There was a time in the 1960’s and 70’s when mathematical psychology was considered an important field. One of the defining tomes produced in that era is the three volume Foundations of Measurement by Krantz, Suppes, Luce and Tversky. Measurement is a particularly tricky problem in the study of the mind. There are two sub-questions here:
What are we (i.e., the scientist studying human or animal psychology) measuring?
What is the mind of the human or animal taking as its starting point in its engagement with the world?
The latter question is the more interesting question for us. For example, most studies of vision assume that the starting point of vision is an array of pixels on a retina or a video camera. Once you take pixels as the starting point, it is clear that tremendous amount of processing is necessary before interesting outputs, such as 3D shape, can be recovered from the retinal inputs. At the other extreme lies Gibson’s theory of direct perception. Gibson argues that the ambient optic array is directly sampled by the moving animal or human being. In other words, what we “measure” gives us all the information about the structure of objects and surfaces in the world. The underlying theory of measurement has a direct impact on our theories of sensory and conceptual systems. To the extent we have an impoverished view of measurement, we have an impoverished view of the mind.