Presenter:  Ehtibar Dzhafarov
Presentation type:  Talk
Presentation date/time:  7/26  11:20-11:45
 
A New Geometry Of Subjective Stimulus Spaces
 
Ehtibar Dzhafarov, Purdue
 
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.