Foundations for Mathematical Systems Science

 Foundations for Mathematical Systems Science: Seven Traditions, One Theorem🔗

Shingai Thornton

Systems theorists, scientists, and engineers have produced numerous formal definitions of "system" since the mid-20th century. These definitions were developed independently, in different decades, using different notation and terminology. Yet they are remarkably compatible. This document investigates how.

The main result: the irreducible categorical content shared by all seven is a single morphism — the walking arrow category 2. A system, in the sense common to every tradition, is relations that depend on things. Everything else — environment, boundary, state, input, output, time, mechanism, feedback — is tradition-specific elaboration.

Seven traditions. One shared categorical structure. Machine-verified in Lean 4.

Contents

  1. 1. Introduction
  2. 2. Seven Definitions, One Tradition
  3. 3. The Commuting Triangle
  4. 4. The Thermostat — One Example, Three Frameworks
  5. 5. Categorification: Making Relationships Visible
  6. 6. The Open Frontier
  7. 7. References