Interpretation as Linear Transformation: A Cognitive-Geometric Model of Belief and Meaning

TL;DR

This paper introduces a geometric model of belief and meaning, representing agents as personalized vector spaces and beliefs as structured vectors, to analyze communication, influence, and belief dynamics across diverse cognitive systems.

Contribution

It presents a novel geometric framework for modeling belief transmission and influence, emphasizing structural compatibility over shared information.

Findings

Beliefs are transmitted via linear maps that preserve structure.

Miscommunication occurs when beliefs fall into null spaces of interpretation maps.

Leadership is characterized by representational reachability, not persuasion.

Abstract

This paper develops a geometric framework for modeling belief, motivation, and influence across cognitively heterogeneous agents. Each agent is represented by a personalized value space, a vector space encoding the internal dimensions through which the agent interprets and evaluates meaning. Beliefs are formalized as structured vectors-abstract beings-whose transmission is mediated by linear interpretation maps. A belief survives communication only if it avoids the null spaces of these maps, yielding a structural criterion for intelligibility, miscommunication, and belief death. Within this framework, I show how belief distortion, motivational drift, counterfactual evaluation, and the limits of mutual understanding arise from purely algebraic constraints. A central result-"the No-Null-Space Leadership Condition"-characterizes leadership as a property of representational reachability rather than persuasion or authority. More broadly, the model explains how abstract beings can propagate, mutate, or disappear as they traverse diverse cognitive geometries. The account unifies insights from conceptual spaces, social epistemology, and AI value alignment by grounding meaning preservation in structural compatibility rather than shared information or rationality. I argue that this cognitive-geometric perspective clarifies the epistemic boundaries of influence in both human and artificial systems, and offers a general foundation for analyzing belief dynamics across heterogeneous agents.