On Aug 12, 2009, at 8:40 AM, David Leal wrote: (01)
> Dear Ingvar,
>
>> "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 completely agree with this quote, and hope that nothing that I
> have said
> conflicts with it. What I have called "magnitude of quantity" is
> called
> "aspect of object" in this quote. (02)
Then what you said does disagree with it. I agree with Ingvar's point:
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. 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. (03)
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. (04)
> 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". (05)
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. (06)
Pat H (07)
>
> 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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>
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> (08)
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