161
6.3.6 Beyond additivity: the example of temperature
Let us first discuss the case of temperature, as characterized and then measured in
thermometric (e.g., Celsius and Fahrenheit) scales. Unlike length, temperature is
not an additive quantity: that is, we do not know how to combine bodies by temperature so that the temperature of the body obtained by combining two bodies at the
same temperature is twice the temperature of each of the combined bodies. This led
Campbell to conclude that “the scale of temperature of the mercury in glass
Centigrade thermometer is quite as arbitrary as that of the instrument with the random marks” (1920: p. 359), so that “the international scale of temperature is as
arbitrary as Mohs’ scale of hardness” (p. 400). Were this correct, values of temperature, such as 23.4 °C, would be only identifiers for ordered classes of indistinguishable temperatures, as are values of Mohs’ hardness. Our question is then: What is a
value of temperature?
The starting point is the same as in the case of length: we assume to be able to
compare bodies by their temperature so as to assess whether two given bodies have
indistinguishable temperatures (in analogy with the comparison depicted in Fig. 6.2)
or whether one body has a greater temperature than the other.
22
On this basis, a (nonarbitrary) scale of temperature (and therefore values of temperature) can be constructed through an empirical procedure, though, admittedly,
not as simply as the one for length. As in the case of length, all assumptions that
follow relate to empirical properties of objects, and non-idealities in the comparisons of such properties are not taken into account.
Let us consider a sequence a i , i = 1, 2, …, of amounts of gas of the same substance, where the ith amount has the known mass M[a i ] = m i and is thermally homogeneous, at the unknown temperature Θ[a i ] = θ i . Let us suppose that any two
amounts of gas a i and a j can be combined into a single amount a i,j , such that
m i,j = m i + m j . It is assumed that a i,j reaches thermal homogeneity and that its temperature θ i,j is only a function of θ i , m i , θ j , and m j (but of course the nonadditivity of
temperature is such that θ i,j ≠ θ i + θ j ). Finally, let us suppose that the temperatures
of any two amounts of gas can be empirically compared by equality and by order,
i.e., whether θ i = θ j or θ i < θ j or θ j < θ i . The hypothesis that temperature is an intensive property (see Sect. 1.2.1) can be tested through some preliminary checks:
• For each m i and m j , if θ i = θ j then θ i,j = θ i = θ j , i.e., thermal homogeneity does not
depend on mass.
• For each m i and m j , if θ i < θ j then θ i < θ i,j < θ j , i.e., thermal composition is internal
independently of mass.
22 For the sake of simplicity, we assume that this construction is done in a context in which sufficiently clear ideas are available about what temperature is and therefore in particular how temperature and heat are related but different properties (note that sometimes temperature is considered to
be the intensity of heat, and this justifies its nonadditivity). The actual historical development of
these ideas was convoluted, and some sorts of “candidate measurements” were instrumental to the
clarification (see Chang, 2004; Sherry, 2011).
6.3 Constructing values of quantities
6.3.6 Beyond additivity: the example of temperature
Let us first discuss the case of temperature, as characterized and then measured in
thermometric (e.g., Celsius and Fahrenheit) scales. Unlike length, temperature is
not an additive quantity: that is, we do not know how to combine bodies by temperature so that the temperature of the body obtained by combining two bodies at the
same temperature is twice the temperature of each of the combined bodies. This led
Campbell to conclude that “the scale of temperature of the mercury in glass
Centigrade thermometer is quite as arbitrary as that of the instrument with the random marks” (1920: p. 359), so that “the international scale of temperature is as
arbitrary as Mohs’ scale of hardness” (p. 400). Were this correct, values of temperature, such as 23.4 °C, would be only identifiers for ordered classes of indistinguishable temperatures, as are values of Mohs’ hardness. Our question is then: What is a
value of temperature?
The starting point is the same as in the case of length: we assume to be able to
compare bodies by their temperature so as to assess whether two given bodies have
indistinguishable temperatures (in analogy with the comparison depicted in Fig. 6.2)
or whether one body has a greater temperature than the other.
22
On this basis, a (nonarbitrary) scale of temperature (and therefore values of temperature) can be constructed through an empirical procedure, though, admittedly,
not as simply as the one for length. As in the case of length, all assumptions that
follow relate to empirical properties of objects, and non-idealities in the comparisons of such properties are not taken into account.
Let us consider a sequence a i , i = 1, 2, …, of amounts of gas of the same substance, where the ith amount has the known mass M[a i ] = m i and is thermally homogeneous, at the unknown temperature Θ[a i ] = θ i . Let us suppose that any two
amounts of gas a i and a j can be combined into a single amount a i,j , such that
m i,j = m i + m j . It is assumed that a i,j reaches thermal homogeneity and that its temperature θ i,j is only a function of θ i , m i , θ j , and m j (but of course the nonadditivity of
temperature is such that θ i,j ≠ θ i + θ j ). Finally, let us suppose that the temperatures
of any two amounts of gas can be empirically compared by equality and by order,
i.e., whether θ i = θ j or θ i < θ j or θ j < θ i . The hypothesis that temperature is an intensive property (see Sect. 1.2.1) can be tested through some preliminary checks:
• For each m i and m j , if θ i = θ j then θ i,j = θ i = θ j , i.e., thermal homogeneity does not
depend on mass.
• For each m i and m j , if θ i < θ j then θ i < θ i,j < θ j , i.e., thermal composition is internal
independently of mass.
22 For the sake of simplicity, we assume that this construction is done in a context in which sufficiently clear ideas are available about what temperature is and therefore in particular how temperature and heat are related but different properties (note that sometimes temperature is considered to
be the intensity of heat, and this justifies its nonadditivity). The actual historical development of
these ideas was convoluted, and some sorts of “candidate measurements” were instrumental to the
clarification (see Chang, 2004; Sherry, 2011).
6.3 Constructing values of quantities