Philosophy-Guided Mathematical Formalism for Complex Systems Modelling

💡 Research Summary

The paper presents a rigorous mathematical formalism for the “allagmatic method,” a framework for modeling complex systems that is inspired by philosophy (Simondon’s individuation, Whitehead’s organism) and cybernetics. The authors argue that traditional analytical approaches based on differential equations are inadequate for capturing the heterogeneous, nonlinear, and locally interacting components characteristic of complex systems. Instead, they propose to describe a system as a tuple consisting of at least one spatial structure and one temporal operation, denoted as a (s + o)-tuple SM = (s₁,…,s_s, o₁,…,o_o).

The spatial part (structures) includes:

• An entity tuple E = (ê₁,…,ê_e), where each entity’s state belongs to a finite set Q (e.g., {0,1}).
• A milieu tuple M = (M̂₁,…,M̂_e), where each M̂_i is an adjacency list of the neighbors of entity ê_i.
• An update‑rule tuple U that stores the logical description of the local transition (e.g., a truth table for cellular automata).
• An adaptation‑rule tuple A and an adaptation‑end tuple P to capture learning or evolutionary processes.
  Additional auxiliary structures (˜s_i) may be added as needed.

The temporal part (operations) comprises:

• An update function φ : Q^{m+1} → Q that maps the current state of an entity and its neighbors to the next state, parameterised by U and the current time step t̄.
• Optionally, an adaptation function ψ ( ḡ, A, P, l ) that implements evolutionary computation, gradient‑based learning, or other optimisation schemes, where ḡ is the current adaptation iteration and l is a loss tolerance.
• Further optional operations (˜o_j).

By formalising all components as tuples, sets, and functions, the authors achieve a one‑to‑one correspondence with their C++ implementation, which uses template meta‑programming to make the entity state type generic and stores the adjacency structure as dynamic vectors (or adjacency lists). The “virtual regime” corresponds to a model with no concrete data types, the “metastable regime” to a model where types and sizes are fixed, and the “actual regime” to a running simulation.

Two concrete case studies illustrate the power of the formalism. First, cellular automata are built by defining E, M, U (and optionally ψ for evolutionary optimisation). The update rule U is encoded as a truth table; an evolutionary algorithm searches the space of possible U to achieve a target output pattern. Second, artificial neural networks are represented with entities as neurons, milieux as weighted adjacency lists, U as activation/propagation rules, and ψ as back‑propagation learning. Both models are instantiated from the same meta‑model, and the authors prove a theorem establishing conditions under which a cellular automaton and a neural network are mathematically equivalent (i.e., they realise the same state‑transition function φ under appropriate encodings of U and ψ).

The formalism yields several notable benefits. Philosophical concepts of individuation and organism are mapped onto precise mathematical structures, allowing the description of systems that evolve both spatially and temporally. The explicit definition of update and adaptation rules enables formal verification, reproducibility, and automated generation of models. The C++ template implementation guarantees type safety while allowing the seamless transition between abstract (virtual), concrete (metastable), and executable (actual) regimes. Moreover, by unifying disparate modelling paradigms—cellular automata and neural networks—within a single meta‑model, the approach facilitates systematic comparison, hybridisation, and extension to other domains such as social or economic systems.

In conclusion, the paper delivers a foundational mathematical language for the allagmatic method, fills a gap left by earlier informal descriptions, and opens avenues for rigorous theoretical studies, automated model synthesis, and cross‑disciplinary applications. Future work is suggested on richer adaptation mechanisms, complexity analysis, and broader validation on real‑world complex systems.