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. (01)
(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). (02)
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. (03)
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. (04)
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.) (05)
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. (06)
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. (07)
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.) (08)
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. (09)
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.) (010)
-Ed (011)
*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. (012)
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'. (013)
This matches the terminology David's draft uses, except that he doesn't
distinguish Q2 from Q4. Do we agree to this terminology? (014)
--
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 (015)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (016)
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