For obvious reasons, psychologists aspire to measure psychological attributes. However, it is reasonable to question whether any psychological attributes are truly quantitative and therefore measurable, despite their common treatment as such. Certainly there are as yet no SI or derived units for any psychological attribute, e.g. intelligence or depression. An often raised philosophical question surrounding this debate concerns whether numbers are real (i.e. whether number is an ontological phenomenon) or whether they are merely representational. For the realist, measurement pertains only to quantitative variables. Adopting a representationalist approach affords much more flexibility in the definition of measurement that is not necessarily dependent on quantitivity. Yet even some representationalists attribute importance to quantitivity – or at least interval scales – in terms of the operations that can validly be applied to such data. What is particularly interesting about this issue is that there exist empirical methods for testing the quantitivity of psychological attributes, e.g. the theory of conjoint measurement (Luce & Tukey, 1964). As such, the question of measurement in psychology is not purely a philosophical one, but also an empirical one. Unfortunately, however, empirical investigations aimed at testing for quantitivity have been rare to date.
While this is not an active area of research for me, it is one that greatly interests me. If it also interests you, then I recommend reading:
Michell, J. (2012). Alfred Binet and the concept of heterogeneous orders. Frontiers in Quantitative Psychology and Measurement, 3: 261.
McGrane, J. A. (2010). Are psychological quantities and measurement relevant in the 21st century? Frontiers in Quantitative Psychology and Measurement. doi: 10.3389/fpsyg.2010.00022
Michell, J. (2009). The psychometricians fallacy: Too clever by half? British Journal of Mathematical and Statistical Psychology, 62: 41-55.
Luce, R.D. & Tukey, J.W. (1964). Simultaneous conjoint measurement: a new scale type of fundamental measurement. Journal of Mathematical Psychology, 1: 1–27.