Dear Ed, (01)
I did not make the structure of the SI Quantity Dimensions a group.
The standard refers to Base Quantities, the "dimensional product" and positive
and negative "dimensional exponents" and a "quantity of dimension one".
I did not make the structure of SI Coherent Derived Units a group.
I merely observe that structure in the SI and state it in more formal
mathematical terms than the standard document does. (02)
The standard does not refer to right or left application of a product: do you
think that Q1*Q2 is ever not equal to Q2*Q1, or (Q1*Q2)*Q3 is ever not equal to
Q1*(Q2*Q3) (when expressed in Coherent Derived Units)? Scalar quantities are
clearly commutative and associative with respect to their multiplication. The
same applies to Quantity Dimensions whenever I do dimensional analysis: do you
have a different experience?
It is closure under the SI specified operations that gives Quantity Dimensions
and Coherent Derived Units both identity elements.
If you were to expand the Base Units by adding more units (and correspondingly,
more Base Quantities), the structure would be pretty much the same except there
would be larger generating sets. (03)
You ask why must it be a group structure.
You may as well ask why does any physical theory have the particular
mathematical structure that it does. The answer is twofold: it is a combination
of Nature's way and the (collective) observer's abstraction of measurement data.
The symbol algebra merely reflects what is in the SI, and I did not create the
SI or its underlying theory. (04)
It seems you raise a question as to whether this structure is necessary.
First, I only claim that it reflects what's in the standard.
I would only claim that the structure is sufficient. One could ask whether all
of the Base Quantities are truly necessary, i.e., truly fundamental. In fact
there are arguments that seven is excessive. For example, following Einstein's
theory, time could be measured in metres, and energy could be measured in mass
units, since they are equivalent modulo factors of the constant speed of light
(t=x/c, E=m*c*c). The mole, in my opinion, is purely for the convenience of
chemists (NOT a bad thing). My observation is that the SI units are sufficient
for what they describe, but not all necessary. I believe that the reason for
its
redundancy is largely convenience: absolute necessity was not the only
consideration. They just didn't strictly follow Albert Einstein's dictum on
simplicity when they made the SI.
If you want to create a structure that has greater conveniences than the SI
provides, e.g., by adding more base units, I have no problem with that. I only
call out the structure of what is in the SI. (05)
Preservation of meaning is important, and I don't mean to exclude that.
That's exactly why I brought up the inherent mathematical structure in the SI.
The discussion here seems to be about formal properties.
Just because the Coherent Derived Units for torque and energy are the same
doesn't mean that I personally advocate that people should fully equate the two.
They are only "dimensionally equivalent", meaning Dim(torque)=Dim(energy),
where
Dim maps a Derived Quantity to a Quantity Dimension.
These mappings for Quantity Dimension and Coherent Derived Unit, Dim() and [],
as applied to quantities, are abstractions, or projections, and, as such,
necessarily ignore some information. Observing the existence of the
abstractions
is not advocacy for throwing out the details altogether. It seems you are
accusing me of killing kittens by observing they belong to the Feline class. I
think you must keep both instances and knowledge of their class membership. (06)
Regards,
Joe C. (07)
Ed Barkmeyer wrote:
> 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.
>
> 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.
>
> -Ed
>
> "We must strive to make things as simple as possible, but no simpler."
> -- Albert Einstein
--
_______________________________
Joseph B. Collins, Ph.D.
Code 5583, Adv. Info. Tech.
Naval Research Laboratory
Washington, DC 20375
(202) 404-7041
(202) 767-1122 (fax)
B34, R221C
_______________________________ (08)
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