Dear all, (01)
Here are some comments and proposals in relation to the discussion that
has taken place since Friday last week. I (Ingvar J) was not able to
participate until now. (02)
ON THE DEFINITION OF ‘SCALE’. (03)
David L: 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. (04)
Ingvar J: In my opinion it is better to speak of ‘isomorphism’ than
‘function’, and (to begin with) better to speak of ‘orders’ than ‘axioms’.
To speak about ‘symbols’, as also has been proposed, seems to me as wrong
as to say that arithmetic is not about ‘numbers’ but ‘numerals’. Here come
my proposals (a quantity that is not a purely ordinal one consists of a
number connected to a standard unit). (05)
Scale-in-itself =def. a set of linearly ordered quantities (includes
‘ordinals’). (06)
Scale =def. a scale-in-itself that is isomorphic to a set of linearly
ordered magnitudes postulated by science or technology.
(Note: the definition presupposes that the magnitudes in question are
comparable, i.e., that they are of the same kind; every scale presupposes
an asymmetric and transitive comparative relation, and only what is of the
same kind can be compared.) (07)
Property scale =def. a scale whose magnitudes constitute a set of linearly
ordered equivalence classes of real and possible property instances.
- When the scale is used in a measurement, the measurement places the
property instance measured in one of the equivalence classes of the scale.
Equivalence classes presuppose a relation of similarity.
(Note: I think the definition has to be modal-intensional. Otherwise, in
order to get a continuous scale, one has to postulate that all the values
of a continuous scale are instantiated at least once in the history of the
universe; a rather remarkable metaphysical ad hoc hypothesis.) (08)
Metric property scale =def. a property scale where a distance measure
between the equivalence classes can be given scientific-technological
relevance.
- Different metric property scales of the same kind-of-quantity have
different standard units; it always holds true: (quantity interval in
scale Y) = (quantity interval in scale X) multiplied by a constant ‘a’.
(Note: don’t confuse ‘quantity intervals’ with the ‘quantity values’
spoken of below.) (09)
Interval scale =def. a metric property scale where the property in
question lacks an absolute physical zero point; transformations of
quantity values between different scales of the same kind-of-quantity
follows the formula: y = ax + b; a, b > 0. (010)
Ratio scale =def. a metric property scale where the property in question
has an absolute physical zero point; transformations of quantity values
between different scales of the same kind-of-quantity follows the formula:
y = ax; a > 0. (011)
MANY SCALES IN RELATION TO ONE KIND-OF-QUANTITY (012)
MW: Another obvious question is the relationship between a scale and a
unit of measure. They are not the same thing, but just how are they
related? (013)
Ed B: 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. (014)
Ingvar J: Any of the equivalence classes mentioned in my proposed
definition of ‘property scale’ can be chosen as the class that corresponds
to the selected standard unit quantity. The choices of standard unit and
corresponding equivalence class are, in contradistinction to the intrinsic
order between the equivalence classes, purely conventional.
- The standard equivalence class can be chosen in two different ways.
Either a concrete member of the class (such as the old standard meter in
Paris) is first chosen, and then automatically everything that is equally
long is regarded as belonging to this class; or the class is given a
theoretical intensional definition (such as the present one: the meter is
the length of the path traveled by light ...).
- Two relations are necessary in order to create a property scale: a
similarity relation is needed for creating the equivalence classes, and an
asymmetric and transitive comparative relation is needed in order to make
the order between the classes explicit. (015)
ON THE DIFFERENCE BETWEEN TEMPORAL DURATION AND TEMPERATURE (016)
Ed B: The absolute zero of time is presumably the Big Bang, but with
respect to absolute time, our current measurement technology is more like
19th century temperature measurement. So our absolute time scale is based
on an arbitrary reference point, like Celsius choice of the freezing
temperature of water. (017)
John S: But for interval measures, such as time and space, there is no
absolute zero, and only intervals are meaningful. The choice of "zero" is
an arbitrary convenience. (018)
David L: A conclusion is that the kinds of quantity for which the BIPM
defines a unit have a zero. Time and position are quite different to time
duration and length and should be considered separately. (019)
Ingvar J: I agree with the last sentence of David. Time in the sense of
the SI system is temporal DURATION, and every such finite duration is
bounded by two time POINTS. Similarly, every finite length (= spatial
extension) is bounded by two spatial points. This means that instances of
both temporal duration and length are, in a certain sense, intervals, but
this sense has nothing with interval scales to do. Each quantity value
(not ‘quantity interval’) both on the temporal duration scale and the
length scale denote an equivalence class of intervals in the instance
sense mentioned, but that does not turn these scales into interval scales.
- Make the following thought experiments. Take a length INSTANCE and make
it in thought shorter and shorter; then you will approach zero, an
absolute physical zero. Similarly, take an INSTANCE of a temporal duration
and make it in thought shorter and shorter; then you will approach zero,
an absolute physical zero. Take then an INSTANCE of a certain gas
temperature, and make it in thought lower and lower. If you regard this
temperature as necessarily caused by the kinetic energy of the molecules
of the gas, then you will approach a zero temperature as the kinetic
energy of the molecules reduces to zero. And since kinetic energy has an
absolute physical zero, so has then by implication temperature. But try
now to forget about statistical thermodynamics. Then it seems to be a
wholly open question whether or not there is a limit to how low a
temperature one can imagine. Therefore, Celsius and Fahrenheit couldn’t
start with a naturally given physical zero point and then just decide what
they wanted to call ‘1 degree C’ and ‘1 degree F’, respectively. They had
to choose TWO temperature magnitudes, ascribe both a quantity, and then
divide the interval between these two quantities into a certain number of
equally long one-degree intervals. (020)
ON THE DIMENSIONS ‘DIMENSIONLESS’ AND ‘DIMENSION ONE’. (021)
Joe C: So, if you propose to enlarge the set of Base Units, an essential
question is, where do you stop? There is an arbitrarily large number of
dimensionless quantities of different Kinds. You may have your favorites,
but what distinguishes them from someone else's favorites? The SI has
chosen to leave the Base Units bounded in number and the mapping "Kind"
loosely defined, I presume so that users may choose to make such
distinctions as they deem necessary by defining as many Kinds as they
want. (022)
Ingvar J: Today, the mole is regarded as a base unit, but the SI system
stipulates that it may not be used if not the kind of elementary entity
that is talked about is mentioned. This means that the mole is not a true
base unit on a par with the others. Rather, it deserves the name
‘parameter base unit’. I am proposing a similar move with respect to the
dimensions ‘dimensionless’ and ‘dimension one’. Let dimension one be a
‘parameter dimension’ that has values such ‘m/m’, kg/kg’, etc., and
stipulate that it may not be used if not its origin is mentioned. The
reason that hardly ever in experimental practice there are any problems
is, I guess, because the experimental context makes the origin implicitly
known to everyone concerned. (023)
BOOKS (024)
Earlier in this discussion, I have mentioned two books, Henry E. Kyburg,
“Theory and Measurement”, and David J. Hand, “Measurement Theory and
Practice”, but there is third one (and much less known) that might be of
even more interest when creating an UoM ontology: René Dybkaer, “An
Ontology on Property for physical, chemical and biological systems”
(Blackwell Munksgaard 2004). (025)
Best,
Ingvar J (026)
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