David Leal wrote: (01)
> 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 (02)
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: (03)
"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." (04)
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). (05)
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. (06)
Ingvar J (07)
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