From Rotations to Teleology: A Quaternionic Model of Semantic Dynamics

This article extends our earlier work From Rotations to Spirals: A Matrix Model of Sentence-Level Semantic Dynamics by introducing the teleological component that had previously been left aside. In the quaternionic extension of the sentence matrix formalism, semantic states are described by a real dimension of actualization and three orthogonal imaginary axes. The axes i and j account for possibilities and goal-orientations, while the real component reflects their stabilization in discourse. Teleological attractors are represented as imaginary targets whose influence is strongest in the distance and vanishes at the point of realization. The resulting dynamics differ fundamentally from gravitational laws: rather than a monotone pull towards what already exists, teleology organizes through the absence of what is not yet realized.Formally, the model is expressed as a drift induced by the gradient of a teleological potential, with feedback on the real component and underdamped oscillations around the attractor. Beyond individual goals, the third imaginary axis k is introduced as the dimension of collective values and norms. These higher-order attractors shape the entire semantic landscape and stabilize discourse over extended timescales. Their detailed mathematical treatment will be developed, however, in a subsequent article.