From Singularities to Graphs

TL;DR

This paper discusses the development of graph-based tools for analyzing singular points on algebraic surfaces, highlighting their historical evolution and classification challenges.

Contribution

It introduces the historical context and the conceptual development of graph calculus for studying algebraic surface singularities.

Findings

Graphs evolved from descriptive tools to formal calculus methods.

Classification problems spurred the development of specialized graph techniques.

The paper bridges mathematical and philosophical perspectives on these tools.

Abstract

In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several classification problems made it necessary to draw them, leading to the elaboration of a special kind of calculus with graphs. This non-technical paper is intended to be readable both by mathematicians and philosophers or historians of mathematics.