> Dear Ed, Ingvar and others,
>
> Ed's initial definition "A quantity scale is a mapping from an ordered set
> of symbols to a kind of quantity" seems very close to me.
>
> This can be made more precise because:
> 1) There has been misunderstanding about the use of the term "symbol". In
> some maths textbooks, the term is used to include numbers. "Element" may
> be
> a better term. (01)
And neither VIM nor the SI brochure are using 'symbol' in the proposed sense. (02)
> 2) Ed has distinguished between two different meanings of kind of
> quantity.
> Kind of quantity Q3 seems most appropriate in this definition.
>
> Hence the definition can be modified to be:
> "A quantity-value scale is a mapping from an ordered set of elements (e.g.
> integers, reals, letters in an alphabet) to a kind of quantity with
> definition Q3." (03)
I would say as follows: in measurements, quantity-values are mapped from a
scale to spatiotemporal elements (Q1), but in constructions of scales,
equivalence classes (Q3) of spatiotemporal magnitudes are mapped to
quantity-values. (04)
Best, Ingvar (05)
> There is also a need for clarity about the term "mapping". Some people
> regard "mapping" and "function" as synonyms. Other people restrict the use
> of the term function to mappings which are total (i.e. defined for each
> member of their domain), and single valued (i.e. which map each member of
> their domain to exactly one member of their range).
>
> I agree with Ed that a constraint that the mapping be single valued is
> useful, but a constraint that it be total is wrong. Where the mapping is
> from the reals to a kind of quantity, it cannot be known whether or not
> there is a member of the kind of quantity for each real.
>
> I believe that Ed's further development of the definition to restrict the
> domain to quantity values is incorrect and may be a mis-reading of the
> VIM.
> The reason for this is as follows:
> 1) from the definition of quantity value, there exists a set of (quantity
> value, magnitude of quantity) pairs;
> 2) a quantity value is a (number, reference) pair, so there exists are set
> of (number, reference, magnitude of quantity) triples;
> 3) for a single reference we can extract a set of (number, magnitude of
> quantity) pairs.
>
> I think that the VIM says that the set of (number, magnitude of quantity)
> pairs is a quantity-value scale that is defined by a quantity value
> reference. A quantity value reference can be a unit, but need not be.
>
> Best regards,
> David
>
> At 15:45 14/08/2009 -0400, you wrote:
>>David Leal wrote:
>>> In your e-mail you have used "scale" as a synonym for "kind of
>>> quantity",
>>> because both temperature and mass are kinds of quantity.
>>>
>>We are still having trouble with consistent terminology. As Matthew
>>pointed out, the relationship between kind of quantity (e.g. length) and
>>quantity scale is one-to-many. A given quantity scale applies to only
>>one kind of quantity, but there can be many different scales for any
>>given kind of quantity.
>>> I have proposed that a scale be defined as a function from a kind of
>>> quantity to a "set of elements with axioms", such as the real numbers.
>>> To be
>>> a valid scale there would have to be constraints upon the nature of
>>> this
>>> function. I have no strong opinion as to whether the term scale should
>>> be
>>> applied to this function, or to its inverse, or to both.
>>>
>>> I don't think that my proposal has consensus, but I am not clear about
>>> further proposals.
>>>
>>David's proposal does NOT have consensus.
>>
>>(As Ingvar corrected me,) the VIM says a 'quantity scale' is an "ordered
>>set of quantity values that express magnitudes of a common kind of
>>quantity". So the base definition is the reverse of David's
>>definition: A quantity scale is a mapping from an ordered set of
>>symbols to quantity magnitudes of a kind of quantity. If we agree that
>>each (instance of) 'kind of quantity' is a subtype of 'quantity
>>magnitude' (my Q3) [*see below], then we can state this as:
>> A quantity scale is a mapping from an ordered set of symbols to a kind
>>of quantity.
>>(which is closer to David's definition).
>>
>>But the VIM makes a stronger requirement. The symbols must be 'quantity
>>values'. 'quantity value' is defined as "the expression of a 'quantity
>>magnitude' as a number and a measurement unit." More carefully, the
>>symbols are called "quantity values" and they have the structure
>>(number, measurement unit), and the scale defines the relationship
>>between each quantity value and a quantity magnitude. That is,
>>'quantity value expresses quantity magnitude' _is_ the mapping that is
>>(defined by) the scale. Put another way, a 'quantity value' has no
>>meaning without reference to a scale that defines the mapping to
>> magnitudes.
>>
>>Now, all magnitudes of the same kind of quantity are "comparable", which
>>I take to mean that the magnitudes of a given kind of quantity are
>>intrinsically ordered. It follows that the "set of symbols" -- the set
>>of quantity values -- is ordered by the mapping to magnitudes and the
>>ordering of the magnitudes.
>>
>>If the measurement unit part of each quantity value is constant across
>>all quantity values of the scale, then intrinsic mathematical ordering
>>of the number parts must be consistent with the ordering of the
>>magnitudes. We may find it convenient to require that the measurement
>>unit part of all quantity values of a given scale is the same, in which
>>case it becomes a property of the scale itself, and each of the quantity
>>values has a representation as a number only. This leads to a model
>>that is closer to the one David propounded, to wit:
>> A quantity scale is a mapping from an (intrinsically) ordered set of
>>numbers to a kind of quantity, in which each number represents a
>>quantity value of the form (n, u) where n is the number and u is the
>>measurement unit associated with the scale.
