163
equal differences of temperature, i.e., if V = f(Θ) and v 1 − v 2 = v 3 − v 4 then it is
because θ 1 − θ 2 = θ 3 − θ 4 .
23
While this development so far involves only properties of objects—temperatures
and volumes
24
—on this basis the construction of a scale of temperatures, and therefore the introduction of values of temperature, is a relatively trivial task. According
to the traditional procedure:
• Two distinct temperatures are identified, θ 1 and θ 2 , each of them being the common, constant temperature of a class of objects, θ 1 = Θ[R 1 ] and θ 2 = Θ[R 2 ], in
analogy with what is discussed in Sect. 6.3.4 about speed of light; θ 1 and θ 2 could
be the temperatures of the freezing point of water and the boiling point of water
in appropriate conditions, respectively.
• The scale identified by θ 1 and θ 2 is given a name, say °C, and a number in the
scale is conventionally assigned to θ 1 and θ 2 , for example 0 °C ≔ θ 1 and 100 °C ≔
θ 2 .
• According to the hypothesis that equal differences of volume are produced by
equal differences of temperature appropriate numbers in the scale are assigned to
all other temperatures: for example, if f(θ 3 ) = [f(θ 1 ) + f(θ 2 )]/2, then
[0 °C + 100 °C]/2 = 50 °C ≔ θ 3 .
The conclusion is then that values of temperature are individual temperatures identified as elements in such a scale.
6.3.7 Beyond additivity: the example of reading
comprehension ability
Let us now discuss the case of reading comprehension ability (RCA), as characterized and then measured by reading tests. Like temperature and unlike length, RCA
is not an additive quantity: that is, we do not know how to combine readers by RCA
so that the RCA of a hypothetical “synthetic reader” is the sum of the RCAs of each
of the combined readers. As above, our question is then: What is a value of RCA
such as, say 150 RCA units? The starting point is the same as in the case of length
and temperature: we assume that we can compare readers by their RCA so as to
23 The condition that this construction applies to multiple bodies/thermometers avoids the problems
of radical operationalism, which would define temperature as what is measured by a given
instrument.
24 Differences of volumes of the relevant bodies have been assumed to be somehow observable.
However, instead of operating on empirical properties it might be more convenient to measure
volumes and then to operate mathematically on the measured values. The change is immaterial
here. Note furthermore that this role of volume as a transduced property that is a function of temperature played an important historical role, as the scientific principle at the basis of the construction of the first thermometers, but is by no means unique. An analogous presentation could be
made, for example, with voltage in place of volume in reference to the thermoelectric effect.
6.3 Constructing values of quantities
equal differences of temperature, i.e., if V = f(Θ) and v 1 − v 2 = v 3 − v 4 then it is
because θ 1 − θ 2 = θ 3 − θ 4 .
23
While this development so far involves only properties of objects—temperatures
and volumes
24
—on this basis the construction of a scale of temperatures, and therefore the introduction of values of temperature, is a relatively trivial task. According
to the traditional procedure:
• Two distinct temperatures are identified, θ 1 and θ 2 , each of them being the common, constant temperature of a class of objects, θ 1 = Θ[R 1 ] and θ 2 = Θ[R 2 ], in
analogy with what is discussed in Sect. 6.3.4 about speed of light; θ 1 and θ 2 could
be the temperatures of the freezing point of water and the boiling point of water
in appropriate conditions, respectively.
• The scale identified by θ 1 and θ 2 is given a name, say °C, and a number in the
scale is conventionally assigned to θ 1 and θ 2 , for example 0 °C ≔ θ 1 and 100 °C ≔
θ 2 .
• According to the hypothesis that equal differences of volume are produced by
equal differences of temperature appropriate numbers in the scale are assigned to
all other temperatures: for example, if f(θ 3 ) = [f(θ 1 ) + f(θ 2 )]/2, then
[0 °C + 100 °C]/2 = 50 °C ≔ θ 3 .
The conclusion is then that values of temperature are individual temperatures identified as elements in such a scale.
6.3.7 Beyond additivity: the example of reading
comprehension ability
Let us now discuss the case of reading comprehension ability (RCA), as characterized and then measured by reading tests. Like temperature and unlike length, RCA
is not an additive quantity: that is, we do not know how to combine readers by RCA
so that the RCA of a hypothetical “synthetic reader” is the sum of the RCAs of each
of the combined readers. As above, our question is then: What is a value of RCA
such as, say 150 RCA units? The starting point is the same as in the case of length
and temperature: we assume that we can compare readers by their RCA so as to
23 The condition that this construction applies to multiple bodies/thermometers avoids the problems
of radical operationalism, which would define temperature as what is measured by a given
instrument.
24 Differences of volumes of the relevant bodies have been assumed to be somehow observable.
However, instead of operating on empirical properties it might be more convenient to measure
volumes and then to operate mathematically on the measured values. The change is immaterial
here. Note furthermore that this role of volume as a transduced property that is a function of temperature played an important historical role, as the scientific principle at the basis of the construction of the first thermometers, but is by no means unique. An analogous presentation could be
made, for example, with voltage in place of volume in reference to the thermoelectric effect.
6.3 Constructing values of quantities