83
claim that “measurement is the process of assigning numbers to represent qualities;
the object of measurement is to enable the powerful weapon of mathematical analysis to be applied to the subject matter of science” (1920: p. 267). The representationalists then formalized the idea of numerical (or, more generally, symbolic)
assignment as the condition that defines measurement: for example, according to
Patrick Suppes, an appropriate representation theorem “makes the theory of finite
weak orderings a theory of measurement, because of its numerical representation”
(2002: p. 59): even though weakly ordered entities do not in general satisfy the
Euclidean conditions on a measure, they are considered measurable because they
can be represented by means of numerical values.
The key concept here is representability, a condition formalized in terms of morphic mappings from empirical entities (either objects or their properties) to informational entities (Krantz, Luce, Suppes, & Tverski, 1971; Suppes, Krantz, Luce, &
Tverski, 1989; Luce, Krantz, Suppes, & Tverski, 1990; Narens & Luce, 1986).
8
According to this view, a precondition for measurement is the availability of some
set of observed empirical relations among objects (e.g., x is greater than y and less
than z), which are then mapped onto a symbolic system in such a way as to preserve
the qualities of their empirical relations. Consistently with positivist principles, this
requires that empirical relations be directly observable, or “identifiable” (Suppes &
Zinnes, 1963: p. 7), though it is not always obvious what this means (cf. Borsboom,
2005; Michell, 1990). Relational systems can possess different sorts of structures,
and the particular sort of mapping of empirical to numerical relations determines
the scale properties; for example, “x is greater than y” can be preserved by assigning
to x any number higher than the number assigned to y, whereas “x is twice y” can be
preserved by assigning to x a number twice that assigned to y. Such differences
formed the basis of Stevens’ system of scale types (most famously, nominal, ordinal, interval, and ratio; see Sect. 6.5), with the important consequence that, according to RTM, even nonquantitative properties could be considered measurable.
Indeed, according to the representational perspective, “the question of measurement” is just “about the possibility of using numbers to describe certain phenomena”, and by representational theories it “has received answers in the form of
testable conditions” (Doignon, 1993: p. 473).
This purely formal interpretation of measurement is attractive for its epistemological simplicity, but the idea that measurement is definitionally equivalent to morphic mapping is so generic that it is unable to distinguish between measurement and
consistent representation (Mari, 2013; Mari, Carbone, & Petri, 2012). Rather, what
representational theories of measurement provide is an abstract framework for scale
construction and meaningfulness of representation (Narens, 1985, 2002), and therefore at most for characterizing conditions of measurability. If, for example, a given
8 Whether such informational entities need to be numbers, with or without measurement units, is
where Stevens, and since him representational theories of measurement, departed from Campbell.
While, as just mentioned, numbers are required “to enable the powerful weapon of mathematical
analysis to be applied to the subject matter of science” according to Campbell, Stevens (1946)
made the representability of properties by means of numbers sufficient, but not necessary, for
measurability.
4.2 Characterizing measurement
claim that “measurement is the process of assigning numbers to represent qualities;
the object of measurement is to enable the powerful weapon of mathematical analysis to be applied to the subject matter of science” (1920: p. 267). The representationalists then formalized the idea of numerical (or, more generally, symbolic)
assignment as the condition that defines measurement: for example, according to
Patrick Suppes, an appropriate representation theorem “makes the theory of finite
weak orderings a theory of measurement, because of its numerical representation”
(2002: p. 59): even though weakly ordered entities do not in general satisfy the
Euclidean conditions on a measure, they are considered measurable because they
can be represented by means of numerical values.
The key concept here is representability, a condition formalized in terms of morphic mappings from empirical entities (either objects or their properties) to informational entities (Krantz, Luce, Suppes, & Tverski, 1971; Suppes, Krantz, Luce, &
Tverski, 1989; Luce, Krantz, Suppes, & Tverski, 1990; Narens & Luce, 1986).
8
According to this view, a precondition for measurement is the availability of some
set of observed empirical relations among objects (e.g., x is greater than y and less
than z), which are then mapped onto a symbolic system in such a way as to preserve
the qualities of their empirical relations. Consistently with positivist principles, this
requires that empirical relations be directly observable, or “identifiable” (Suppes &
Zinnes, 1963: p. 7), though it is not always obvious what this means (cf. Borsboom,
2005; Michell, 1990). Relational systems can possess different sorts of structures,
and the particular sort of mapping of empirical to numerical relations determines
the scale properties; for example, “x is greater than y” can be preserved by assigning
to x any number higher than the number assigned to y, whereas “x is twice y” can be
preserved by assigning to x a number twice that assigned to y. Such differences
formed the basis of Stevens’ system of scale types (most famously, nominal, ordinal, interval, and ratio; see Sect. 6.5), with the important consequence that, according to RTM, even nonquantitative properties could be considered measurable.
Indeed, according to the representational perspective, “the question of measurement” is just “about the possibility of using numbers to describe certain phenomena”, and by representational theories it “has received answers in the form of
testable conditions” (Doignon, 1993: p. 473).
This purely formal interpretation of measurement is attractive for its epistemological simplicity, but the idea that measurement is definitionally equivalent to morphic mapping is so generic that it is unable to distinguish between measurement and
consistent representation (Mari, 2013; Mari, Carbone, & Petri, 2012). Rather, what
representational theories of measurement provide is an abstract framework for scale
construction and meaningfulness of representation (Narens, 1985, 2002), and therefore at most for characterizing conditions of measurability. If, for example, a given
8 Whether such informational entities need to be numbers, with or without measurement units, is
where Stevens, and since him representational theories of measurement, departed from Campbell.
While, as just mentioned, numbers are required “to enable the powerful weapon of mathematical
analysis to be applied to the subject matter of science” according to Campbell, Stevens (1946)
made the representability of properties by means of numbers sufficient, but not necessary, for
measurability.
4.2 Characterizing measurement