A Geometric Theory of Cognition

TL;DR

This paper introduces a unified geometric framework for cognition, modeling diverse mental processes as emergent from a single mathematical principle involving Riemannian manifolds and gradient flows.

Contribution

It presents a novel mathematical model that unifies various cognitive functions through geometric principles, avoiding modular architectures and explaining dual-process effects naturally.

Findings

Cognitive phenomena emerge from gradient flows on a Riemannian manifold.

Dual-process effects are explained by metric-induced anisotropies.

Simulations demonstrate the model's ability to replicate canonical cognitive tasks.

Abstract

Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified mathematical framework in which diverse cognitive processes emerge from a single geometric principle. We represent the cognitive state as a point on a differentiable manifold endowed with a learned Riemannian metric that encodes representational constraints, computational costs, and structural relations among cognitive variables. A scalar cognitive potential combines predictive accuracy, structural parsimony, task utility, and normative or logical requirements. Cognition unfolds as the Riemannian gradient flow of this potential, providing a universal dynamical law from which a broad range of psychological phenomena arise. Classical dual-process effects--rapid intuitive responses and slower deliberative reasoning--emerge naturally from metric-induced anisotropies that generate intrinsic time-scale separations and geometric phase transitions, without invoking modular or hybrid architectures. We derive analytical conditions for these regimes and demonstrate their behavioural signatures through simulations of canonical cognitive tasks. Together, these results establish a geometric foundation for cognition and suggest guiding principles for the development of more general and human-like artificial intelligence systems.