97
observable, either directly or through their transduction to observable properties).
But while these constraints may have seemed reasonable for the measurement of
properties that Campbell would later refer to as extensive—i.e., properties that could
be empirically concatenated, such as spatial distance, mass, and volume, and which
could therefore be demonstrated to physically satisfy the Euclidean requirement of
property-related additive divisibility—they became a matter of controversy with
respect to the so-called intensive quantities, like temperature and density, and even
more so with nonphysical properties such as those of interest to psychophysicists
(i.e., intensity of sensations) such as Gustav Fechner in the late nineteenth century
(see, e.g., Gescheider, 2013).
22
Difficulties with issues such as these were a major part of the motivation for the
formation of the Ferguson Committee, as described previously, which was charged
with studying the possibility of providing “quantitative estimates of sensory events”.
The committee largely endorsed the Euclidean perspective on additive divisibility
being a necessary condition of measurability, with the consequence that “the main
point against [for example] the measurability of the intensity of a sensation was the
impossibility of satisfactorily defining an addition operation for it” (Rossi, 2007:
p. 551; see also Sect. 6.5). Indeed, additivity is at the basis of what Campbell called
“fundamental measurement” (1920: p. 267), from which, in his account, any other
form of measurement needs to be derived, corresponding to the possibility of obtaining an intensive quantity (e.g., density) as a function of extensive quantities (e.g.,
mass and volume).
Of course, in general, nonphysical properties are not empirically additive. In the
face of the possible conclusion that this simply precludes the possibility that such
properties could ever be measured, two paths were explored.
The first path started from the observation that if a nonadditive quantity like
density is acknowledged to be measurable, it is because even in the Euclidean context empirical additivity is not necessary for measurability. Rather, what is required
is the possibility for one to meaningfully interpret the relation Q[a] = x q ref (see Sect.
2.2.4), for example L[a] = 1.2345 m, as a ratio of the two involved properties—the
length of the object a and the metre—i.e., x = Q[a]/q ref .
23
Hence, the critical condition
22 As in definition 1 of Book 5 of the Elements, as previously quoted, the condition for a quantity
(a “magnitude” in the traditional translation) “to measure” another quantity is that the first is a part
of the second: “a magnitude is a part of a(nother) magnitude, the less of the greater, when it measures the greater” (Euclid, 2008). But for a property P which makes objects a, b, …, comparable
through an order relation (or least as a partial order), it is clear that P[a] < P[b] does not generally
mean that the property P of a is a part of the property of b. Indeed, it is additivity that guarantees
this meaningfulness.
23 This condition can be generalized by admitting that sometimes the zero of Q is not an intrinsic
feature of Q, so that the numerical value x in the relation Q[a] = x q ref is determined only when a
zero property q 0 is set for Q, as x = (Q[a] − q 0 )/(q ref − q 0 ) (for example, this was the case of temperature before the introduction of thermodynamic temperature and its measurement in kelvins,
and is the case of position along a line, which, differently from length, can be measured only having chosen a reference/zero position). Since, in most cases, nonphysical properties do not have an
intrinsic or obvious zero, this generalization—leading to what Stevens called an “interval scale”
(1946)—proved to be very important for the development of measurability conditions for nonphysical properties.
4.4 An interpretive framework
observable, either directly or through their transduction to observable properties).
But while these constraints may have seemed reasonable for the measurement of
properties that Campbell would later refer to as extensive—i.e., properties that could
be empirically concatenated, such as spatial distance, mass, and volume, and which
could therefore be demonstrated to physically satisfy the Euclidean requirement of
property-related additive divisibility—they became a matter of controversy with
respect to the so-called intensive quantities, like temperature and density, and even
more so with nonphysical properties such as those of interest to psychophysicists
(i.e., intensity of sensations) such as Gustav Fechner in the late nineteenth century
(see, e.g., Gescheider, 2013).
22
Difficulties with issues such as these were a major part of the motivation for the
formation of the Ferguson Committee, as described previously, which was charged
with studying the possibility of providing “quantitative estimates of sensory events”.
The committee largely endorsed the Euclidean perspective on additive divisibility
being a necessary condition of measurability, with the consequence that “the main
point against [for example] the measurability of the intensity of a sensation was the
impossibility of satisfactorily defining an addition operation for it” (Rossi, 2007:
p. 551; see also Sect. 6.5). Indeed, additivity is at the basis of what Campbell called
“fundamental measurement” (1920: p. 267), from which, in his account, any other
form of measurement needs to be derived, corresponding to the possibility of obtaining an intensive quantity (e.g., density) as a function of extensive quantities (e.g.,
mass and volume).
Of course, in general, nonphysical properties are not empirically additive. In the
face of the possible conclusion that this simply precludes the possibility that such
properties could ever be measured, two paths were explored.
The first path started from the observation that if a nonadditive quantity like
density is acknowledged to be measurable, it is because even in the Euclidean context empirical additivity is not necessary for measurability. Rather, what is required
is the possibility for one to meaningfully interpret the relation Q[a] = x q ref (see Sect.
2.2.4), for example L[a] = 1.2345 m, as a ratio of the two involved properties—the
length of the object a and the metre—i.e., x = Q[a]/q ref .
23
Hence, the critical condition
22 As in definition 1 of Book 5 of the Elements, as previously quoted, the condition for a quantity
(a “magnitude” in the traditional translation) “to measure” another quantity is that the first is a part
of the second: “a magnitude is a part of a(nother) magnitude, the less of the greater, when it measures the greater” (Euclid, 2008). But for a property P which makes objects a, b, …, comparable
through an order relation (or least as a partial order), it is clear that P[a] < P[b] does not generally
mean that the property P of a is a part of the property of b. Indeed, it is additivity that guarantees
this meaningfulness.
23 This condition can be generalized by admitting that sometimes the zero of Q is not an intrinsic
feature of Q, so that the numerical value x in the relation Q[a] = x q ref is determined only when a
zero property q 0 is set for Q, as x = (Q[a] − q 0 )/(q ref − q 0 ) (for example, this was the case of temperature before the introduction of thermodynamic temperature and its measurement in kelvins,
and is the case of position along a line, which, differently from length, can be measured only having chosen a reference/zero position). Since, in most cases, nonphysical properties do not have an
intrinsic or obvious zero, this generalization—leading to what Stevens called an “interval scale”
(1946)—proved to be very important for the development of measurability conditions for nonphysical properties.
4.4 An interpretive framework