On Aug 13, 2009, at 4:00 AM, ingvar_johansson wrote: (01)
> 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.
>
> Pat H:
>> A purely formal difference must be between two things that have
>> formal
>> descriptions.
>
> 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." (02)
All because these things do have formal - mathematical - descriptions,
as you handily go on to demonstrate: (03)
> 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].)
>
>> 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.
>
> 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 (04)
> 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)
Our purpose here is to further the art of metrology, but to create an
ontology. By definition, such an ontology is a formal description - in
a formalism - of the concepts we are here discussing. I do not care
whether or not this is viewed as a 'reduction' , but it WILL be
couched ONLY in logic. (If, that is, it ever gets done, which I am
beginning to doubt.) If you find this unsatisfactory, or if your
methodological intuitions lead you to be uncomfortable with it, then
you should perhaps not be engaged in this effort. I myself will find
it vastly more satisfying than endless discussions in English which
come to no conclusion. However, I have no more spare time to spend on
this now, and will not for some weeks. (06)
If I might make a general observation before leaving: I have the sense
from these discussions that several of us have a feeling that the
ontologists in this forum are straying from the correct methodological
path of metrology. And of course they - we - are, because writing an
ontology of metrology is not doing metrology. (One can substitute
almost any other science or human activity for 'metrology' in that
sentence). In this effort, metrologists or philosophers of metrology
are what are technically known as SMEs: Subject Matter Experts. Your
job is to tell us what your concepts are, not to instruct us in how to
write axioms or to debate the philosophical foundations of our
technical area. This is not the place to be having a debate in the
philosophy of logic. (07)
See y'all in September. (08)
Pat (09)
>
>>> 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)."
>
>> Pat
>> PS, the above seems to imply (since rescaling is a linear operation)
>> that ratio is a special case of interval. Is that correct?
>
> An interval scale in physics has from a PHYSICAL point of view no
> absolute
> zero point, whereas a ratio scale has. (010)
However, if you check the above, no mention is made there of an
absolute zero. Hence my question. (011)
> So, if this feature is made part of
> the definitions of the terms (012)
Which, in the above, it is not. Apparently, Hand's definitions are
faulty. (013)
> , ratio scales cannot be regarded as special
> cases of interval scales.
>
> Ingvar
>
>
> (014)
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