Joe Collins wrote: (01)
> I, however, do not advocate a theory different from the SI.
> I have no problem with the Quantity Dimension "one" (or "dimensionless"). (02)
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. (03)
> 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. (04)
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'. (05)
> 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. (06)
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." (07)
> 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. (08)
> Are you proposing that your theory be incorporated in an UoM ontology in
> preference to the long-standing, internationally agreed upon SI? (09)
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. (010)
Best,
Ingvar (011)
> 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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