A mathematical framework of intelligence and consciousness based on Riemannian Geometry

Abstract

Understanding intelligence is a central pursuit in neuroscience, cognitive science, and artificial intelligence. Intelligence encompasses learning, problem-solving, creativity, and even consciousness. Recent advancements in geometric analysis have revealed new insights into high-dimensional information representation and organisation, exposing intrinsic data structures and dynamic processes within neural and artificial systems. However, a comprehensive framework that unifies the static and dynamic aspects of intelligence is still lacking. This manuscript proposes a mathematical framework based on Riemannian geometry to describe the structure and dynamics of intelligence and consciousness. Intelligence elements are conceptualised as tokens embedded in a high-dimensional space. The learned token embeddings capture the interconnections of tokens across various scenarios and tasks, forming manifolds in the intelligence space. Thought flow is depicted as the sequential activation of tokens along geodesics within these manifolds. During the navigation of geodesics, consciousness, as a self-referential process, perceives the thought flow, evaluates it against predictions, and provides feedback through prediction errors, adjusting the geodesic: non-zero prediction errors, such as learning, lead to the restructuring of the curved manifolds, thus changing the geodesic of thought flow. This dynamic interaction integrates new information, evolves the geometry and facilitates learning. The geometry of intelligence guides consciousness, and consciousness structures the geometry of intelligence. By integrating geometric concepts, this proposed theory offers a unified, mathematically framework for describing the structure and dynamics of intelligence and consciousness. Applicable to biological and artificial intelligence, this framework may pave the way for future research and empirical validation.

Figures (2)

• Figure 1: The flow chart illustrates the transition of consciousness cycle at t to the next cycle at $t+\Delta t$: perception, prediction, evaluation, and adjustment. The left side represents the process at time $t$, where external input $I(t)$ is integrated with prediction $g(t)$ to obtain the perception $f(t)$. The evaluation $f(t) - g(t)$ is used to adjust the thought flow. The right side represents the process at time $t + \Delta t$, showing the continuous loop of feedback and adaptation.
• Figure 2: A simplified representation of manifolds in intelligence space. A. In the intelligence space, tokens are coherently connected to form smooth and stable manifolds. Some manifolds remain isolated, indicating that they are not explainable by others. No geodesic paths exist between these isolated manifolds. B. As learning progresses, the manifolds expand through the addition of new tokens. Consequently, more manifolds become interconnected, forming larger smooth and stable structures that facilitate longer and more complex geodesic navigation along them.