Most of mathematics has its roots in physics, having grown from the internal logic of physics problems. By contrast, the mathematics used in psychology is primarily "ready-made": it is adopted and adapted instead of growing from the internal logic of substantive issues. A recent development in Fechnerian Scaling breaks with this tradition by constructing a mathematical theory specifically aimed at the oldest problem of scientific psychology: the reconstruction of subjective distances among stimuli from their discriminability. The central notion of the theory is that of a dissimilarity function, which very likely captures all empirical measures of dissimilarity, whether "direct or computed from discrimination probabilities. The subjective distance between stimuli x and y is defined by the smallest amount of accumulated dissimilarity as one "moves" from x to y and back through intermediate stimuli. If stimuli form an arc-connected space in the topology induced by a dissimilarity function, the latter is used to construct a new mathematical theory for computing lengths of continuous paths. Most of the fundamental results of the traditional metric-based path length theory (additivity, lower semicontinuity, etc.) turn out to hold in the general dissimilarity-based path length theory. The triangle inequality and symmetry are not therefore essential for these results. In special arc-connected spaces (e.g., Euclidean n-spaces) the theory specializes to traditional versions of Finsler geometry. The latter, however, is arrived at rather than borrowed ready-made, distinguishing thereby the present development from the previous uses of differential geometry in psychology. |