>>(But I'm not sure we want this definition. The intent is that the scale
>>is a mapping from quantity values to magnitudes, and this latter
>>definition is more about representation than intent.)
>>
>>We want to require that the mapping is a function, i.e., that each
>>quantity value is mapped to exactly one magnitude.
>>We want to require that the mapping is 1-to-1 ("injective"), i.e., that
>>different quantity values always map to different magnitudes, stated:
>>if m is a quantity scale and v1 v2 are quantity values in S and m(v1) =
>>m(v2) then v1=v2.
>>
>>Now, if the cardinality of the quantity value set S is equal to the
>>cardinality of the magnitude set Q, it is possible that the mapping can
>>be onto ("surjective"), i.e., for every q in Q, there is some v in S
>>such that m(v) = q. (But if they are both infinite sets, the mapping
>>doesn't have to be onto.) But if Q has more instances than S, then the
>>scale mapping m can never be onto: there is always some q in Q such
>>that no v in S is mapped to q. For example, if S is countable (which is
>>typical of scales in use) and Q is uncountable (like the Real numbers),
>>then the scale mapping is not onto. And that means the inverse mapping
>>from Q to S (which is what David postulates) is not defined by m.
>>It is convenient to define that mapping by "interval", i.e., the
>>"inverse mapping" f: Q -> S is defined by:
>> f(q) = that value v in S such that m(v) <= q and for all x such that
>>m(x) <= q, v >= x. That is, f(q) is the largest value in S that maps to
>>a magnitude less than or equal to q.
>>For example, if your scale S has only centimetre marks, then f maps the
>>quantity that would be 38 mm to 3cm.
>>
>>The reason for doing all this is that S is the set of all real number
>>multiples of some unit only in theory. All actual scales are discrete.
>>There is a finest level of distinction they can make, which we may call
>>the "granularity", and they can't distinguish values that are closer
>>than the granularity. (In the above example, the granularity of the
>>scale is 1cm.)
>>
>>We can also say that a scale is "regular" if it has some granularity g
>>that is a magnitude in Q and S is the set of all (non-negative, if
>>necessary) integer multiples of g. It is not a requirement that g be a
>>standard/conventional measurement unit. For example, g could be 1/60
>>second. Whether a scale is regular or not is not entirely a choice of
>>the scale designer. As John Sowa pointed out, the physical properties
>>of the quantity kind Q dictate to some extent what kinds of scales are
>>useful.
>>
>>I am clearly proposing the model of 'scale' above as an alternative to
>>David's proposed model. But I am sure there are a number of
>>inaccuracies in it. Consider this "running it up a flagpole". (The
>>other discussion we have been having is about how the units will be
>>defined and what quantity value (0, unit) means.)
>>
>>-Ed
>>
>>*The VIM seems to say that each instance of 'kind of quantity' is a
>>subtype of 'particular quantity'. That is, the instances of a given
>>'kind of quantity', such as length, are the "tropes" -- the lengths of
>>specific things. By definition all the instances of a given 'kind of
>>quantity' are comparable. What we are saying is that for each VIM 'kind
>>of quantity' there is a set of equivalence classes of particular
>>quantities (tropes) such that all the members of each equivalence class
>>are "equal in that quantity", and we are calling each equivalence class
>>a "(quantity) magnitude". So all the things that have the same length
>>are members of one "length magnitude", e.g., the magnitude that is
>>called "5cm". And now we are saying that for our model, each instance
>>of a UOM 'kind of quantity' is the class of all magnitudes of that VIM
>>'kind of quantity', e.g., "length" is the UOM 'kind of quantity' that is
>>the class of all 'length magnitudes'. So every instance of UOM 'kind of
>>quantity' is a subtype of 'quantity magnitude'. This is not a major
>>departure from the VIM, but it is a difference.
>>
>>Per my previous "concept set",
>> Q1 is 'particular quantity', the class of individual quantifiable
>>properties of individual things (the tropes).
>>Example: the length of my thumb (and no one else's thumb)
>> Q2 is VIM 'kind of quantity', the class whose members are subtypes of
>>Q1 in which all the individual properties are comparable.
>>Example: length, seen as the class of properties to which "the length of
>>my thumb" belongs.
>> Q3 is '(quantity) magnitude', the class whose members are equivalence
>>classes, each of which comprises all properties in Q1 that compare as
>> equal.
>>Example: the length magnitude called "5cm", which includes the length of
>>my thumb, the height of my coffee cup, and so on.
>> Q4 is UOM 'kind of quantity', the class whose members are subtypes of
>>Q3 in which all the members of all the magnitude equivalence classes
>>belong to the same Q2, the same VIM 'kind of quantity'.
>>Example: length, seen as the class of magnitudes named "5cm" and "4
>>miles" and the like.
>> 'measurement unit' is a subclass of 'quantity magnitude', each
>>measurement unit is a specific magnitude of a specific UOM 'kind of
>>quantity'.
>>
>>This matches the terminology David's draft uses, except that he doesn't
>>distinguish Q2 from Q4. Do we agree to this terminology?
>>
>>--
>>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."
>>
>>
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>>
>
> ============================================================
> David Leal
> CAESAR Systems Limited
> registered office: 29 Somertrees Avenue, Lee, London SE12 0BS
> registered in England no. 2422371
> tel: +44 (0)20 8857 1095
> mob: +44 (0)77 0702 6926
> e-mail: david.leal@xxxxxxxxxxxxxxxxxxx
> web site: http://www.caesarsystems.co.uk
> ============================================================
>
>
>
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