Dear Pat, (01)
Thank you for your reply some further comments are interleaved below. (02)
Best regards,
David (03)
At 17:47 05/01/2010 -0600, you wrote:
>
>On Jan 5, 2010, at 12:09 PM, David Leal wrote:
>
>> Dear Pat,
>>
>> Thank you for this useful and clear note. I am not clear what the
>> procedure
>> will be for raising issues or suggesting clarifications, but I have
>> a few
>> comments on your approach to scale.
>
>Email is fine. I'm not a great believer in procedures :-)
>
>>
>> Classification of a scale
>> -------------------------
>> Taking a scale to be a mapping between numbers and "quantity values
>> of a
>> particular kind", there is only a little we can say about the nature
>> of the
>> mapping before assuming something about the structure of the "quantity
>> values of a particular kind".
>
>True, which is why I am avoiding saying very much about these
>mappings, other than that they exist and what their domains and ranges
>are. (04)
Mostly I agree, but my motiviation for the initial comments was that perhaps
in a few cases you seek to make statements about scales :
1) which are more correctly statements about "quantity values of a
particular kind"; and
2) which imply statements about the "quantity values of a particular kind"
that are their range that should be made explicit. (05)
>> Cardinality
>> -----------
>> Without structure we can just talk about cardinality. It is
>> reasonable for
>> us to insist that in a scale, a number maps to only one quantity
>> value.
>
>Hmm, not sure about that. There can be 'coarse' scales, for example,
>which do not make distinctions between different quantity values.
>Qualitative reasoning often uses {negative, zero, positive} as a crude
>measure scale, and yet is capable of quite surprisingly sophisticated
>analyses.
>
>> However a quantity value can map to more than one number. An example
>> of this
>> the mapping between real numbers and angle. If the range of the real
>> numbers is [-180, +180], then there is an angle which maps to both
>> -180 and
>> +180. We can get the problem to go away in this simple case by
>> making the
>> interval of numbers open at one end, but this may be inconvenient.
>
>It is arbitrary, but it can always be done, I think. (Do you agree?) I
>would prefer to say that in cases like this, the scale is simply
>underspecified as stated, and the actual scale is either [-180, 180)
>or (-180, 180] (06)
Yes. I think that we can always fix up in 1D. I raised the issue, just so
that we do not forget it. (07)
>
>> In more complicated cases, even this does not work. Consider angle
>> in 3D -
>> i.e. a direction defined by the Euler angles theta and phi as in
>> http://conventions.cnb.uam.es/Submit/Euler.jpg . (We have ruled non-
>> scalar
>> quantities out of scope, but it is useful to consider a more general
>> case.)
>> The same direction is defined for theta=0, whatever the value of phi.
>>
>> The source of the problem is the same for angle in 2D or 3D. It is
>> because
>> the space of quantity values is topologically equivalent to a circle
>> or
>> sphere, whilst the space of the numbers is topologically equivalent
>> to a
>> line or square.
>
>We should describe the structure of our scales so as to eliminate this
>kind of ambiguity. Perhaps the results will not be conventional
>numbers, but then scale values are often not treated as purely
>numerical in any case. For example, we can simply declare that in the
>Euler angle scale (writing (Euler theta phi) for such a 3d angle) that
>
>(forall (x y)(= (Euler 0 x)(Euler 0 y))
>
>thereby defining the actual scale as a mathematical object with the
>correct topology. (08)
I do not have a definition for "topology of a function". The space of
vectors of numbers has topology and the space of directions has topology. We
need to say something about the function from one to the other. In this case
there are some specific statements to be made about the function as you show
above. My point is that the function cannot be simply classified as
"continuous" and "bijection". (09)
>>
>> Order
>> -----
>> You cannot classify a scale as an OrderScale unless the quantity
>> values have
>> an order.
>
>Sure I can :-). I can simply assign things to points on an order scale
>according to some fixed but arbitrary rule. Rockwell hardness is
>pretty arbitrary, in fact. (010)
Here I think we disagree. Of course, if I identify a set of things by
numbers I am assigning an order to them, but this is not interesting. But if
the things that are identified already have an order, independent of
identification, then we can ask whether or not the assigned numbers are
consistent with that order. (011)
>> The property that we are looking for in a scale is that the order
>> of the numbers is consistent with the order of the quantity values
>> that they
>> represent.
