184
However, it would be fallacious to conflate empirical properties and mathematical
variables, or to assume out of hand that the presence of either implies the existence
of the other: there can be empirical properties without corresponding mathematical
models (for example, because we are unaware of the very existence of such properties, e.g., blood type prior to 1900), and there can be mathematical variables without
corresponding empirical properties (for example, the variables in generic mathematical equations such as y = mx + b).
Although this distinction may seem obvious when presented in these terms, conventions in terminology and modes of discourse may sometimes obfuscate it, as
when the term “variable” is used to refer both to an empirical property and a mathematical variable (which is common in the literature on “latent variable modeling”,
for example, see McGrane & Maul, 2020), or when, as described in the GUM, “for
economy of notation […] the same symbol is used for the [property] and for the
random variable that represents the possible outcome of an observation of that
[property]” (JCGM, 2008: 4.1.1).
As a consequence, it cannot be assumed out of hand that any given feature of a
mathematical variable is shared by the empirical property that the variable claims to
model. For example, some physical quantities are customarily and effectively modeled as real-valued functions (which is a precondition for modeling the dynamics of
such quantities by means of differential equations), but assuming that all features of
real numbers apply to the quantities they purport to model could, for example, lead
to the conclusion that a given quantity is dense in the way that real numbers are,
which in many cases is known to be false, as in the case of quantized quantities such
as electrical charge. Analogously, properties are customarily and effectively modeled as continuous random variables for a variety of purposes, but, again, this does
not guarantee that all features of continuous random variables hold true for the modeled properties (see also, e.g., Borsboom, 2006; McGrane & Maul, 2020), even for
models that fit the data according to commonly accepted criteria (see, e.g., Maraun,
1996; Maul, 2017; Michell, 2000, 2004).
With respect to the confusion between a knowable entity and what we know of it
(i.e., the concept that we have of it), a particularly pernicious class of properties are
those considered to be (in some sense) constructed, as was previously discussed in
Sect. 4.5: one might infer from the fact that “concepts such as compassion and
prejudice are […] created from […] the conceptions of all those who have ever used
these terms” that they therefore “cannot be observed directly or indirectly, because
they don’t exist” (Babbie, 2013, p. 167). This fallaciously conflates the concepts we
have of psychosocial properties such as compassion with the empirical referents of
those concepts.
48
That is, if compassion, prejudice, and other psychosocial properliarly, in the terminology of representationalism, the term “qualitative” is used to refer to the structure of properties even when they are quantities.)
48 Again, as discussed in Sect. 4.5 (and at more length in a variety of sources such as Mislevy,
2018), there are many important differences in the ontological character of psychosocial properties
compared to (classical) physical properties, including the facts that their existence depends on
human consciousness (with all the ontological challenges this entails; see, e.g., Dennett, 1991;
6 Values, scales, and the existence of properties
However, it would be fallacious to conflate empirical properties and mathematical
variables, or to assume out of hand that the presence of either implies the existence
of the other: there can be empirical properties without corresponding mathematical
models (for example, because we are unaware of the very existence of such properties, e.g., blood type prior to 1900), and there can be mathematical variables without
corresponding empirical properties (for example, the variables in generic mathematical equations such as y = mx + b).
Although this distinction may seem obvious when presented in these terms, conventions in terminology and modes of discourse may sometimes obfuscate it, as
when the term “variable” is used to refer both to an empirical property and a mathematical variable (which is common in the literature on “latent variable modeling”,
for example, see McGrane & Maul, 2020), or when, as described in the GUM, “for
economy of notation […] the same symbol is used for the [property] and for the
random variable that represents the possible outcome of an observation of that
[property]” (JCGM, 2008: 4.1.1).
As a consequence, it cannot be assumed out of hand that any given feature of a
mathematical variable is shared by the empirical property that the variable claims to
model. For example, some physical quantities are customarily and effectively modeled as real-valued functions (which is a precondition for modeling the dynamics of
such quantities by means of differential equations), but assuming that all features of
real numbers apply to the quantities they purport to model could, for example, lead
to the conclusion that a given quantity is dense in the way that real numbers are,
which in many cases is known to be false, as in the case of quantized quantities such
as electrical charge. Analogously, properties are customarily and effectively modeled as continuous random variables for a variety of purposes, but, again, this does
not guarantee that all features of continuous random variables hold true for the modeled properties (see also, e.g., Borsboom, 2006; McGrane & Maul, 2020), even for
models that fit the data according to commonly accepted criteria (see, e.g., Maraun,
1996; Maul, 2017; Michell, 2000, 2004).
With respect to the confusion between a knowable entity and what we know of it
(i.e., the concept that we have of it), a particularly pernicious class of properties are
those considered to be (in some sense) constructed, as was previously discussed in
Sect. 4.5: one might infer from the fact that “concepts such as compassion and
prejudice are […] created from […] the conceptions of all those who have ever used
these terms” that they therefore “cannot be observed directly or indirectly, because
they don’t exist” (Babbie, 2013, p. 167). This fallaciously conflates the concepts we
have of psychosocial properties such as compassion with the empirical referents of
those concepts.
48
That is, if compassion, prejudice, and other psychosocial properliarly, in the terminology of representationalism, the term “qualitative” is used to refer to the structure of properties even when they are quantities.)
48 Again, as discussed in Sect. 4.5 (and at more length in a variety of sources such as Mislevy,
2018), there are many important differences in the ontological character of psychosocial properties
compared to (classical) physical properties, including the facts that their existence depends on
human consciousness (with all the ontological challenges this entails; see, e.g., Dennett, 1991;
6 Values, scales, and the existence of properties