181
Condition (C1) seems obvious, particularly in the context of representational
theories of measurement where it may be considered the premise of representation
theorems.
42
For example, if the property of an object is compared with the property
of another object and the former is observed to be greater than the latter, the value
of the former should be greater than the value of the latter. However, the meaning of
(C1) is based on the nontrivial acknowledgment that properties of objects may also
be compared independently of their evaluation, and therefore that the comparison
has features which are independent of the evaluation. The condition that the property of one object is greater than the property of another object might be in some
sense observable, and in this case does not require such properties to be evaluated.
This gives support to the position that NOIR is a feature not only of the ways in
which properties are evaluated, but also of properties as such, via what we know
about the ways in which they can be compared.
43
In Michell’s words, “the existence
of the empirical relations numerically represented must be logically independent of
the numerical assignments made. That is, these empirical relations must be such that
it is always possible (in principle, at least) to demonstrate their existence without
first making numerical assignments” (Michell, 1999: p. 167). For sure, any such
ontic claim may be updated, and in particular improved—for example when a metric is discovered to apply to what was previously considered to be a nonquantitative
property—but this is just in agreement with the general understanding that empirical knowledge is always revisable.
Condition (C2) has more complex implications: How can we be sure that a relation among values does not correspond to a still-unobserved relation among properties of objects? The point here is not about accepting or refusing “proscriptions”, in
the sense of Velleman and Wilkinson (1993) and as already discussed in Sect. 6.5.1,
but about acknowledging that through evaluation some features of properties might
be discovered. For example, historically, the idea that temperature can be evaluated
on an interval scale was formulated as the result of its evaluation by means of thermometers, not via the comparison of temperatures of objects in terms of their distances/intervals. As documented by Chang (2004), a crucial problem was in the
confirmation of the preliminary hypothesis that the evaluation is linear (in this case,
that thermometers have a linear behavior in transducing temperatures to lengths), so
that divisions in the scale of values (in this case, of length in the capillary) can be
treated as evidence of correspondingly proportional divisions in the scale of properties of objects (in this case, of temperatures).
44
Such an inference is then justified on
42 For example, Fred Roberts describes what he calls “the representation problem” as follows:
“Given a particular numerical relational system Y, find conditions on an observed relational system
X (necessary and) sufficient for the existence of a homomorphism from X into Y” (1979: p. 54).
43 It seems paradoxical that representationalism—a weak position about the epistemic state of measurement, as also discussed in Chap. 4—assumes some strong ontic requirements on properties.
44 This hypothesis of linearity can be empirically corroborated by ascertaining that different temperatures produce proportional changes in different thermometers, operating according to different
transduction effects. Four conceptual (though not necessarily historical) stages may be envisioned
to such a process:
6.5 Generalizing the framework to nonquantitative properties
Condition (C1) seems obvious, particularly in the context of representational
theories of measurement where it may be considered the premise of representation
theorems.
42
For example, if the property of an object is compared with the property
of another object and the former is observed to be greater than the latter, the value
of the former should be greater than the value of the latter. However, the meaning of
(C1) is based on the nontrivial acknowledgment that properties of objects may also
be compared independently of their evaluation, and therefore that the comparison
has features which are independent of the evaluation. The condition that the property of one object is greater than the property of another object might be in some
sense observable, and in this case does not require such properties to be evaluated.
This gives support to the position that NOIR is a feature not only of the ways in
which properties are evaluated, but also of properties as such, via what we know
about the ways in which they can be compared.
43
In Michell’s words, “the existence
of the empirical relations numerically represented must be logically independent of
the numerical assignments made. That is, these empirical relations must be such that
it is always possible (in principle, at least) to demonstrate their existence without
first making numerical assignments” (Michell, 1999: p. 167). For sure, any such
ontic claim may be updated, and in particular improved—for example when a metric is discovered to apply to what was previously considered to be a nonquantitative
property—but this is just in agreement with the general understanding that empirical knowledge is always revisable.
Condition (C2) has more complex implications: How can we be sure that a relation among values does not correspond to a still-unobserved relation among properties of objects? The point here is not about accepting or refusing “proscriptions”, in
the sense of Velleman and Wilkinson (1993) and as already discussed in Sect. 6.5.1,
but about acknowledging that through evaluation some features of properties might
be discovered. For example, historically, the idea that temperature can be evaluated
on an interval scale was formulated as the result of its evaluation by means of thermometers, not via the comparison of temperatures of objects in terms of their distances/intervals. As documented by Chang (2004), a crucial problem was in the
confirmation of the preliminary hypothesis that the evaluation is linear (in this case,
that thermometers have a linear behavior in transducing temperatures to lengths), so
that divisions in the scale of values (in this case, of length in the capillary) can be
treated as evidence of correspondingly proportional divisions in the scale of properties of objects (in this case, of temperatures).
44
Such an inference is then justified on
42 For example, Fred Roberts describes what he calls “the representation problem” as follows:
“Given a particular numerical relational system Y, find conditions on an observed relational system
X (necessary and) sufficient for the existence of a homomorphism from X into Y” (1979: p. 54).
43 It seems paradoxical that representationalism—a weak position about the epistemic state of measurement, as also discussed in Chap. 4—assumes some strong ontic requirements on properties.
44 This hypothesis of linearity can be empirically corroborated by ascertaining that different temperatures produce proportional changes in different thermometers, operating according to different
transduction effects. Four conceptual (though not necessarily historical) stages may be envisioned
to such a process:
6.5 Generalizing the framework to nonquantitative properties