Infinite graphs in systematic biology, with an application to the species problem

TL;DR

This paper explores the implications of assuming an infinite biosphere in systematic biology, using infinite graph theory to shed light on the species problem and proposing new mathematical insights.

Contribution

It introduces a mathematical framework based on infinite graphs to analyze biological classification and addresses foundational questions about biology's role in generating new theorems.

Findings

Mathematically formalizes the assumption of an eternal biosphere

Provides a dualization of a law by Knight and Darwin

Sketches a decomposition involving internodons

Abstract

We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels' question, "Can biology lead to new theorems?"