Seven Traditions

A system is a relation on abstract sets: S ⊆ ×{Vᵢ : i ∈ I}. In the canonical input-output case: S ⊆ I × O. The system is the relation — not a thing that has a relation.

The most general and parsimonious base. The decomposition theorem — systems of dimension greater than 2 require internal states — is a foundational result not derived elsewhere. Every other definition on this page can be mapped back to this one.

A system is the 7-tuple Σ = ⟨T, X, Ω, Y, Q, δ, λ⟩ — time set, input values, admissible input functions, output values, states, state transition function, and readout function. A generalized Moore machine over continuous or discrete time.

Admissible input function space Ω as a first-class component — time and trajectory structure are explicit. Coupling theory with closure proofs. Subsumes Turing machines, sequential machines, and ODEs in one definition.

A system is a structured reconstruction from observed variables and their relationships. Given a set of variables V with ranges S₁, …, Sₙ, a system at the generative level is G = ⟨V, R⟩ where R ⊆ S₁ × … × Sₙ is the constraint relation on their joint state space.

The only full account of inverse system identification — given observed behavior, what system generated it? His epistemological hierarchy (source → data → generative → structure → metasystem) provides a rigorous ladder from raw observation to structural explanation. Integrated uncertainty measures into systems theory.

A system is the 4-tuple σ = ⟨C, E, S, M⟩ — composition (parts), environment (what it interacts with), structure (relations among parts and between parts and environment), and mechanism (the processes by which the system functions and changes).

The only account that includes mechanism as an ontological primitive — not just what a system does or is made of, but how it works causally. Emergence defined precisely: properties of the whole not present in the parts, derivable from M acting on C under S.

A system is a cardinal distinction on a variety of dimensional distinctions. Formally: S ⊆ ∏ Xᵢ where the dimensional distinctions (the Xᵢ) are constructivist acts of observation, and the cardinal constraint (S ⊊ X) is classically Boolean.

The only account of what kind of functional relation inhabits a system — rules (contingent, selected) vs. laws (necessary, discovered). Bridges classical realism and constructivism. Proves that active control systems necessarily involve semantic relations, connecting systems theory to biosemiotics.

A system is the 8-tuple Sₗ = {C, N, I, B, K, H}ₗ — components, networks, interfaces, boundary, knowledge (stocks), and hierarchy, indexed by level. Grounded in physical process quantities: stocks, flows, and feedback topology.

The only account grounded in physical process quantities. The strongest alignment with system dynamics and simulation practice. The only one with a developed pedagogy — his textbook is how most contemporary systems scientists first encounter formal definitions.

A deterministic system is S = ⟨State, Out, In, expose, update⟩ — a lens in a cartesian category. The expose function reads state into outputs; the update function takes state and inputs to the next state. Systems compose via lens composition.

The only account that proves a general compositionality theorem — behaviors of composite systems computed from component behaviors. The doctrine framework is the only meta-level account of why definitions differ. Lens identification makes composition a first-class mathematical operation.