The Carpenter's Mind Unbound by Tools: The Structural Necessity of Meta-Theory for Intelligence

This paper establishes the structural necessity for intelligence theory to possess a meta-theoretical framework that transcends specific formal tools. Intelligence operates universally across natural understanding, formal construction, artificial creation, and social practice. If intelligence were constrained by particular tools such as calculus, set theory, or logical operators, it would become dysfunctional in domains where those tools are inapplicable. Therefore, intelligence theory must encompass all tools while remaining subordinate to none. We present Noology as the unique formal system satisfying this requirement. Noology adopts consistency (Consensus) as an absolute requirement, employs mathematical language to describe consistency, yet remains independent of any mathematical axiomatic system. The system is built on three primitive notions: Ordo (non-commutative operational order), Consensus (absence of contradiction), and Arbitrium (external configuration authority). This paper demonstrates that the universal operability of intelligence structurally necessitates a meta-theoretical architecture. We introduce a three-tier framework: Tier-1 (Noology) as the regulative layer, Tier-2 (Cognitional Mechanics with M₃(ℂ) algebra) as the executive substrate, and Tier-3 (MUT/GUT) as the projective display layer for physics and mathematics. We prove that conventional tool-specific theories—logical positivism, computationalism, and physicalism—fail because they recognize only one element of the tripartite structure while excluding others. Through phenomenological classification of four intelligence domains and dominance analysis, we show that all intelligent operations are weighted combinations of Ordo, Consensus, and Arbitrium. The meta-theoretical structure generates a structural boundary for artificial systems: since AI cannot originate Arbitrium or bear irreversible responsibility, Artificial General Intelligence is definitionally incoherent. Intelligence amplification remains viable by preserving human Arbitrium as the source of configuration while employing AI as an operational instrument. This framework redirects cognitional inquiry from simulating autonomy in machines to understanding the geometry of human judgment itself.