>
>Surely that is about the appropriateness of the scale to measure the
>quantity, rather than about the structure of the scale itself. A
>scale, considered in isolation, is simply a mathematical abstraction. (012)
A function does not have a structure unless its domain and range have a
structure. All statements about the structure of a function are about how
structures within the domain map to structures within the range and vice versa. (013)
Without a structure for both the domain and range, a function is just a set
of pairs and all we can say about it is cardinality. (014)
>> If we formalise scale as a function, then there is already a term
>> for this - "increasing function" see
>> http://mathworld.wolfram.com/IncreasingFunction.html .
>>
>> 2D angle is almost certainly within scope but is only ordered locally.
>
>? I do not understand this. Seems to me that angles have no boundary
>in either direction. Engineers talk of rotations through angles
>greater than 360. Angles used to measure relative orientation of
>lines in space are of course in the range (0 360) or perhaps (-180,
>180), but angle has other uses. (015)
I am making a trivial point. Angles 10.0, 10.1 and 10.2 degrees seem to be
ordered. But is angle 120 degrees greater than or less than the angle 240
degrees - both it is 120 degrees behind and 240 degrees ahead. (016)
>
>> Hence
>> the scale for angle is not an increasing function everywhere.
>> However we can
>> choose a function which looks locally like an increasing function
>> almost
>> everywhere.
>>
>> 3D angle is not ordered. We still have a requirement for a "good
>> function",
>> so this must be expressed in terms of continuity.
>>
>> Continuity
>> ----------
>> We cannot define continuity for a scale unless we have a concept of
>> closeness between quantity values.
>
>Again, the *scale* can be continuous even when the quantity is not
>(though I agree this would be a highly improper use in most
>circumstances.) (017)
No. The usual definition of "continuous" for a function does not allow this.
What is your definition of continuous? (018)
>> This relies neither upon an order for the
>> quantity values nor upon a "zero quantity value". Intuitivly a
>> scale is a
>> continuous function if closeness between numbers (or vectors of
>> numbers)
>> implies closeness between corresponding quantity values.
>
>But how is one to express this independently of any scale? I am
>reluctant to assert that quantity values themselves satisfy any
>particular algebraic or topological axioms. What is the meaning of
>subtracting one Rockwell hardness from another? (019)
This is the key issue. In practice our scientific and engineering
mathematics only works because we do. We blithely write down equations
involving quantity values forgetting that we have implicitly assumed
algebraic and topological axioms for them. (020)
The use of equations involving quantity values is specifically recommended
by the NIST "Guide for the use of SI"
<http://physics.nist.gov/Pubs/SP811/sec07.html> clause 7.11. Exactly the
same text is reproduced in the ISO Directives 3. (021)
There is no meaning for subtracting one Rockwell hardness from another.
Therefore if a class of function relies upon the existence of arithmetic
operations on the domain, this class of function cannot be applied to
Rockwell hardness. We cannot state that Rockwell hardness is linear - this
is nonsense. (022)
>> There is a definition of continuous function which works for any
>> dimension
>> of the domain and range - see
>> http://mathworld.wolfram.com/ContinuousFunction.html . The
>> definition relies
>> upon our ability to define an "open set" of quantity values. Hence
>> we assume
>> a topological structure for the quantity values.
>
>Right, but again, it seems like hubris to assert topological
>properties of real things like quantity values. (?) Bear in mind that
>at this level of generality, a set of quantity values can be almost
>anything, including things like the pharmaceutical 'amounts of
>dilution' and other unscientific exotica. (023)
It is hubris! The metrologists are cautious, but practical scientists and
engineers make assumptions. I assert that it is not possible to define a
continuous scale without an ability to define an open set of quantity
values, and not possible to define a linear scale without an ability to do
arithmetic with quantity values. Therefore a useful ontology has to have two
levels: (024)
- a hubris-free level which may identify a scale, and specify little more
than its cardinality;
- a hubris level which classifies the scale as continuous or linear (say). (025)
>>
>> Linearity
>> ---------
>> Whether or not a quantity value can be multiplied by a real number is
>> nothing to do with scale. The statement that Ta = 2.Tb , where Ta
>> and Tb are
>> values of thermodynamic temperature, is well defined irrespective of
>> scale.
