Simple Matroids and Alfred North Whitehead's theory of dimension (1906)

TL;DR

This paper establishes a correspondence between simple matroids and Whitehead's generalized theory of dimension, linking combinatorial and geometric frameworks from early 20th-century mathematics.

Contribution

It demonstrates that finite, phi-maximal geometrical systems in Whitehead's sense are equivalent to simple matroids, bridging two mathematical concepts.

Findings

Finite, phi-maximal Whitehead systems correspond to simple matroids.

Every simple matroid can be viewed as a phi-maximal Whitehead system.

The work generalizes Whitehead's three-dimensional axiom to finite dimensions.

Abstract

We give a correspondence between simple matroids and a reconstruction of Alfred North Whitehead's theory of dimension, as developed in "On Mathematical Concepts of the Material World" (1906). In brief, if a geometrical system in the generalized sense of Whitehead has finite ground set and is phi-maximal, then it is a simple matroid. Here "generalized" means that Whitehead's three-dimensional axiom is replaced by finite-dimensionality. Conversely, every simple matroid is a phi-maximal geometrical system in the generalized sense of Whitehead.