154
as birthday[b], while his or her height in a given time t as height[b, t]: in this way
we acknowledge that birthday is time invariant, whereas height is time variant.
In this sense, the definition ℓ ref ≔ L[r], where the identifier “ℓ ref ” is not indexed
with time, assumes that the length L[r] is time invariant. Since quantities of objects
are instead usually subject to variations, this is a strong assumption: of course,
assigning a name to the quantity of an object does not make it stable.
11
The consequence of choosing the length of an object r as a reference length ℓ ref ,
thus under the condition of its stability, is that ℓ ref can also be considered to be the
length of any other sufficiently stable object having the same length as r. This allows
the assessment of L[a] ≈ x ℓ ref not only by means of L[a] ≈ x L[r] but also by means
of L[a] ≈ x L[r′], for any sufficiently stable r′ in a class of objects such that
L[r′] ≈ L[r]. Hence the choice of referring to a length through an identifier as “ℓ ref ”
(for example “metre”—note: it is not “metre in a given time t”) assumes that the
referenced length is both space and time invariant: according to the conceptual
framework introduced in Sect. 6.1, it is an individual length, identified by L[r], L[r′],
… but abstracted from any particular object.
6.3.3 Alternative reference quantities and their relations,
i.e., scale transformations
The only condition for having singled out r as a reference object is that its length is
stable. Hence nothing precludes the independent choice of an alternative reference
object, r*, whose length L[r*] is distinguishable from L[r] and defines a new reference length (for example the foot instead of the metre):
l ref :=
∗
∗
L r
A new rod a* can be now calibrated with respect to ℓ ref* , exactly as was done before
for the rod a with respect to ℓ ref , so that the same object b could be compared in its
length with both the rod a and the rod a*. Different relations of indistinguishability
are then obtained, L[b] ≈ x ℓ ref and L[b] ≈ x′ ℓ ref* , with x ≠ x′, as exemplified in
Fig. 6.6.
The lengths marked in this way on the rods a and a* can be compared, which is
particularly interesting because such lengths are indexed by numbers, attributed
according to the hypothesis of empirical additivity, such that the length 2 ℓ ref is
L[r] ⊕ L[r] and so on. Hence the hypothesis that the lengths marked on two rods
have been additively constructed can be experimentally validated, by finding the
factor k such that ℓ ref ≈ k ℓ ref* (in the example in Fig. 6.6, k = 0.5) and then checking
11 This problem is arguably even more pernicious in the human sciences, wherein properties commonly vary not only by time but also by sociocultural-historical context, as also discussed in Sect.
4.4.
6 Values, scales, and the existence of properties
as birthday[b], while his or her height in a given time t as height[b, t]: in this way
we acknowledge that birthday is time invariant, whereas height is time variant.
In this sense, the definition ℓ ref ≔ L[r], where the identifier “ℓ ref ” is not indexed
with time, assumes that the length L[r] is time invariant. Since quantities of objects
are instead usually subject to variations, this is a strong assumption: of course,
assigning a name to the quantity of an object does not make it stable.
11
The consequence of choosing the length of an object r as a reference length ℓ ref ,
thus under the condition of its stability, is that ℓ ref can also be considered to be the
length of any other sufficiently stable object having the same length as r. This allows
the assessment of L[a] ≈ x ℓ ref not only by means of L[a] ≈ x L[r] but also by means
of L[a] ≈ x L[r′], for any sufficiently stable r′ in a class of objects such that
L[r′] ≈ L[r]. Hence the choice of referring to a length through an identifier as “ℓ ref ”
(for example “metre”—note: it is not “metre in a given time t”) assumes that the
referenced length is both space and time invariant: according to the conceptual
framework introduced in Sect. 6.1, it is an individual length, identified by L[r], L[r′],
… but abstracted from any particular object.
6.3.3 Alternative reference quantities and their relations,
i.e., scale transformations
The only condition for having singled out r as a reference object is that its length is
stable. Hence nothing precludes the independent choice of an alternative reference
object, r*, whose length L[r*] is distinguishable from L[r] and defines a new reference length (for example the foot instead of the metre):
l ref :=
∗
∗
L r
A new rod a* can be now calibrated with respect to ℓ ref* , exactly as was done before
for the rod a with respect to ℓ ref , so that the same object b could be compared in its
length with both the rod a and the rod a*. Different relations of indistinguishability
are then obtained, L[b] ≈ x ℓ ref and L[b] ≈ x′ ℓ ref* , with x ≠ x′, as exemplified in
Fig. 6.6.
The lengths marked in this way on the rods a and a* can be compared, which is
particularly interesting because such lengths are indexed by numbers, attributed
according to the hypothesis of empirical additivity, such that the length 2 ℓ ref is
L[r] ⊕ L[r] and so on. Hence the hypothesis that the lengths marked on two rods
have been additively constructed can be experimentally validated, by finding the
factor k such that ℓ ref ≈ k ℓ ref* (in the example in Fig. 6.6, k = 0.5) and then checking
11 This problem is arguably even more pernicious in the human sciences, wherein properties commonly vary not only by time but also by sociocultural-historical context, as also discussed in Sect.
4.4.
6 Values, scales, and the existence of properties