Gunther Schadow wrote: (01)
> Joe Collins wrote:
>> Dear Ingvar,
>> Assume you have a beam.
>> The beam has a length, l, a volume, V, a surface area, A, and a second
>> moment of
>> area, I.
>> The properties l, V/A, and sqrt(sqrt(I)) all have quantity dimension of
>> Are they all of the same kind?
>> How do you know? By what process does one decide?
> This is exactly my point. Thanks Joe for giving such rich set of
> fresh examples. (02)
I am sorry, before I sent my mail I should have repeated that I regard all
multiplications and divisions of dimensions as giving rise to new and
un-reducible dimensions. For instance, I regard V/A as being the
un-reducible dimension volume per area, not length. And I regard as false
the VIM/SI view that relative magnitudes such as relative length and mass
fraction have the dimension one. At the end of the this mail, I have
inserted some paragraphs about this issue; they are from an unpublished
paper of mine. (03)
> And this is the heart of the answers to the other related issues
>>> Gunther S (to Pat H): “so we agree that 1 N.m = 1 N.m when we talk
>>> units and there is no such thing as "N.m moment of force" as a unit?
>>> There is of course the Quantity moment of force 1 N.m, but the unit is
>>> still N.m without knowing anything about torque vs. energy.”
>>> IJ-comment: Every real unit presupposes a scale for a kind-of-quantity;
>>> unit with no reference at all to a scale or quantity is meaningless.
>>> I have proposed to call a ‘nominal unit’, is a unit that necessarily
>>> refers to other units, i.e., to real units with scales for
>>> Gunther S: “The VIM speaks about Quantities and measurement and a
>>> bit about Units. The SI speaks a lot about Units. I don't think that
>>> either one argues that Units contain the detail of the Quantities.”
>>> IJ-comment: I think you are wrong. VIM’s definition 1.9 says:
>>> unit = real scalar quantity, defined and adopted by convention, with
>>> any OTHER QUANTITY OF THE SAME KIND CAN BE COMPARED. Definition 1.10
>>> base unit = measurement unit that is adopted by convention FOR A BASE
>>> QUANTITY. The SI brochure says (p. 103): The terms quantity and unit
>>> defined in VIM. That is (at least in my interpretation), both VIM and
>>> SI are stating the view that ‘Units presuppose the detail of the
>>> Quantities’. Or in my words, once again: every real unit presupposes a
>>> scale for a kind-of-quantity.
> I am not doubting that in a system of quantities and units, every
> base unit is tied to one base kind of quantity. And in that respect,
> all of Joe C's examples are of kind of quantity "length". But it
> is length at a very high level of abstraction where kind of quantity
> is 1:1 related to dimension. They are all in dimension L, and their
> base kind of quantity is "length" but we would be hard pressed to
> be happy calling the quantity "surface areic volume" (V/A) a "length". (04)
In my opinion (see above), a "surface areic volume" is not a "length". So,
here we agree. (05)
> This is where IMO the VIM and SI have not completely succeeded to
> formally lay out the problem space. We use "quantity" and "kind
> of quantity" is if we understood what it means, but they didn't
> really define those very well. (06)
In my opinion, VIM can and should be interpreted as if "kind-of-quantity"
is defined by means of the notion of "significant physical-chemical
comparability"; and this latter notion has to be taken as being primitive. (07)
> I believe that "kind of quantity" for VIM and SI is often this very
> high-level kind where everything measured in 1 m is a "length".
> They use the term "quantity" to mean the more specific quantitative
> property universal, but still "quantity" is used as universal. Notably
> the VIM isn't very good in applying "universal" and "particular"
> systematically and so "quantity" is still ambiguous and therefore
> "kind of quantity" is ambiguous too. (08)
I agree wholeheartedly with the last sentence, but I think Ed Barkmeyer's
distinctions between Q1, Q2, Q3, and Q4 has made this clear. (09)
> -Gunther (010)
Here come the paragraphs I talked about at the beginning of the mail: (011)
Let me frame in my own words what I regard as the central objection to the
Both relative length (original dimension: length/length) and mass fraction
(original dimension: mass/mass) should according to the present SI system
be measured by the same metrological unit, one, and be ascribed the same
dimension, dimension one. Now, if one knows only that A and B has the mass
fraction 5 mass/mass and that C and D has the fraction 3 mass/mass, then
it is impossible to draw any conclusions about the fractions A/C and A/D;
and the same is of course true if we exchange mass/mass for length/length.
Similarly, if A and B has the mass fraction 5 mass/mass and C and D has
the relative length 3 length/length, it is impossible to draw any
conclusions about a ratio between A and C and A and D. Nonetheless there
is a relevant difference between the last example and the first two ones,
which brings home Emerson’s point.
Even if not known, there is always a mass fraction (and a relative length)
also between B and C; and as soon as this fraction becomes known, the mass
fractions (and, alternatively, the relative lengths) A/C and A/D can be
calculated and in a significant way compared with that of A/B. However,
mass fractions and relative lengths can never in such a sense be compared
with each other; it is always impossible to compare in a physical-chemical
meaningful way A/B mass/mass with that of A/D length/length. This means
that if both mass fraction and relative length are measured by the unit
one, then this unit is (in the light of Sections 1 and 2) a nominal unit.
Consequently, the dimension one is only a nominal dimension.
The fact that the so-called dimensionless quantities are not completely
dimensionless is in passing noted in the SI brochure (which, remember,
claims that dimensionless quantities have the unit one): “In a few cases,
however, a special name is given to the unit one, in order to facilitate
the identification of the quantity involved.” Why, I ask, is there a
problem with “the identification of the quantity involved” if all
dimensionless quantities are the same kind of quantities? Why is there a
need to distinguish one kind of dimensionless quantity from another?
Answer: because the so-called dimensionless quantities are not in fact
What then about multiplications of dimensions and units that do not give
rise to relative kind-of-quantities? What about, for instance, L2 with
unit m2, L T-1 with unit m s-1, and M L-3 with unit kg m-3? How can they
be dimensions and units for area, speed, and density, respectively? In my
opinion, out of context, they cannot. When area is ascribed the dimension
L2, the context makes it clear that the two L has to be orthogonal to each
other, something which is not and cannot be said in the SI, since it
restricts itself to scalar quantities. When speed is ascribed the
dimension L T-1, the context has it that L and T are related to one and
the same movement. When density is ascribed the dimension M L-3, the
context makes it clear that M and L-3 are properties of one and the same
material thing. In abstraction, these metrological multiplications have no
specific physical-chemical significance. (012)
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