Mathematical self-determination theory I: Real representation

In two parts, MSDT1 this paper and MSDT2 the follow-up paper, we treat the topic of mathematical self-determination theory. MSDT1 considers the real representation, MSDT2 the affine space representation. The aim of the two papers is to lay the mathematical foundations of self-determination motivation theory. Self-determination theory was proposed by Deci and Ryan, which is a popular theory of motivation. The fundamental concepts are extrinsic and intrinsic motivation, amotivation, their type of regulation, locus of causality, and especially, self-determination. First, we give a geometric description of its concepts for the regulated case (no amotivation), as the unit 1-simplex. Thereby, we derive a symmetric definition of self-determination. Second, we extend the geometric description to the regulated and unregulated case, based on a more general ternary model, in internal motivation, external motivation, and amotivation. We define gradations of amotivation (and motivation), as 1-simplexes parallel to the unit 1-simplex. The ternary representation implies the types of strong, weak, and general self-determination, as partial orders on the motivation space. Third, we study the order, lattice, and algebraic properties of self-determination. In a version of polar coordinates, strong self-determination turns out to be a complete lattice on angular line segments, weak self-determination is a complete lattice on radial line segments, and general self-determination entails a complete lattice on the entire motivation space. In addition, the modified polar coordinates are employed to obtain necessary and sufficient conditions for strong, weak, and general self-determination. We propose measures for the strength of an ordinal dependency in self-determination, which are partial metrics on the motivation space.