On Jul 15, 2009, at 6:42 PM, Ed Barkmeyer wrote: (01)
> David Leal wrote:
>
>>>> Probably we need to look at two "kinds of quantity":
>>>> - categories such as that which includes Ed Barkmeyer's height,
>>>> width of the
>>>> Thames at London Bridge, the diameter of the earth's orbit;
>>>> - categories such as that which includes ultimate tensile
>>>> strength, yield
>>>> strength in tension, yield strength in compression (all are
>>>> stresses).
>> Can anyone elaborate on the distinction here? I don't follow it.
>
> The distinction I see is between 'particular quantities' in the
> first group, and categories of particular quantity in the second. (02)
I guess I was assuming there was more to it than this. I can speak of
the ultimate tensile strength of a given material (or object?) just as
I can speak of the width of my third finger, and that is just as
particular a value. David, correct me if I have this wrong, but were
you meaning to draw a distinction between for example the category of
lengths, and the category of strengths (stresses?)? If so, can you
expand on what you see as the difference between them? (03)
Pat H (04)
> Each of the examples in the first group is an individual physical
> phenomenon; no example in the second group is a particular quantity
> -- each of the examples in the second group is a class of
> phenomena. So if I understand David correctly, one 'kind of
> quantity' is a class whose instances are individual phenomena; the
> other 'kind of quantity' is a class whose instances are classes of
> phenomena.
>
> VIM has only one category of the second kind: 'kind of quantity'.
> The instances of 'kind of quantity' are categories of the first
> kind: each instance of 'kind of quantity' is a subclass of the
> general class 'particular quantity'.
> (forall (k)(if (kind-of-quantity k)
> (forall (x) (if (k x) (particular-quantity x)))
> ))
>
> "length", as the class of phenomena that are one-dimensional spatial
> displacements, is a 'kind of quantity'. That is, "length" is a
> subtype of 'particular quantity' and an instance of the class 'kind
> of quantity'.
>
> Further, a 'kind-of-quantity' has the property that all particular
> quantities of the same kind are comparable.
> (forall (k)(if (kind-of-quantity k)
> (forall (x y) (if (and (k x) (k y))
> (comparable x y)))
> ))
>
>>> There is a set of categories that are
>>> 'kinds of quantity', such that all instances of any 'kind of
>>> quantity'
>>> category are comparable and no pair of instances from two different
>>> kinds of quantity are comparable.
>> Why the second requirement? If length and width are different
>> quantities (why not?) they are nevertheless comparable.
>
> As I said above, the VIM only introduces one class whose instances
> are subtypes of 'particular quantity', but that class does not
> include all subtypes of 'particular quantity'! To make this clearer
> we might define 'kind of quantity' with only the axioms above, and
> then 'length' and 'width' and 'circumference' and 'diameter', etc.,
> would all be instances of it.
>
> But then we need the concept VIM has: 'special kind of quantity',
> which is a subtype of 'kind of quantity' with the additional
> property that no pair of instances from two different special kinds
> of quantity are comparable.
> (forall (s1 s2)
> (if (and (special-kind-of-quantity s1)
> (special-kind-of-quantity s2))
> (forall (x y) (if (and (s1 x) (s2 y))
> (not (comparable x y))
> ))
> ))
>
> The reason for this is the concept 'system of measurements'. A
> system of measurements selects a set of special kinds of quantity to
> be base quantities (reference dimensions), and it assigns to each
> 'base quantity' a particular measurement unit to be the reference
> unit for all instances of that 'base quantity'.
>
> So 'length' is a 'base quantity' or 'reference dimension' of SI and
> the MKS system assigns 'metre' as the reference unit for 'length'.
> MKS does not assign units to 'width' and 'diameter' and
> 'circumference', etc., because they are all subtypes of 'length' and
> can all be measured in metres.
>
> Now, velocity is not a base quantity; it is a 'derived quantity'.
> 'Derived quantities' are all the special kinds of quantity that are
> not base quantities. Each derived quantity can be defined/expressed
> as a relationship among base quantities, i.e., in terms of the
> reference dimensions. So velocity can be described as length/
> duration.
>
> This is the idea that underlies the International System of Units
> (SI).
>
> We need these "maximal subtypes" of 'particular quantity' to create
> a uniform scheme for magnitudes of things that can be compared.
> But the VIM has no need to deal with all the other subtypes -- those
> are just the particular relations, like 'circumference' and
> 'ultimate tensile strength', which are irrelevant to the system of
> measurements and the system of units.
>
> Pat noted this, but he devised an alternative model:
>
>> It would make sense to allow each KOQ to have an associated
>> dimension, but the association be many-one, so that length, width,
>> distance, etc. all are different KOQs but all associated with the
>> one physical dimension, which is spatial distance. But that
>> requires relaxing the 'incomparable with other kinds' restriction
>> you mention above. NOt sure about this.
>
> The VIM model is 1-to-1 by choosing a maximal KOQ that represents
> the dimension. That makes the KOQ and the dimension the same
> concept. The alternative model, which Pat sketched, separates the
> two concepts: KOQ and dimension, and makes the relationship many-
> to-1. In that model, there is no concept for the maximal SKOQ, but
> the comparability property is then based on the dimension, not the
> KOQ; so the reason for the SKOQ may also disappear.
>
> I hope this is clearer.
>
> One other observation: SI treats mass and energy as 'special kinds
> of quantity', with the property that no mass is comparable to an
> energy. We have known for almost 100 years that this is not correct
> at the atomic particle level. But there have been as yet no
> international commerce measurement issues that have had to be
> concerned about this. So, an ontology that supports SI implicitly
> supports only Newtonian physics! (But mass is a base quantity,
> while energy is a derived quantity, so perhaps that can somehow be
> fudged.)
>
> -Ed
>
> --
> 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
>
> "The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
> (05)
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