I would like to share a question from a philosophy professor that I received privately concerning this article. The inquirer asks:

“So, then, some magnitude—or quantity of extension—characterizes every existent, no? So the thought that today is a nice day is quantifiable, given that the thought is an existent?”

As I wrote in “Universals and Measurement,” I suppose that every concrete stands in measurable relations to other concretes. I supposed also that

“Finest objectivity requires measurement scales appropriate to the magnitude structures to which they are applied. What does appropriate mean in this context? It means that all of the mathematical structure of the measurement scale is needed to capture the concept-class magnitude structure of concretes under consideration. It means as well that all the magnitude structure pertinent to the concept class is describable in terms of the mathematical structure of the measurement scale.” (JARS 5(2), p. 276)

Which type is appropriate is not up to us, but is fundamentally tuned to the kind of magnitude structure being measured. There is an inclusive hierarchy of types and structures. The types of (one-dimensional) measurement are these: absolute, ordinal, hyperordinal, interval, and ratio. In “Universals and Measurement” I discussed the different magnitude structures to which some of these measurement types are appropriate. I discussed also the levels in multidimensional measurement types and their magnitude structures (geometry).

Most concept classes are multidimensional. An example would be the class animal (metazoan). To get the dimensions, we begin with the definition. A general-purpose definition of animal would be: a multicellular living being capable of nervous sensation and muscular locomotion. Surely the mathematically determinate form of the concept class animal is multidimensional (cf. Rand in IOE, 16, 24–25, 42).

I don’t know a good definition of thought that is well-agreed upon and not quickly circular. Perhaps a reader will help us out with a good definition. I imaging it will need to work upward from some definitions of perception, categorical perception, schematization, conceptualization, and predication.

The thought that today is a nice day occurs in a region of space, and for many aspects of that space and many aspects of the neurological processes necessary for the thought, it is the case that the measurement type appropriate to the magnitude is ratio-scale. But for other aspects of that space and those processes, ratio-scale is not appropriate. Scales with lesser structure are appropriate to these still perfectly physical aspects, because that is their magnitude character, not because we don’t know how to measure them with ratio scale. (My authority, I should perhaps say, is the 3-volume work Foundations of Measurement by P. Suppes et al. and the related work in the journals.)

The type of measurement appropriate to a thought is unknown to me (and perhaps to anyone so far), but I would bet a coke that it is not ratio-scale (nor an n-dimensional Euclidean space). I stress that, gentle Professor, because of your use of the word extension. That rings of what is called extensive measurement, which is a sub-class of ratio-scale measurement.

In addition to that Q&A, I should correct an error in the RoR article. I botched the third line of the illustrations of hierarchies of structure in geometry. The inclusive hierarchical dependencies would be more obvious if I broke apart each of those three lines into their two wings, which are independent anyway. Here, this is better, and it contains all the information of the three lines in the article:

{[(Ordered) Affine] Euclidean}

{[(Ordered) Absolute] Euclidean}

{[(Ordered) Absolute] Hyperbolic}

{[(Projective) Affine] Euclidean}

[(Projective) Elliptic]