152
relations between L[a] and L[r], by considering, together with iteration, L[a] ≈ n
L[r], the inverse operation of partition (as the terms “iteration” and “partition” are
used by Weyl, 1949: p. 30), such that L[r] is assumed to be constituted of n′ indistinguishable lengths L[c], so that L[r] ≈ n′ L[c]. By combining the two operations,
a length is obtained as n/n′ L[r]. In varying the ratio n/n′ a set of lengths is thus
obtained, and while the construction starts from the length of a given object, r, each
entity n/n′ L[r] is a length constructed without an object that bears it: What sort of
entities are they, then? While leaving this question open for the moment, let us point
out that all these relations involve only quantities of objects, and are obtained by
experimentally comparing objects.
Suppose now that the length L[r] is agreed to be taken as a reference quantity and
given an identifier for convenience, say “ℓ ref ” (or, for example “metre”). The reference length ℓ ref is then defined as the length of the object r
l ref := L r
[ ]
and r can be called a reference object. The indistinguishability relation L[a] ≈ 2 L[r]
can then also be written as
L a
[ ] ≈ 2 l ref
This shows that the following relations
L a L r L r
[ ] ≈ [ ]⊕ [ ]
′
L a L r L r
L r L r
[ ] ≈ [ ]⊕ [ ]
[ ] ≈ [ ]
(
)
′
provided that
L a
L r
[ ] ≈ [ ] (
)
2
ashorthand of the previous relation
L a
[ ] ≈
(
)
2 l
l
ref
r ef
according to the definition of
all refer to the same empirical situation and only differ in the way the information
is conveyed: in terms of the distinction between senses and referents of expressions
(as explained in Sect. 5.3.2), the senses of the involved expressions are different, but
their referent is the same. All these relations—including the last one—involve
lengths, and the difference between the length L[r] ⊕ L[r′] and the length 2 ℓ ref is
only about how such lengths are identified.
A rod a can be now calibrated in terms of its length with respect to ℓ ref by aligning
the left ends of a and r and placing a mark on the rod a at the other end of the rod r.
Additional marks can be placed on the rod a, using geometrical methods that implement the iteration and partition methods mentioned above, to denote multiples of
ℓ ref , as depicted in Fig. 6.4.
Common measuring instruments of length, such as metersticks and tape measures, are constructed and then calibrated in this way: indeed, the rod a can be
6 Values, scales, and the existence of properties
relations between L[a] and L[r], by considering, together with iteration, L[a] ≈ n
L[r], the inverse operation of partition (as the terms “iteration” and “partition” are
used by Weyl, 1949: p. 30), such that L[r] is assumed to be constituted of n′ indistinguishable lengths L[c], so that L[r] ≈ n′ L[c]. By combining the two operations,
a length is obtained as n/n′ L[r]. In varying the ratio n/n′ a set of lengths is thus
obtained, and while the construction starts from the length of a given object, r, each
entity n/n′ L[r] is a length constructed without an object that bears it: What sort of
entities are they, then? While leaving this question open for the moment, let us point
out that all these relations involve only quantities of objects, and are obtained by
experimentally comparing objects.
Suppose now that the length L[r] is agreed to be taken as a reference quantity and
given an identifier for convenience, say “ℓ ref ” (or, for example “metre”). The reference length ℓ ref is then defined as the length of the object r
l ref := L r
[ ]
and r can be called a reference object. The indistinguishability relation L[a] ≈ 2 L[r]
can then also be written as
L a
[ ] ≈ 2 l ref
This shows that the following relations
L a L r L r
[ ] ≈ [ ]⊕ [ ]
′
L a L r L r
L r L r
[ ] ≈ [ ]⊕ [ ]
[ ] ≈ [ ]
(
)
′
provided that
L a
L r
[ ] ≈ [ ] (
)
2
ashorthand of the previous relation
L a
[ ] ≈
(
)
2 l
l
ref
r ef
according to the definition of
all refer to the same empirical situation and only differ in the way the information
is conveyed: in terms of the distinction between senses and referents of expressions
(as explained in Sect. 5.3.2), the senses of the involved expressions are different, but
their referent is the same. All these relations—including the last one—involve
lengths, and the difference between the length L[r] ⊕ L[r′] and the length 2 ℓ ref is
only about how such lengths are identified.
A rod a can be now calibrated in terms of its length with respect to ℓ ref by aligning
the left ends of a and r and placing a mark on the rod a at the other end of the rod r.
Additional marks can be placed on the rod a, using geometrical methods that implement the iteration and partition methods mentioned above, to denote multiples of
ℓ ref , as depicted in Fig. 6.4.
Common measuring instruments of length, such as metersticks and tape measures, are constructed and then calibrated in this way: indeed, the rod a can be
6 Values, scales, and the existence of properties