Interpretation as Linear Transformation: A Cognitive-Geometric Model of Belief and Meaning
Chainarong Amornbunchornvej
TL;DR
The paper introduces a cognitive-geometric framework in which each agent has a personalized value space and beliefs are modeled as structured vectors (abstract beings) within that space. Communication is mediated by linear interpretation maps, and a belief survives transfer only if it avoids a recipient's null space, providing a structural criterion for intelligibility, miscommunication, and belief death. It unifies belief dynamics, motivational drift, and innovation through geometric concepts such as forward/backward consistency, local coherence, and convex-hull expansion, with No-null-Space Leadership redefining leadership as a property of representational reachability rather than authority. The framework offers insights for human and AI alignment, conceptual spaces, marketing, social identity, and emotion, while acknowledging limitations related to linearity and empirical estimation. Overall, it reframes influence and understanding as geometry- rather than content-driven, enabling principled analysis of belief transmission across heterogeneous minds.
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.