Arbran mathic philosophy of Protas, corresponding to Platonism or, more specifically, Platonic realism. Protism states that the Hylaean Theoric World is the domain of cnoöns - pure geometric forms, theorems and ideas, and that theors in the Arbran Causal Domain are influenced by their perception of such cnoöns. The movement of pure ideas from the Hylaean Theoric World is called Hylaean flow. The concept is illustrated in Calca 3: Complex Versus Simple Protism.
Simple Protism involves two cosmi; one more purely Hylaean than the other. Simple Protism is illustrated by two boxes, one representing the Hylaean Theoric World (HTW) and the other the Arbran Causal Domain (ACD). An arrow leads from the HTW to the ACD, representing the flow of information or causality from one cosmos to another. The flow is unidirectional. This type of diagram is called a Directed Acyclic Graph (DAG).
Complex Protism, developed by Suur Uthentine and her fid Fraa Erasmas (the namesake of protagonist Erasmas) at Saunt Baritoe's in the fourteenth century A.R., involves multiple cosmi, arranged in some order from more purely Hylaean to less purely Hylaean. With Complex Protism, Hylaean flow exists between multiple cosmi in any topologies that are acyclical. Hylaean theorics can only flow from more purely Hylaean cosmi to less purely Hylaean cosmi. The calca gives examples of some characteristic topologies such as the Freight Train, the Firing Squad and the Strider. A fully generalized DAG is called the Wick and includes multiple HTW's and inhabited cosmi, always arranged in an acylclical (non-recursive) topology.
Arbre as a Model of Protism
Given that Mathic Society is more "Hylaean" than Saecular society, the world of Arbre is in the form of a Directed Acyclic Graph. Each concent relates to the Saecular world in a state of Simple Protism. The Mathic Society collectively relates to the Saecular world through the influence of all the concents. This relationship could be represented as a Firing Squad type of DAG. The Millenarian, Centenarian, Decenarian and Unarian maths of any single concent are arranged in order of decreasing Hylaean purity as a Freight Train type of DAG. When the Firing Squad and Freight Train DAG's are combined and interconnected, and the relationships between concents are included, they form a more generalized DAG called the Reverse Delta, representing all of Arbre and its concents. The flow of information between the concents and the Saecular world is unidirectional, except at Apert. The rules that separate Mathic Society from Saecular act to restrict this non-Hylaean flow.
This model assumes that Millenarian maths are generally more purely Hylaean. In other words, their work reveals more perfect geometric forms, theorems and ideas ( cnoöns). This may not have been the case with the Millenarian math where one Apert revealed a new arboreal post-human primate species. It certainly was true of the Millenarian math at Saunt Edhar where Fraa Jad made polycosmic manipulation his vocation.
The model is not purely representative of the state of affairs on Arbre, since the Saecular Society is able to interact and influence the Mathic Society during Apert (normally) and voco (during an emergence). Likewise at Apert the less purely Hylaean maths can have an upstream causal relationship with the maths that have longer attention spans. However, this "imperfection" of the model does suggest recursion, as seen in the ability of Rhetors and Incanters to manipulate the Hylaean flow between cosmi, which Fraa Jad does during the invasion of Daban Urnud.