Pacotte tree networks, graph theory and projective geometry

TL;DR

This paper revisits Julien Pacotte's original concept of tree networks, clarifying their properties using graph theory and projective geometry, and explores how they could reconstruct all of mathematics from empirical structures.

Contribution

It redefines Pacotte's tree networks beyond common interpretations and demonstrates their potential to serve as foundational structures for reconstructing mathematics.

Findings

Pacotte's tree networks are more general than traditional trees.

These networks can be formalized using graph theory concepts.

They can be used to reconstruct mathematical foundations from empirical data.

Abstract

The notion of tree network has sparked renewed interest in recent years, particularly in computer science and biology (neural network). However, this notion is usually interpreted in an extremely restrictive way: essentially linked to data processing, today tree networks are hybrid network topologies in which star networks are generally interconnected via bus networks. These networks are, most often, hierarchical and regular, and each of their nodes can have an arbitrary number of child nodes. At the outset, however, the notion of tree network, introduced in 1936 by Belgian physicist Julien Pacotte, was quite different: more general and, at the same time, more constrained, it should also serve an ambitious objective: the reconstruction of mathematics from concrete empirical structures. Usually poorly commented on and poorly understood (especially by philosophers), it had no real posterity. In this article, we first try to clarify this notion of "tree network" in the sense of Julien Pacotte, which makes it possible to eliminate the bad interpretations to which this notion has given rise. To this end, we use the language and concepts of graph theory and formalize the main properties of these networks which, contrary to popular belief, are not, in general, trees. In a second part, we then try to follow and explain, step by step, how Pacotte intended, using concepts borrowed from projective geometry, to reconstruct all of mathematics from such a network.