Alfred Binet and the concept of heterogeneous orders

In a comment, hitherto unremarked upon, Alfred Binet, well known for constructing the first intelligence scale, claimed that his scale did not measure intelligence, but only enabled classification with respect to a hierarchy of intellectual qualities. Attempting to understand the reasoning behind this comment leads to an historical excursion, beginning with the ancient mathematician, Euclid and ending with the modern French philosopher, Henri Bergson. As Euclid explained (Heath, 1908), magnitudes constituting a given quantitative attribute are all of the same kind (i.e., homogeneous), but his criterion covered only extensive magnitudes. Duns Scotus (Cross, 1998) included intensive magnitudes by considering differences, which raised the possibility (later considered by Sutherland, 2004) of ordered attributes with heterogeneous differences between degrees ("heterogeneous orders"). Of necessity, such attributes are non-measurable. Subsequently, this became a basis for the "quantity objection" to psychological measurement, as developed first by Tannery (1875a,b) and then by Bergson (1889). It follows that for attributes investigated in science, there are three structural possibilities: (1) classificatory attributes (with heterogeneous differences between categories); (2) heterogeneous orders (with heterogeneous differences between degrees); and (3) quantitative attributes (with thoroughly homogeneous differences between magnitudes). Measurement is possible only with attributes of kind (3) and, as far as we know, psychological attributes are exclusively of kinds (1) or (2). However, contrary to the known facts, psychometricians, for their own special reasons insist that test scores provide measurements.