172
that “we now know precisely why some attributes are measurable and some not:
what makes the difference is possession of quantitative structure” (p. 59). Along the
same vein Jan De Boer claimed that “Helmholtz and the mathematician Hölder are
usually seen as the initiators of the axiomatic treatment of what is often called the
theory of measurement” (De Boer, 1995: p. 407). But even just a glance at the scope
of Hölder’s axioms shows that they do not relate to any experimental process, as
would be expected from von Helmholtz’s own words—“the most fruitful, most certain, and most exact of all known scientific methods” (von Helmholtz, 1887: p. 1)—
and confirmed by the way the VIM defines: a “process of
experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity” (JCGM, 2012: 2.1). Indeed, Hölder himself admitted that “by
‘axioms of arithmetic’ has been meant what I prefer to call ‘axioms of quantity’”
(p. 237), so that measurement is involved in them only insofar “the theory of the
measurement” is equated to “the modern theory of proportion” (p. 241), thus confirming the purely mathematical nature of the treatment.
In Sect. 3.4.2 we argued that this superposition of conditions of being measurable and being quantitative derives from a confusion between, an
empirical concept, and, a mathematical concept. The conclusion is simple to state: what is to be found in Euclid’s Elements and what Hölder considered
“the modern theory of proportion” is not a theory of measurement but a theory of
measure, where measures are taken to be continuous quantities. From this, one
might assume that only properties modeled as measures are measurable—this is
what we take to be the position of what Michell (1990) calls “the classical theory of
measurement”, as rooted in Euclid’s geometry—but his sharp tenet that “without
ratios of magnitudes there is no measurement” (p. 16) cannot be maintained without
this strong and basically arbitrary assumption.
The problems generated by this confusion are not just lexical or semantic. A
well-grounded distinction between quantitative and nonquantitative properties
would be a key target, at least as a means to identify and justify possible differences
in inferential processes and their results, and therefore in the kind of the information
they produce. The basic intuition about the distinction remains, e.g., that individuals
can be compared in such a way that the height of a person can be one-third greater
than the height of another, or a difference on an interval scale can be one-third
greater than another difference, whereas the blood type of that person cannot be
one-third greater than the blood type of another. This intuition needs a persuasive
explanation, which ultimately would be beneficial for a better identification of the
conditions of measurability. In fact, the mentioned confusion is a good reason for
developing this analysis as a key component of an ontology and an epistemology of
properties: once an appropriate classification of types of properties has been established, whether only quantities are measurable might be thought of as simply an
arbitrary lexical choice (Mari, Maul, Torres Irribarra, & Wilson, 2017).
32
32 Given this, the reader will not find here the proposal of a clear-cut criterion to distinguish between
quantities and non-quantities. At least since Hölder’s (1901) paper, several axiomatizations of
quantities have been proposed (e.g., Mundy, 1987; Suppes, 1951; Suppes & Zanotti, 1992), and
6 Values, scales, and the existence of properties
that “we now know precisely why some attributes are measurable and some not:
what makes the difference is possession of quantitative structure” (p. 59). Along the
same vein Jan De Boer claimed that “Helmholtz and the mathematician Hölder are
usually seen as the initiators of the axiomatic treatment of what is often called the
theory of measurement” (De Boer, 1995: p. 407). But even just a glance at the scope
of Hölder’s axioms shows that they do not relate to any experimental process, as
would be expected from von Helmholtz’s own words—“the most fruitful, most certain, and most exact of all known scientific methods” (von Helmholtz, 1887: p. 1)—
and confirmed by the way the VIM defines
experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity” (JCGM, 2012: 2.1). Indeed, Hölder himself admitted that “by
‘axioms of arithmetic’ has been meant what I prefer to call ‘axioms of quantity’”
(p. 237), so that measurement is involved in them only insofar “the theory of the
measurement” is equated to “the modern theory of proportion” (p. 241), thus confirming the purely mathematical nature of the treatment.
In Sect. 3.4.2 we argued that this superposition of conditions of being measurable and being quantitative derives from a confusion between
empirical concept, and
“the modern theory of proportion” is not a theory of measurement but a theory of
measure, where measures are taken to be continuous quantities. From this, one
might assume that only properties modeled as measures are measurable—this is
what we take to be the position of what Michell (1990) calls “the classical theory of
measurement”, as rooted in Euclid’s geometry—but his sharp tenet that “without
ratios of magnitudes there is no measurement” (p. 16) cannot be maintained without
this strong and basically arbitrary assumption.
The problems generated by this confusion are not just lexical or semantic. A
well-grounded distinction between quantitative and nonquantitative properties
would be a key target, at least as a means to identify and justify possible differences
in inferential processes and their results, and therefore in the kind of the information
they produce. The basic intuition about the distinction remains, e.g., that individuals
can be compared in such a way that the height of a person can be one-third greater
than the height of another, or a difference on an interval scale can be one-third
greater than another difference, whereas the blood type of that person cannot be
one-third greater than the blood type of another. This intuition needs a persuasive
explanation, which ultimately would be beneficial for a better identification of the
conditions of measurability. In fact, the mentioned confusion is a good reason for
developing this analysis as a key component of an ontology and an epistemology of
properties: once an appropriate classification of types of properties has been established, whether only quantities are measurable might be thought of as simply an
arbitrary lexical choice (Mari, Maul, Torres Irribarra, & Wilson, 2017).
32
32 Given this, the reader will not find here the proposal of a clear-cut criterion to distinguish between
quantities and non-quantities. At least since Hölder’s (1901) paper, several axiomatizations of
quantities have been proposed (e.g., Mundy, 1987; Suppes, 1951; Suppes & Zanotti, 1992), and
6 Values, scales, and the existence of properties