David, let me try to clear the ground here a little. I think we may be
talking at cross purposes. (01)
I can define a purely mathematical structure which is a set of, let me
call them 'measures', together with an ordering < and, say, an
addition operator, so we can define a multiplication-by-natural-number
operator: (02)
(times (n+1) measure) = measure + (times n measure) (03)
Such structures may be dense or not, continuous (having limits) or
not, etc..; we can classify them in various ways, again purely
mathematically. This is what I mean by a 'scale'. And then, almost as
an afterthought, we can *use* one of these to record values of a
quantity. That involves the assumption of a mapping from the scale
measures to the values, call it a measure mapping. And then we can ask
about the properties of this measure mapping. (04)
The only such property that I have sought to capture is that is is
functional, so we can describe a length quantity value by writing a
scale mapping and a scale reading, such as (meter 3.2). Any more
advanced structure on the space of values is to be inferred from the
structure of the space(s) of readings used to measure it. (BTW, even
this is a simplification, of course, since measurements are always
approximate. I would like to return to this issue later, but ignore
it for now.) (05)
Your presumption, I take it, is that the measure mappings themselves
should be asserted to be order-preserving, continuous, etc.., and that
this therefore requires that each space of quantity values be assigned
an appropriate mathematical structure; so we should say, for each
quantity type, what its topology is. Temperatures and lengths lie on a
semi-continuum, for example, with an absolute zero. We could go this
route, but I am nervous about it. First. I am not convinced that it
can always be done, that every quantity comes with a clearly defined
topology. (Bear in mind that there are probably thousands of quantity
types in general use, far more than the small set of basic physical
quantities; and that many in actual use, such as Rockwell hardness,
measure something that is in fact defined by the measuring apparatus
itself, and may have no purely physical definition which would allow
anyone to theorize about its 'true' structure.) Second, in the cases
where it can be done, it seems like overkill, since we will be obliged
to describe the scale topology and the quantity topology and then
assert that they are isomorphic (or perhaps that the scale is a
homomorph of the quantity) but once that is established, we can use
the scale to refer to the quantity, just as everyone does in the
practical world. Third, I don't see the need for it, at least in an
initial ontology, in that things like scale conversions can be
expressed without introducing the 'physical' structure on quantities.
We have no way to refer to an actual individual value other than by
using a scale reading, in any case. And fourth, it can always be added
later in the cases (like thermodynamic temperature) where it is well-
defined and of utility or importance, as a detailed ontology about
that particular scale. (06)
Pat (07)
PS. Your point about the ordering of angles is well taken. The moral I
draw from it is that maybe a scale should not even be assumed to have
the order structure, necessarily. So in the most basic, general,
sense, a scale is simply a *set* of measures, basically a recording
convention; and then we will have a first category of 'ordered' for
the usual case of a total order on measurements. (08)
PPS. On terminology. Is there a widely accepted classification of
scales into linear, affine, etc..? I simply made up some terminology,
but it would be more satisfactory to use a standard if one is available. (09)
--------- (010)
On Jan 6, 2010, at 5:06 AM, David Leal wrote: (011)
> Dear Pat,
>
> Thank you for your reply some further comments are interleaved below.
>
> Best regards,
> David
>
> 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.
>
> 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.
>
>>> 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]
>
> Yes. I think that we can always fix up in 1D. I raised the issue,
> just so
> that we do not forget it.
>
>>
>>> 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.
>
> I do not have a definition for "topology of a function". (012)
The notion is quite acceptable. The set of continuous functions
between two topspaces itself has a natural topology. (013)
> 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".
>
>>>
>>> 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.
>
> 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. (014)
We do agree, I was just expressing myself rhetorically. But OK,
suppose we introduce this independent ordering on quantity values,
then our ontology now has two orderings, one of them 'intrinsic' and
the other, to coin a phrase, 'metric', ie defined on the scale
elements, and there is an isomorphism between them. What has been
gained by this? The description is more complicated, and users have
more distinctions to remember, but nothing can be said that cannot be
said without the duplication; and we now have considerable extra
complication, since not all quantity values do have the required
structure, so we have to classify quantity value kinds as well as
scales. (015)
>
>>> 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.
>
> 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.
>
> 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. (016)
Well, a function *is* a set of pairs, in all cases. And it is
functional, ie a many-to-one mapping everywhere defined, which is not
completely vacuous. In many cases, such as this, this is all that is
really required. (017)
>
>>> 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.
>
> 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. (018)
Ah, I see. Again, I would consider this to be an issue in how to
define the ordering function on the angle scale. There might be many
such scales, differing in their ordering, just as fahrenheit and
centigrade differ in their unit size and zero point. (019)
>
>>
>>> 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.)
>
> No. The usual definition of "continuous" for a function does not
> allow this.
> What is your definition of continuous? (020)
I was here speaking informally, but the topological one will do. There
is no requirement that the function be continuous, only the space of
scale values. (021)
>
>>> 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?
>
> 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.
>
> 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.
>
> 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.
>
>>> 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.
>
> 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:
>
> - 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).
>
>>>
>>> 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.
>
> I agree.
>
>>> 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?
>
> 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>.
>
> 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.
>
>>> 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.
>
> My point is that the scales do not have a mathematical structure
> that is
> independent of the mathematical structures of their domain and range.
>
>>>
>>> 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.
>
> Agreed, but I am unhappy about merging:
> - scales derived from a unit; and
> - scales which are linear.
>
> The onus is upon you to prove that the two classes are equivalent.
>
>>> 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.
>
> 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>.
>
>>>
>>> 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
>>> ============================================================
>>>
>>>
>>>
>>
>> ------------------------------------------------------------
>> IHMC (850)434 8903 or (650)494
>> 3973
>> 40 South Alcaniz St. (850)202 4416 office
>> Pensacola (850)202 4440 fax
>> FL 32502 (850)291 0667 mobile
>> phayesAT-SIGNihmc.us http://www.ihmc.us/users/phayes
>>
>>
>>
>>
>>
>>
>>
>
> ============================================================
> 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
> ============================================================
>
>
> (022)
------------------------------------------------------------
IHMC (850)434 8903 or (650)494 3973
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