Joe Collins wrote:
> Dear Ingvar,
>
> From what you respond, I gather that:
>
> 1) You believe that the dimensionless dimension is overloaded and/or
>conflates
> many distinctly different kinds of derived quantities. I would agree with
>that.
>
> 2) The distinction of multiplication may be summarized in the following,
> (assuming all quantities are in Coherent Derived Units):
>
> Q3 = Q1 * Q2
> = ({Q1}*[Q1]) * ({Q2}*[Q2])
> = ({Q1}*{Q2}) * ([Q1]*[Q2])
> = {Q3} * [Q3]
>
> Where curly braces means "numerical value" and rectangular brackets mean
> Measurement Unit.
>
> You believe that the multiplication of numerical values, or normal
> multiplication, is distinct from the multiplication of units.
> I agree that multiplication of numerical values is the normal (Real number
>field
> or Rational number field) multiplication and that the multiplication of units
>is
> given by the abelian group multiplication operation I cited earlier (wherein
>the
> Base Units form a generating set).
>
> An inportant point is that they are consistent, meaning that we really only
>need
> to use one symbol in the above equation set. In reality, there are three
> multiplication operations: a) between quantities; b) between units, and; c)
> between numerical values. The quantity multiplication is a combination of the
> other two, since the set of scalar quantities are something like the outer
> product of the Reals (or Rationals, if you prefer) and the Coherent Derived
>Units.
>
> Nevertheless, a group requires an identity operator, so the Coherent Derived
> Units have a dimensionless unit, represented using a "1", and the Quantity
> Dimensions needs a member called the "dimensionless Quantity Dimension".
>
>
These are both true only after you assert that they must be abelian
groups. Why would the Quantity Dimension have to be a group? You
make the units a group so as to do symbol algebra on unit symbols, but
there is no reason to do the same for Quantity Dimensions. The only
thing we do with Quantity Dimension expressions is use them to state
relationships of a derived dimension to other dimensions. For that, it
may or may not help to have association and commutativity, but it is not
clear why we would need an identity.
> So, if you propose to enlarge the set of Base Units, where do you stop?
With the Base Quantity "count"/"ones". It is the only measurement
quantity in common use that is not derivable from the SI Base Quantities.
> There is an arbitrarily large number of dimensionless
> quantities of different Kinds.
Yes. There is an arbitrarily large number of Derived Quantities, some
of which are "dimensionless" in your terms, but have very clear
derivation expressions in terms of base quantities and perhaps other
derived quantities. The "base quantity" "ones" isn't derived. It
doesn't have any such expression that conveys anything about its meaning.
> You may have your favorites, but what
> distinguishes them from someone else's favorites? The SI has chosen to leave
>the
> Base Units bounded in number and the mapping "Kind" loosely defined, I
>presume
> so that users may choose to make such distinctions as they deem necessary by
> defining as many Kinds as they want.
>
Of course. The expectation is that new derived quantities will arise as
scientific and engineering disciplines expand. (01)
The fundamental problem with your approach is that no derived quantity
is "dimensionless" in the sense of lacking a relationship to any base
quantity. The fact that your symbol algebra discards relationships
without apparent harm to the calculations doesn't mean that you have
discovered a meaningful simplification of the Derived Quantity concept.
At the level of defining the Kind of Quantity, the simplification
(reduction to lowest terms) LOSES meaning. And at that level, the loss
of meaning is loss of knowledge. Energy is not the same thing as
Torque, and if you simplify the derivation expressions for the Quantity,
you can't distinguish them. If you simplify the unit arithmetic you get
mathematically correct results, which will be physically correct results
only if you interpret the mathematical results correctly. That is the
difference. (02)
-Ed (03)
"We must strive to make things as simple as possible, but no simpler."
-- Albert Einstein (04)
--
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694 (05)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (06)
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