Dear Ed, Ingvar and others, (01)
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. (02)
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.
2) Ed has distinguished between two different meanings of kind of quantity.
Kind of quantity Q3 seems most appropriate in this definition. (03)
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." (04)
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). (05)
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. (06)
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. (07)
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. (08)
Best regards,
David (09)
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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> (010)
============================================================
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
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e-mail: david.leal@xxxxxxxxxxxxxxxxxxx
web site: http://www.caesarsystems.co.uk
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