Dear Ingvar, (01)
From what you respond, I gather that: (02)
1) You believe that the dimensionless dimension is overloaded and/or conflates
many distinctly different kinds of derived quantities. I would agree with that. (03)
2) The distinction of multiplication may be summarized in the following,
(assuming all quantities are in Coherent Derived Units): (04)
Q3 = Q1 * Q2
= ({Q1}*[Q1]) * ({Q2}*[Q2])
= ({Q1}*{Q2}) * ([Q1]*[Q2])
= {Q3} * [Q3] (05)
Where curly braces means "numerical value" and rectangular brackets mean
Measurement Unit. (06)
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). (07)
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. (08)
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". (09)
So, if you propose to enlarge the set of Base Units, an essential question is,
where do you stop? There is an arbitrarily large number of dimensionless
quantities of different Kinds. 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. (010)
Regards,
Joe C. (011)
ingvar_johansson wrote:
> Joe Collins wrote:
>
>> I, however, do not advocate a theory different from the SI.
>> I have no problem with the Quantity Dimension "one" (or "dimensionless").
>
> To be 'dimensionless' or to have the 'dimension one' is not exactly the
> same thing, even though the SI system and VIM now and then write as if
> they were the same.
>
>> The quantity "angle" is physically measurable, has the natural unit
>> radian, and
>> is naturally dimensionless. There's nothing unnatural about the so-called
>> dimensionless Quantity Dimension.
>
> There is nothing wrong with the practical use of radian, but in my (and
> some others) opinion, its dimension is 'plane angle', neither 'dimension
> one' nor 'dimensionless'.
>
>> Multiplication and division are well defined arithmetic operations and
>> their
>> application to the physical theory of units and quantities in the SI is
>> consistent with their usual mathematical meaning.
>
> Here is a paragraph from my paper:
> "What I will stress is this: metrological multiplication importantly
> differs from arithmetic multiplication.
> Arithmetic multiplications of integers (which I think is enough to
> discuss here) have always a very clear-cut connection to arithmetic
> addition; a multiplication can be regarded as repeated addition.
> Metrological multiplications, however, have no relations to any
> corresponding metrological additions. Furthermore, they cannot have,
> since there simply is no such thing as meaningful additions of physical
> dimensions or metrological units. For instance, whereas the
> multiplication 3 ∙ 5 is equivalent to the repeated addition 3 + 3
> + 3 + 3 + 3 (or 5 + 5 + 5), neither for m ∙ m (L ∙ L) nor
> for s ∙ A (T ∙ I) is there any corresponding metrological
> addition. In additions such as 3m + 3m + 3m + 3m + 3m the metrological
> units are not added. In the exemplification, five quantities of the
> same dimension and metrological unit are added, but the dimensions and
> units themselves are not added."
>
>> I see no reason to make
>> a
>> distinction if there is no substantive difference. What new mathematics is
>> required to re-define multiplication and division? It is true that the
>> theory of
>> units and quantities is a "physical theory" which is distinct from a
>> mathematical theory. Were it to fail as a theory, it would only mean that
>> we
>> would have to apply a different formalism. Pragmatically, I find this
>> unlikely.
>
>> Are you proposing that your theory be incorporated in an UoM ontology in
>> preference to the long-standing, internationally agreed upon SI?
>
> No, by no means! Of course an UoM ontology should be consistent with the
> SI system. In the paper I have mentioned, I am arguing for changes in the
> SI system.
>
> Best,
> Ingvar
>
>> Regards,
>> Joe C.
>> ingvar_johansson wrote:
>>> Joe Collins wrote:
>>>
>>>> There is an inherent mathematical structure to the SI units and
>>>> dimensions
>>>> beyond scale which are defined in the SI.
>>>>
>>>> The Base Quantities, length, mass, time, electric current, temperature,
>>>> amount
>>>> of substance, and luminous intensity, whose dimensions, L, M, T, I, Θ,
>>>> N,
>>>> J,
>>>> form a generating set, using the operations of multiplication and
>>>> multiplicative
>>>> inverse, for the Quantity Dimensions.
>>> In my paper "Two Changes in the International System of Units?", which I
>>> mentioned in my former mail, I argue that it is important not to
>>> conflate
>>> *metrological* multiplication and division of dimensions with
>>> *arithmetical* multiplication and division. If one does, one ends up in
>>> proposing (as VIM and the SI system do) the dimension 'dimension one'.
>>>
>>> Ingvar
>> --
>> _______________________________
>> 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
>> _______________________________
>>
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>
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> (012)
--
_______________________________
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
_______________________________ (013)
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