Modern Definition and Ancient Definition

TL;DR

This paper compares ancient and modern definitions, exploring their differences and similarities, and investigates potential connections through topos theory, revealing overlooked questions about modern mathematical practice.

Contribution

It provides a comparative analysis of ancient and modern definitional practices and suggests a possible interpretative link via topos theory.

Findings

Ancient definitions raise questions overlooked in modern logic.

Potential for co-interpretation between ancient and modern practices.

Highlights gaps in current philosophical and mathematical literature.

Abstract

In this essay we examine some aspects of the classical theory of definition as codified in Aristotle's \emph{Topics} and Porphyry's \emph{Eisagog\^e} in the light of the way definition is carried out in modern mathematical practice. Our goal is to contribute to the understanding of the alleged gap existing between ancient and modern logic and science as well as the reasons behind allegations of inadequacy and lack of sophistication in the ancient theory of definition. Also to investigate the possibility of a co-interpretation between modern mathematical definitional practice and ancient definitional practice in particular in the light of topos theory. We find the ancient definitional practice asks relevant and overlooked questions about modern mathematical practice which apparently have escaped current philosophical and mathematical logical literature. We also present some general considerations about the structure and development of theories as these relate to the theory of definition.