>
>True, but that is because we have a sophisticated understanding of
>exactly what underlies the concept of temperature. But the Fahrenheit
>and Centigrade scales were invented and used long before this
>understanding was developed, so they cannot depend upon it for their
>basic ontological description, surely. (026)
I agree. (027)
>> The concept we require is "linear function", in the narrow sense - see
>> http://mathworld.wolfram.com/LinearFunction.html. Hence if x = 2.y,
>> then
>> (the quantity value represented by x) = 2.(the quantity value
>> represented by
>> y). Unfortunately the term "linear" is sometimes used to include
>> affine
>> functions - see http://mathworld.wolfram.com/AffineFunction.html .
>> Kelvin
>> and Rankine are linear scales.
>
>How is it known that the 'degree' between, say, 0F and 1F, is the same
>"amount" of temperature as that between, say, 1000F and 1001F? Does
>this assertion even have a clear physical meaning? (028)
Yes it does. This is because Fahrenheit is defined with respect to Kelvin,
and Kelvin a scale for "thermodynamic temperature" which is defined
abstractly in terms of the laws of themodynamics. This abstract approach
does not lead to sufficient precision, so alongside this we have
International Temperature Scale 90
<http://www.bipm.org/en/publications/its-90.html>. (029)
There is no garantee that the thermodynamic temperature difference betweeen
0K (ITS90) and 1K(ITS90) is the same as the difference between 1000K (ITS90)
and 1001K(ITS90), but it is believed to be close. (030)
>> Celsius and Fahrenheit are affine but not linear.
>>
>> The definition of linearity relies upon our ability to define addition
>> between quality values and the multiplication of a quantity value by
>> a real
>> number.
>
>The latter depends upon the former, in fact. 3 x foo is (foo + foo +
>foo)
>
>> Hence we assume an algebraic structure for the quantity values.
>
>Yes, we could do that, for some quantity values. I had thought to
>characterize the value types by referring to the mathematical
>structures of the scales used to measure them, rather than the values
>themselves. (031)
My point is that the scales do not have a mathematical structure that is
independent of the mathematical structures of their domain and range. (032)
>>
>> Conversion between scales
>> -------------------------
>> Because scales are functions, a scale conversions is not expressed
>> by an
>> equation, such as (= (meter 1)(foot 3.281)).
>
>This is sufficient when the scales are both unit scales. I will spell
>this out in full. (033)
Agreed, but I am unhappy about merging:
- scales derived from a unit; and
- scales which are linear. (034)
The onus is upon you to prove that the two classes are equivalent. (035)
>> The counter example is (=
>> (celsius -40)(fahrenheit -40)). This is true but not enough.
>>
>> The relationship between two scales is itself a function. Consider
>> celsius:R
>> -> T and fahrenheit:R -> T .
>
>These are not unit scales. The general conversion is an equation with
>a zero shift, yes. (036)
No, the general conversion is a function (albeit one derived trivally from
an equation with a zero shift in this case). Being pedantic is our job. The
conversion from IPTS68 (International Practical Temperature Scale 68 -
predecessor to ITS90) to ITS90 is not trivial - see
<http://www.its-90.com/table6.html>. (037)
>>
>> We can define a function X:R -> R, such that X(x) (5/9)*(x-32) for
>> all real x.
>>
>> Hence we can say that Fahrenheit(x) = Celsius(X(x)). Or Fahrenheit =
>> Celsius*X, where * is the composition operator - see
>> http://mathworld.wolfram.com/Composition.html .
>>
>> Best regards from a conservative applied mathematician,
>> David
>>
>> At 02:47 04/01/2010 -0600, you wrote:
>>> I have put up some preliminary notes at
>> http://ontolog.cim3.net/cgi-bin/wiki.pl?UoM_Ontology_Standard_CLIF_Draft
>>>
>>> Pat
>>
>> ============================================================
>> 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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>
>
>
>
>
>
> (038)
============================================================
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
============================================================ (039)
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