120
4. The comparison of O-entities among themselves and the comparison of V-entities
among themselves are different processes:
• For comparing O-entities among themselves we compare properties of
objects, thus by means of an empirical process, such as the one performed by
means of a balance and leading to the conclusion that the mass of rod a is less
than the mass of rod b.
• For comparing V-entities among themselves we compare numbers (assuming their unit is the same), thus by means of a mathematical process, such as
the one that leads to the conclusion that 1.23 kg is less than 2.34 kg.
5. A correspondence can be established between O-entities and V-entities:
• O-entities can be made to correspond to V-entities through a process that
can be generically called “evaluation” (see Sect. 2.2.4 for discussion of this
lexical choice), leading to the association of a value with the given property
of an object; measurement is then a specific kind of evaluation: the mass of
rod a can be evaluated as 0.12 kg.
• V-entities can be made to correspond to O-entities through a process that
can be generically called “realization”, leading to the selection or construction of an object such that one of its properties is associated with the given
value: 0.12 kg can be realized by the mass of rod a.
6. Via these correspondences, if O-entities are evaluated, then the obtained
V-entities can be compared among themselves, and this comparison conveys
information about the relevant O-entities, so that the O-entities do not themselves need to be directly compared (as in stage (3) above): if the mass of rod a
is evaluated as 0.12 kg and the mass of rod b is evaluated as 3.45 kg, the comparison that 0.12 kg is less than 3.45 kg leads to the inference that the mass of
rod a is less than the mass of rod b. Nevertheless, comparing O-entities among
themselves and comparing V-entities among themselves remain different
processes.
7. Hence O-entities and V-entities are at the same time
• analogous in some respects, because they are comparable: both O-entities
and V-entities can be thought of as properties, but
• different in some other respects, because the ways in which we compare
them among themselves are different: O-entities are properties identified
through objects that have them, whereas V-entities are properties identified
through numbers that multiply units.
12
12 We have not been able to find a term to designate the entities obtained by multiplying or dividing
a unit by a number which is not necessarily integer (a multiple of a unit is a “measurement unit
obtained by multiplying a given measurement unit by an integer greater than one” (JCGM, 2012:
1.17), and a submultiple of a unit is a “measurement unit obtained by dividing a given measurement unit by an integer greater than one” (JCGM, 2012: 1.18, emphasis added)). Hence we maintain the term “multiple” with this broader meaning: if u is a unit and x is a nonnegative real number,
x u is a multiple of u.
5 What is measured?
4. The comparison of O-entities among themselves and the comparison of V-entities
among themselves are different processes:
• For comparing O-entities among themselves we compare properties of
objects, thus by means of an empirical process, such as the one performed by
means of a balance and leading to the conclusion that the mass of rod a is less
than the mass of rod b.
• For comparing V-entities among themselves we compare numbers (assuming their unit is the same), thus by means of a mathematical process, such as
the one that leads to the conclusion that 1.23 kg is less than 2.34 kg.
5. A correspondence can be established between O-entities and V-entities:
• O-entities can be made to correspond to V-entities through a process that
can be generically called “evaluation” (see Sect. 2.2.4 for discussion of this
lexical choice), leading to the association of a value with the given property
of an object; measurement is then a specific kind of evaluation: the mass of
rod a can be evaluated as 0.12 kg.
• V-entities can be made to correspond to O-entities through a process that
can be generically called “realization”, leading to the selection or construction of an object such that one of its properties is associated with the given
value: 0.12 kg can be realized by the mass of rod a.
6. Via these correspondences, if O-entities are evaluated, then the obtained
V-entities can be compared among themselves, and this comparison conveys
information about the relevant O-entities, so that the O-entities do not themselves need to be directly compared (as in stage (3) above): if the mass of rod a
is evaluated as 0.12 kg and the mass of rod b is evaluated as 3.45 kg, the comparison that 0.12 kg is less than 3.45 kg leads to the inference that the mass of
rod a is less than the mass of rod b. Nevertheless, comparing O-entities among
themselves and comparing V-entities among themselves remain different
processes.
7. Hence O-entities and V-entities are at the same time
• analogous in some respects, because they are comparable: both O-entities
and V-entities can be thought of as properties, but
• different in some other respects, because the ways in which we compare
them among themselves are different: O-entities are properties identified
through objects that have them, whereas V-entities are properties identified
through numbers that multiply units.
12
12 We have not been able to find a term to designate the entities obtained by multiplying or dividing
a unit by a number which is not necessarily integer (a multiple of a unit is a “measurement unit
obtained by multiplying a given measurement unit by an integer greater than one” (JCGM, 2012:
1.17), and a submultiple of a unit is a “measurement unit obtained by dividing a given measurement unit by an integer greater than one” (JCGM, 2012: 1.18, emphasis added)). Hence we maintain the term “multiple” with this broader meaning: if u is a unit and x is a nonnegative real number,
x u is a multiple of u.
5 What is measured?