67
appears plausible, so that a theory of measurement and a theory of measure would
be more or less the same thing, or at least inherently related. This position is evidenced, for example, in the claim that “to understand measurement theory it is necessary to revisit the theory of integration and, particular, Lebesgue measure theory”
(Sawyer, Sankey, & Lombardo, 2013: p. 90).
However, suspicions about the equivalence of and
might arise from a sufficiently careful reading of Euclid’s work itself, which is not
really about measurement as we intend it. For example, in the introduction to an
English translation of the Elements one can read that “in the geometrical constructions employed in the Elements […] empirical proofs by means of measurement are
strictly forbidden” (Euclid, 2008: introductory notes). Let us indeed compare the
abovementioned definition by Euclid, with the following one, now from Book 7 of
the Elements: “a number is part of a(nother) number, the lesser of the greater, when
it measures the greater” (VII.3, emphasis added). While the two quoted sentences
refer to different entities (magnitudes, μεγέθη, and numbers, ἀριθμοὶ), in both the
relation is said that one entity measures (καταμετρῇ) the other. Hence, as derived
from the Euclidean tradition, “to measure” does not necessarily have an empirical
connotation, and the Euclidean is coextensive with
reminder. So, 2 is a measure of 4, of 8, or of any even number; and 3 is a measure
of 6, or of 9, or of 12, etc.” (Hutton, 1795). The conclusion is that “measure” has (at
least) two distinct meanings: one is empirical, and is indeed related to measurement,
and the other is mathematical; this twofoldness, already recognized by Bunge
(1973), has often been confused.
21
Perhaps unsurprisingly, on this conceptual basis a “measure theory” has developed, where “a measure is an extended real valued, non negative, and countably
additive set function μ, defined on a ring R, and such that μ(0) = 0” (Halmos, 1950:
p. 30): that is, it is a mathematical entity. Whether and how measure theory is related
to a theory of measurement, and more generally to measurement science, is an issue
that we discuss in Chap. 6, in the section about the measurability of nonquantitative
(and specifically nonadditive) properties. But it should be clear now that “measure”
is not just a synonym of “measurement” and, most importantly, that “quantification”
is not just a synonym of “measurement”: not every quantification is a measurement,
and it could be accepted that nonquantitative properties may also be measured.
Thus, one can see the wisdom in the VIM’s avoidance of any use of the noun “mea21 Consider, as a significant case, what Michell and Ernst wrote on this matter: “there are two sides
to measurement theory: one side (emphasized in the modern era) at the interface with experimental
science, the other side (emphasized in the classical) at the interface with quantitative theory”
(1996: p. 236). But these are two sides of measure, not measurement. This sentence is excerpted
from the introduction that Michell wrote to the English translation of the 1901 paper by Hölder on
the axioms of quantity. This confusion was worsened by their choice of translating in the title of
Hölder’s paper the German noun “mass” as “measurement” rather than “measure”: they used “The
axioms of quantity and the theory of measurement”, instead of “The axioms of quantity and the
theory of measure” (Mari et al., 2017).
3.4 The conceptual context
appears plausible, so that a theory of measurement and a theory of measure would
be more or less the same thing, or at least inherently related. This position is evidenced, for example, in the claim that “to understand measurement theory it is necessary to revisit the theory of integration and, particular, Lebesgue measure theory”
(Sawyer, Sankey, & Lombardo, 2013: p. 90).
However, suspicions about the equivalence of
might arise from a sufficiently careful reading of Euclid’s work itself, which is not
really about measurement as we intend it. For example, in the introduction to an
English translation of the Elements one can read that “in the geometrical constructions employed in the Elements […] empirical proofs by means of measurement are
strictly forbidden” (Euclid, 2008: introductory notes). Let us indeed compare the
abovementioned definition by Euclid, with the following one, now from Book 7 of
the Elements: “a number is part of a(nother) number, the lesser of the greater, when
it measures the greater” (VII.3, emphasis added). While the two quoted sentences
refer to different entities (magnitudes, μεγέθη, and numbers, ἀριθμοὶ), in both the
relation is said that one entity measures (καταμετρῇ) the other. Hence, as derived
from the Euclidean tradition, “to measure” does not necessarily have an empirical
connotation, and the Euclidean
reminder. So, 2 is a measure of 4, of 8, or of any even number; and 3 is a measure
of 6, or of 9, or of 12, etc.” (Hutton, 1795). The conclusion is that “measure” has (at
least) two distinct meanings: one is empirical, and is indeed related to measurement,
and the other is mathematical; this twofoldness, already recognized by Bunge
(1973), has often been confused.
21
Perhaps unsurprisingly, on this conceptual basis a “measure theory” has developed, where “a measure is an extended real valued, non negative, and countably
additive set function μ, defined on a ring R, and such that μ(0) = 0” (Halmos, 1950:
p. 30): that is, it is a mathematical entity. Whether and how measure theory is related
to a theory of measurement, and more generally to measurement science, is an issue
that we discuss in Chap. 6, in the section about the measurability of nonquantitative
(and specifically nonadditive) properties. But it should be clear now that “measure”
is not just a synonym of “measurement” and, most importantly, that “quantification”
is not just a synonym of “measurement”: not every quantification is a measurement,
and it could be accepted that nonquantitative properties may also be measured.
Thus, one can see the wisdom in the VIM’s avoidance of any use of the noun “mea21 Consider, as a significant case, what Michell and Ernst wrote on this matter: “there are two sides
to measurement theory: one side (emphasized in the modern era) at the interface with experimental
science, the other side (emphasized in the classical) at the interface with quantitative theory”
(1996: p. 236). But these are two sides of measure, not measurement. This sentence is excerpted
from the introduction that Michell wrote to the English translation of the 1901 paper by Hölder on
the axioms of quantity. This confusion was worsened by their choice of translating in the title of
Hölder’s paper the German noun “mass” as “measurement” rather than “measure”: they used “The
axioms of quantity and the theory of measurement”, instead of “The axioms of quantity and the
theory of measure” (Mari et al., 2017).
3.4 The conceptual context