On Aug 12, 2009, at 11:23 AM, ingvar_johansson wrote: (01)
> Pat H wrote:
>
>> there seems to be a notion of 'scale' involved here which is purely
>> mathematical, and the distinctions between the types of them are made
>> purely mathematically. And these distinctions seem to be prior to any
>> use of these scales to measure any particular quantity or magnitude
>> or
>> aspect.
>
> Following Stevens, I would say that there is no purely
> mathematical-metrological notion of scale, but when the ordinary (02)
> empirical-scientific concept of scale has been understood, one
> realizes
> that there are purely formal differences between different types of
> scales. (03)
A purely formal difference must be between two things that have formal
descriptions. All I wish to do is to try to characterize these formal
differences by writing the formal descriptions. It is not very helpful
to be told that this is impossible a priori. (04)
I have to say, the result of this first tiny step towards attempting
to gain some precision in this discussion is not at all encouraging. (05)
> Here is a another quotation from the book by Hand that I have
> mentioned (to be find in the mails below):
>
> "Given some numerical assignment which properly reflects the
> relationship
> between the objects in terms of the attribute under consideration,
> Stevens
> defined the scale as nominal if any one-to-one onto transformation
> of this
> assignment also leads to a legitimate numerical assignment. He
> defined it
> as ordinal if any monotonic (order-preserving) transformation led to a
> legitimate assignment. It was interval if any linear transformation
> (x ->
> ax + b) led to another legitimate assignment. And it was ratio if any
> rescaling operation (x -> ax, a > 0) led to another legitimate
> assignment
> (p. 41)." (06)
One sees what he means, but words such as "properly" "under
consideration" and "legitimate" do not pick out ideas that we want to
try to incorporate into a foundational ontology. (07)
Pat (08)
PS, the above seems to imply (since rescaling is a linear operation)
that ratio is a special case of interval. Is that correct? (09)
>
> Ingvar
>
>> David's own formulations of the different scales make no
>> reference to the magnitudes involved, AFAIKS. So I suggest we first
>> characterize these purely structural distinctions, and then apply
>> them
>> to things like magnitudes.
>>
>> BTW, on a slightly side note: there is no such object as a 'partial
>> isomorphism'. What there can be is an isomorphism between structures
>> which only partially describe the underlying reality. But
>> 'isomorphism' is a mathematical term, and should be used in its
>> mathematical sense, especially here where we intend to formalize it
>> in
>> the very near future.
>>
>>> We have started at a even lower level than this quote. Our "first
>>> place" is
>>> that there is a one-to-one correspondance between the "aspects of
>>> object"
>>> and symbols (restricted to numbers in the quote). Only afterwards do
>>> we
>>> consider isomorphisms between "properties of the numeral series" and
>>> "empirical operations that we can perform with the aspects of
>>> objects".
>>
>> These arguments about what is 'first' and what comes 'afterwards'
>> have
>> no value. The question to ask is, what can be described without
>> referring to what else? The scale distinctions we are trying to
>> capture here seem to be describable without any reference to the
>> underlying magnitudes being measured.
>>
>> Pat H
>>
>>
>>>
>>> My proposed definition of ordinal scale: a scale where both Q and S
>>> are
>>> ordered, such that:
>>>
>>> f(q1) > f(q2) if and only if q1 > q2
>>>
>>> is exactly in line with this quote. The comparison q1 > q2 is an
>>> "empirical
>>> operation that we can perform with aspects of objects". The
>>> comparison f(q1)
>>>> f(q2) is a "property of the numeral series". The function f is an
>>> isomorphism with respect to order.
>>>
>>> Best regards,
>>> David
>>>
>>> At 14:41 12/08/2009 +0200, you wrote:
>>>> David Leal wrote:
>>>>
>>>>> I agree except for one thing - a scale is not a set of items/
>>>>> symbols in
>>>>> itself, but a mapping from a set of "magnitudes of quantity" to a
>>>>> set of
>>>>> items/symbols. Hence re-expressing the consensus in these terms we
>>>>> have:
>>>>>
>>>>> scale: a mapping f from Q (set of magnitudes of quantity) to S
>>>>> (set of
>>>>> symbols - commonly numbers), such that:
>>>>>
>>>>> f(q1) = f(q2) if and only if q1 = q2
>>>>
>>>> Whatever kind of definition of 'scale' the information sciences in
>>>> the end
>>>> will find good and useful, everyone ought to be aware of the fact
>>>> that the
>>>> definition above is not what one finds in traditional philosophy of
>>>> science literature on measurement (whose terminology, BTW, I have
>>>> been
>>>> using). Here is a quotation from the man (S. S. Stevens) who first
>>>> made
>>>> the distinctions between nominal, ordinal, interval, and ratio
>>>> scales
>>>> explicit:
>>>>
>>>> "Scales are possible in the first place only because there exists
>>>> an
>>>> isomorphism between the properties of the numeral series and the
>>>> empirical
>>>> operations that we can perform with the aspects of objects. This
>>>> isomorphism is, of course, only partial. Not *all* the properties
>>>> of
>>>> number and not *all* the properties of objects can be paired off
>>>> in a
>>>> systematic correspondence. But *some* properties of objects can be
>>>> related
>>>> by semantical rules to *some* properties of the numeral series."
>>>>
>>>> I have taken the quotation from the latest overview book of
>>>> measurement
>>>> that I know of: D. J. Hand, "Measurement Theory and Practice. The
>>>> World
>>>> Through Quantification" (Arnold 2004; quotation p. 41).
>>>>
>>>> If one accepts such a definition of 'scale' (which I do), then
>>>> David
>>>> Leal's term 'set of magnitudes of quantity' is already implicitly
>>>> presupposing a scale. Without a scale (in the traditional sense)
>>>> there can
>>>> be no magnitudes.
>>>>
>>>> Ingvar J
>>>>
>>>>
>>>>
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>>>>
>>>
>>> ============================================================
>>> David Leal
>>> CAESAR Systems Limited
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