148
somehow be expressed by means of linguistic entities to be communicated, but they
are not, in themselves, expressions.
Second, sometimes values are said to be symbols, or identifiers, which stand for
or represent objects or quantities of objects.
5
Of course, they may well be used as
such, but this does not solve the problem of what values are. Indeed, stating that x
is a symbol of y does not say anything about what x is. In this sense, Napoleon can
be a symbol of political power, and a sphere can be a symbol of perfection, but this
does not change the fact that Napoleon was a human being and a sphere is a geometric object. “To be a symbol” is just convenient shorthand for “to be used as a symbol”. Hence values may be used as symbols to represent quantities of objects, but a
definition of phrased as “symbol such that …” is ontologically
vacuous.
6.2.2 Values of properties cannot be discarded in contemporary
measurement
At this point we need to face the possible objection that values are not needed at all,
and therefore our whole problem can be dismissed as immaterial. At least two analogous arguments can be made in support of this position.
One argument is that most equations and the related explanations that appear in
the literature on, for example, physics do not even mention units: while often introduced as relations among general quantities, physical laws are also interpreted as
equations that relate numerical values of such quantities, under the assumption that
their units are consistently chosen in a system of units. Hence it would seem that,
after a system of units has been chosen, values can be discarded, and instead one
need to only report numbers, instead of values (e.g., 1.2345 instead of 1.2345 m),
for conveying information about quantities of objects.
The second argument starts from the supposition that measurement produces
numbers rather than values. As mentioned above, this seems to be assumed in particular by representational theories of measurement (see, e.g., Krantz et al., 1971),
which usually formalize measurement as a mapping from objects or properties of
objects (see also Sect. 5.2.5) to numbers
6
by maintaining the unit implicit in the
5 For example, André Weyl wrote that “measurement permits things … to be represented conceptually, by means of symbols” (1949: p. 144). While not false, this claim is by no means characteristic
of measurement in particular, and therefore is not very informative.
6 The fact that distinct objects can have the same quantity, e.g., the same length, and therefore are
mapped to the same number, makes the quantity-related mapping non-injective, thus a homomorphism. What Louis Narens wrote (1985: p. 7) on this matter is interesting (note that he uses the
term “scale” to refer to such mappings): “I often prefer to change the character of the representational theory a little and consider a scale to be an isomorphism between the empirical or qualitative
situation and some mathematical situation. The primary reason for this is that isomorphisms preserve truth whereas homomorphisms do not.” According to the ontology we are proposing, a way
for making the mapping injective, and therefore an isomorphism, is to assume that its domain is the
6 Values, scales, and the existence of properties
somehow be expressed by means of linguistic entities to be communicated, but they
are not, in themselves, expressions.
Second, sometimes values are said to be symbols, or identifiers, which stand for
or represent objects or quantities of objects.
5
Of course, they may well be used as
such, but this does not solve the problem of what values are. Indeed, stating that x
is a symbol of y does not say anything about what x is. In this sense, Napoleon can
be a symbol of political power, and a sphere can be a symbol of perfection, but this
does not change the fact that Napoleon was a human being and a sphere is a geometric object. “To be a symbol” is just convenient shorthand for “to be used as a symbol”. Hence values may be used as symbols to represent quantities of objects, but a
definition of
vacuous.
6.2.2 Values of properties cannot be discarded in contemporary
measurement
At this point we need to face the possible objection that values are not needed at all,
and therefore our whole problem can be dismissed as immaterial. At least two analogous arguments can be made in support of this position.
One argument is that most equations and the related explanations that appear in
the literature on, for example, physics do not even mention units: while often introduced as relations among general quantities, physical laws are also interpreted as
equations that relate numerical values of such quantities, under the assumption that
their units are consistently chosen in a system of units. Hence it would seem that,
after a system of units has been chosen, values can be discarded, and instead one
need to only report numbers, instead of values (e.g., 1.2345 instead of 1.2345 m),
for conveying information about quantities of objects.
The second argument starts from the supposition that measurement produces
numbers rather than values. As mentioned above, this seems to be assumed in particular by representational theories of measurement (see, e.g., Krantz et al., 1971),
which usually formalize measurement as a mapping from objects or properties of
objects (see also Sect. 5.2.5) to numbers
6
by maintaining the unit implicit in the
5 For example, André Weyl wrote that “measurement permits things … to be represented conceptually, by means of symbols” (1949: p. 144). While not false, this claim is by no means characteristic
of measurement in particular, and therefore is not very informative.
6 The fact that distinct objects can have the same quantity, e.g., the same length, and therefore are
mapped to the same number, makes the quantity-related mapping non-injective, thus a homomorphism. What Louis Narens wrote (1985: p. 7) on this matter is interesting (note that he uses the
term “scale” to refer to such mappings): “I often prefer to change the character of the representational theory a little and consider a scale to be an isomorphism between the empirical or qualitative
situation and some mathematical situation. The primary reason for this is that isomorphisms preserve truth whereas homomorphisms do not.” According to the ontology we are proposing, a way
for making the mapping injective, and therefore an isomorphism, is to assume that its domain is the
6 Values, scales, and the existence of properties