Simple matroids and Alfred North Whitehead’s theory of dimension (1906)

Abstract

We give a correspondence between simple matroids and a reconstruction of Whitehead's theory of dimension, as found in On Mathematical Concepts of the Material World (MW), published in 1906 [8]. In brief, if \((E,\phi)\) is a geometrical system in the generalized sense of Whitehead, where \(E\) is finite and \(\phi\)-maximal, then \(E\) is a simple matroid. (Generalized here means that Whitehead's three-dimensional axiom has been replaced with finite-dimensionality.) Within the context of the generalized geometrical axioms, \(\phi\)-maximal subsets correspond with simple submatroids. Conversely, every simple matroid is a \(\phi\)-maximal geometrical system in the generalized sense of Whitehead.