This post will begin with the causal set model of cosmology, an approach which can be modeled by partially ordered sets (directed acyclic graphs) of spacetime “events.” Actually, it will begin with a generalization of this model with category theory leveraging concepts such as functors, natural transformations, and colimits. We assume that in the limit, such relational “augmented abstract block diagrams,” borrowing Rosen’s terminology, exhibit maximal entailment- that is, few morphisms will themselves be unentailed by yet other morphisms when compared with the cardinality of the diagram.
The post will attempt to bridge the gap between such a static, category-theoretic, relational and observer-independent formalism of explanation (I daresay, reality itself) and the dynamic, ergodic, biosemiotic model of self-organization that models evolution by the communication of learning agents (“anticipatory systems” in Rosen’s parlance, or autopoietic systems) performing two tasks: 1) pattern recognition / data compression / model building, and 2) maximization of Fisher information or, in anthropomorphic terms, maximizing future possibilities. In effect, we will be reconciling Tegmark’s Mathematical Universe Hypothesis (i.e. that relations between numbers — in an abstract sense, patterns themselves — are the only things that objectively exist in “reality” irrespective of the presence of an observer or the names issued to their entities) with the learning-agent-based models of organization and emergence provided by Markovian biosemiotics.
It seems the first task is to complete the analogy binding data compression to estimation theory (i.e., the entropy maximization that takes place during data compression is mathematically isomorphic to Fisher information maximization… the strategic positioning of a sensor in order to be able to observe recursion- i.e., differential equations- in sampled data). Of course, Frieden’s Extreme Physical Information provides us with a model for how the intentional agents communicate with one another and the open systems in which they reside. With a complete analogy, the appearance of intentionality (that is, any localized combination of model building and maximization of future possibilities) can then be understood as a natural result of the process of communication. Communication, here, should be understood as the transmission of probability distributions from one location to another such that the two distributions become identical. The meaning of “location” and “transmission” is what we will need to derive from the static, category-theoretic model of cosmology.
The process of communication and conversation can be understood as a movement toward thermodynamic equilibrium; consider a Chinese tangram puzzle — the individual shapes can be shuffled about but will only fall into place corresponding to an emergent configuration that was predetermined by their interaction with each other and with the environment. The static geometry of the pieces and the constraints imposed by their environment predetermined their resting places. In a sense, the pieces of the tangram had no say in their ultimate distribution and ordering. It was only the way their temporal form interacted with environment that determined their fate- neither the form of any entity nor the environment are alone sufficient to determine outcome. This post is trying to understand how the final configuration of the tangram comes about from the shaking of the pieces… what causes the appearance of the shaking (appearance of dynamics as measured by a consensus of first-person, subjective perspectives) given some third-person perspective of a static category-theoretic augmented abstract block diagram?
To transition from a maximally entailed augmented abstract block diagram specification in category theory — complete with functors and natural transformations which themselves enable learning and indeed formally model the concept of modeling — to the thermodynamic equilibrium process of communicating probability distributions, we refer to an anecdote. Here, I suggest the reader to investigate the discussion of layered architectures using Algebraic Higher Order (AHO) nets in the context of formally modeling mobile ad-hoc networks (MANETs). Indeed, such self-configuring networks of intelligent radios are quite analogous to what I’ve described above as “learning-agent-based models of organization and emergence provided by Markovian biosemiotics.” The layered architectures using AHO nets appear to provide the sought-after analogy with category theory. Introductory details may be found in “Formal Modeling and Analysis of Mobile Ad Hoc Networks and Communication Based Systems using Graph and Net Technologies” by Kathrin Hoffmann, Hochschule für Angewandte Wissenschaften, Hamburg, Germany. A related text entitled “Petri Net Technology for Communication-Based Systems” was published by Springer in 2003. Ms. Hoffman’s analysis of concurrency and partial order in the context of applying category theory to distributed configuration of intelligent radios reminds me of the recent articles I shared by Tommaso Bolognesi which discuss the use of process algebra in the context of algorithmic causal sets and observers: “Event patterns: from process algebra to algorithmic causal sets” and “Internal observers in causet-based algorithmic spacetime.”
MANET technology is now being investigated and developed in order to enable the emerging “Internet of Things” and to respond to the growth of complex information networks. It seems this commercial source of funding may inadvertently help answer open questions in cosmology, biology, and general intelligence.
Lowly Ocean 