11
if
then the alcohol stays in its rest position
if
4 a
k
x
> @ 1
0
,
,
,
4 4 a
k
x
> @ t 1
1
,
,
.
then the alcohol elongates toa fixed position
®
°
°
¯
°
°
(1.2)
Let us call such a transducer a “Boolean thermometer”, whereas “linear thermometer” will be the term for any transducer behaving according to Eq. (1.1). (The
principle of transduction for a Boolean thermometer is not important here: we might
suppose, for example, that the substance enters the tube only when it reaches its
boiling temperature—as such, it could be interpreted as a calibrated thermoscope.)
While the behavior of a linear thermometer is mathematically modeled as a continuous, linear function in the relevant range, Eq. (1.2) defines a function whose
range is discrete, and in fact binary. A second major difference between Eqs. (1.1)
and (1.2) is related to the dimension of the parameter k: while in the case of linear
thermometers, dim Θ/k = dim x = L, and therefore dim k = ΘL
−1
, Eq. (1.2) assumes
that dim Θ/k = 1 (i.e., a quantity of unit one—sometimes the term “dimensionless
quantity” is used in this case), so that dim k = dim Θ for Boolean thermometers. The
fact that in this case the parameter k is dimensionally homogeneous to a temperature
has the important consequence that it can be interpreted as a “threshold temperature”, such that the substance elongates in the tube only if the applied temperature
is greater than the threshold. This interpretation is crucial for what follows, as it
allows the comparison of the involved quantities not only through ratios (Θ/k > 1)
but also through differences (Θ − k > 0) and orderings (Θ > k), and therefore makes
it possible to place values of the measurand and the parameter of the measuring
instrument on the same scale.
Calibrating such a Boolean thermometer requires one to apply increasing temperatures whose values are known and register the value Θ′ of the temperature that
makes the substance elongate, so that k = Θ′. If we then apply the temperature Θ[a]
of an object a to this calibrated Boolean thermometer, and we obtain the indication
Fig. 1.2 The relationship between Θ and x (the transduction function) for thermometers as given
in Eqs. (1.1) and (1.2) (scaled values)
1.2 Some familiar and not-so-familiar contexts for measurement
if
then the alcohol stays in its rest position
if
4 a
k
x
> @ 1
0
,
,
,
4 4 a
k
x
> @ t 1
1
,
,
.
then the alcohol elongates toa fixed position
®
°
°
¯
°
°
(1.2)
Let us call such a transducer a “Boolean thermometer”, whereas “linear thermometer” will be the term for any transducer behaving according to Eq. (1.1). (The
principle of transduction for a Boolean thermometer is not important here: we might
suppose, for example, that the substance enters the tube only when it reaches its
boiling temperature—as such, it could be interpreted as a calibrated thermoscope.)
While the behavior of a linear thermometer is mathematically modeled as a continuous, linear function in the relevant range, Eq. (1.2) defines a function whose
range is discrete, and in fact binary. A second major difference between Eqs. (1.1)
and (1.2) is related to the dimension of the parameter k: while in the case of linear
thermometers, dim Θ/k = dim x = L, and therefore dim k = ΘL
−1
, Eq. (1.2) assumes
that dim Θ/k = 1 (i.e., a quantity of unit one—sometimes the term “dimensionless
quantity” is used in this case), so that dim k = dim Θ for Boolean thermometers. The
fact that in this case the parameter k is dimensionally homogeneous to a temperature
has the important consequence that it can be interpreted as a “threshold temperature”, such that the substance elongates in the tube only if the applied temperature
is greater than the threshold. This interpretation is crucial for what follows, as it
allows the comparison of the involved quantities not only through ratios (Θ/k > 1)
but also through differences (Θ − k > 0) and orderings (Θ > k), and therefore makes
it possible to place values of the measurand and the parameter of the measuring
instrument on the same scale.
Calibrating such a Boolean thermometer requires one to apply increasing temperatures whose values are known and register the value Θ′ of the temperature that
makes the substance elongate, so that k = Θ′. If we then apply the temperature Θ[a]
of an object a to this calibrated Boolean thermometer, and we obtain the indication
Fig. 1.2 The relationship between Θ and x (the transduction function) for thermometers as given
in Eqs. (1.1) and (1.2) (scaled values)
1.2 Some familiar and not-so-familiar contexts for measurement