of difference. It explores how mathematics formalizes absence through two fundamental concepts: the number zero and the empty set. The text traces the historical evolution of zero from a positional marker in Babylonian systems to an independent algebraic entity in Indian mathematics, emphasizing its structural roles as a neutral and absorbing element. Parallel to this, the empty set is analyzed not as a void, but as an existing axiomatic object that serves as the foundation for constructing the number system in modern set theory. Furthermore, the operational utility of active nothing is demonstrated through its role in algebraic manipulation -- such as adding zero to reveal structure -- and in physical conservation laws. Finally, drawing on the philosophy of Nikolaus von Kues and the discovery of non-Euclidean geometry, the paper interprets mathematics as a free, creative act of the human mind emerging ex nihilo.