Pat H wrote: (01)
> 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. (02)
Following Stevens, I would say that there is no purely
mathematical-metrological notion of scale, but when the ordinary
empirical-scientific concept of scale has been understood, one realizes
that there are purely formal differences between different types of
scales. Here is a another quotation from the book by Hand that I have
mentioned (to be find in the mails below): (03)
"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)." (04)
Ingvar (05)
> 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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