Mathematics

This module develops the formal mathematical foundations of relationality. The work is the construction and analysis of a specific hierarchy of mathematical structures that instantiate the claim that relations are prior to entities.

The hierarchy proceeds through three levels:

1. Semiotic Universe — a complete Heyting algebra with modal closure and Heyting-comonadic trace, extended with typed lambda calculus and three closure operators (semantic, syntactic, fusion) whose composite yields a least fixed point. This is the pure mathematical bedrock: the structure of signs, meaning, and formal inference.

2. Interactive Semioverse — extends the Semiotic Universe with external handles (Things), interaction terms, semantic closures of interactions (footprints), failure semantics, provenance, and sheaf semantics. This is the structure of how signs interact with external reality.

3. Agential Semioverse — extends the Interactive Semioverse with agent profiles (role, goals, policy, skills, tools, memory), tool signatures, skill calculus, and execution semantics. This is the structure of how agents act within the semioverse.

The tools drawn upon include higher topos theory, homotopy theory, sheaf theory, modal logic, category theory, and stratified directed infinity-sites.

Formal objects

• Mathematical Universes
  • Semiotic Universe
  • Interactive Semioverse
  • Agential Semioverse

Topic areas

• Foundations
• Logic
• Modal Logic
• Category Theory
• Homotopy Theory
• Higher Topos Theory
• Sheaf Theory
• Probability
• Statistics

Related

• Relationality — the philosophical project this mathematics instantiates
• Technology — the ASR specification, which implements this formal model as a repository