Theory of Knowledge as Recursive Fibration

J. Rogers, SE Ohio, 14 Jul 2025, 1319

1. Introduction

This document presents a unified epistemological framework in which all knowledge, including mathematics, physics, and social constructs, is modeled as a system of recursive fibrations. At the heart of this framework is the insight that knowledge is constructed by projecting universal, dimensionless states through structured conceptual spaces defined by measurement axes. Crucially, the very process of expanding, redefining, or coordinating these axes is itself modeled within the same system, enabling a theory that is self-referential, extensible, and socially aware.

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2. Categories of Knowledge Construction

Let:

• $\mathcal{S}_u$ be the category of Universal States (dimensionless, context-free knowledge states).

• $\mathcal{A}$ be the category of Conceptual Axes (abstract measurement axes without unit systems).

• $\mathcal{U}$ be the category of Unit Systems (coordinate charts assigning scales to axes).

Define a total category $\mathcal{E}$ of Structured Knowledge objects, with a fibration:
$p : \mathcal{E} \to \mathcal{B} \quad \text{where} \quad \mathcal{B} = \mathcal{A} \times \mathcal{U}$
Each fiber $p^{-1}(A, U)$ consists of knowledge projections under conceptual axis set $A$ and unit system $U$.

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3. Formula Forge Functor

There exists a functor:
$\Lambda : \text{Hom}_{\mathcal{S}_u}(X, Y) \to \text{Hom}_{\mathcal{E}}(\Lambda(X), \Lambda(Y))$
which maps dimensionless, scale-free relationships into coordinate-specific physical laws (e.g., $E = f \cdot h$, $F = G M_1 M_2 / r^2$, etc.). The structure of physical constants ($h, c, G, k_B$) appears as cocycle data ensuring commutativity and unit coherence in the lifted morphisms.

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4. Recursive Fibration Layer

Define a higher-order fibration:
$q : \mathcal{B} \to \mathcal{C}$
where $\mathcal{C}$ is the category of Epistemic Meta-Structures, each object of which is a conceptual framework permitting certain kinds of conceptual axis formation, deletion, transformation, and social enforcement.

Each object in $\mathcal{C}$ encodes a social or philosophical structure (e.g., physics, law, culture, religion), and each morphism is a transformation of conceptual freedom (e.g., revolutions, reforms, paradigm shifts).

The composite fibration:
$\mathcal{E} \xrightarrow{p} \mathcal{B} \xrightarrow{q} \mathcal{C}$
models knowledge as a structure projected through a conceptual layer ($\mathcal{B}$) which itself is governed by broader socio-epistemic conditions ($\mathcal{C}$).

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5. Soft and Hard Sciences as Fiber Cohesion

A domain is "hard" when the fiber over $(A, U)$ in $\mathcal{E}$ is consistent and closed under transformation.
A domain is "soft" when its unit or axis structures shift easily, or when morphisms in $\mathcal{B}$ or $\mathcal{C}$ are frequent and ambiguous.

Thus, the distinction between physics and sociology is not in the content of their laws, but in the rigidity of their conceptual fibrations.

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6. Social Control and Knowledge Governance

Control over measurement axes ($\mathcal{A}$) or permitted unit systems ($\mathcal{U}$) constitutes control over what knowledge is recognized, formalized, or validated. Cultural norms, laws, institutions, and educational systems define permissible regions of $\mathcal{B}$, and by extension, what projections into $\mathcal{E}$ are deemed legitimate.

Paradigm shifts are thus modeled as nontrivial morphisms in $\mathcal{C}$ inducing base change in $\mathcal{B}$, and requiring re-lifting of structure into $\mathcal{E}$.

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7. Self-Reference and Meta-Epistemology

By incorporating axis construction as a modeled process, the theory becomes self-referential: it models not just knowledge, but the conditions of its own construction. This supports a recursive understanding of learning, where conceptual systems evolve via internal axis expansion and are subject to constraints imposed by their embedding meta-fibrations.

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8. Applications and Implications

• Physics: Derive laws via dimensional analysis as structured coordinate projections.

• Law and Politics: Model interpretive frameworks, legal axioms, or ideological units.

• Education: Understand curricular domains as conceptual bases with controlled access.

• AI and NLP: Treat conceptual embeddings as categorical fibrations over linguistic spaces.

• Myth, Culture, and Art: Analyze symbolic structures as coordinate systems on the substrate of shared human experience.

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9. Conclusion

This framework redefines knowledge not as a collection of facts, but as a process of structured projection through recursive layers of conceptual space along scales we create and apply arbitrary scales to. By unifying the mathematics of physics with the socio-political machinery of epistemology, it opens a new path toward both understanding and transforming the nature of human inquiry.