Ingvar J:
>> 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. (01)
Pat H:
> A purely formal difference must be between two things that have formal
> descriptions. (02)
This tautology does not make the following sentence meaningless: "The
(substantive) statements 'all electrons have the same charge' and 'some
people are red-haired' are formally different." The formal descriptions
can be implicit. Similarly, it makes good sense to say: "The pre-Kelvin
temperature scales and the length scales are formally different." This
means among other things (remember that the choice of standard unit is
conventional), that the transformation formula between Celsius and
Fahrenheit has the form '[°F] = 9/5 × [°C] + 32', but that between yard
and meter has the form '[yard] = 0.91 × [meter]'. Put more generally, the
first case fits the form 'y -> ax + b', and the latter 'y -> ax' (compare
the quotation from Hand three paragraphs below).
(What at a first glance may be confusing is the fact that INTERVALS of
interval scales follow the form exemplified by ratios such as
[interval °C] = 9/5 × [interval °F].) (03)
> 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)
Pat, I have only said (check above) "that there is no purely
mathematical-metrological notion of scale". If you create and investigate
a purely mathematical notion of scale, then you leave metrology and the
empirical sciences behind. But I have no objections to such an
undertaking. Just be clear about what you are doing. Metrology can just as
little as mathematical physics be reduced to only logic and mathematics. (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)
> Pat
> PS, the above seems to imply (since rescaling is a linear operation)
> that ratio is a special case of interval. Is that correct? (07)
An interval scale in physics has from a PHYSICAL point of view no absolute
zero point, whereas a ratio scale has. So, if this feature is made part of
the definitions of the terms, ratio scales cannot be regarded as special
cases of interval scales. (08)
Ingvar (09)
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