On Jan 6, 2010, at 11:07 AM, David Leal wrote: (01)
> Dear Pat,
>
> We were very much at cross purposes.
>
> When you said "Scales are formalized as functions from numbers to
> quantity
> values", I thought ah-ha I agree with that. This is the conclusion
> that
> Matthew and I came to when working on quantities, units and scales
> for ISO
> 15926, and it is consistent with my understanding of
> instrumentation. (I
> used to be involved with instrumentation many years ago, before
> working on
> FEA, which is why I am interested in a formal representation for
> ITS90.) (02)
I was careless there, I have to confess. I should have said, from a
structured set of measures (frequently numbers) to quantity values. (03)
>
> In this e-mail, I think you are suggesting that we work with three
> types of
> thing (04)
No, just two. (05)
> :
> - numbers (which have the structure defined by their axioms);
> - measures (which have some other explicitly defined structure);
> - quantity values (with whatever structure they actually have in
> nature).
>
> Presumably you also suggest two mappings: numbers <-> measures, and
> measures
> <-> quantity values. (06)
I would actually just have measures and values. The numbers are a
special (but very common) case of measures. But one can have scales
with entries other than numbers, eg vectors. I would like to keep this
as open-ended as possible. (07)
> At present, I do not see the utility of the concept of "measure". It
> does
> not seem to be used in practice. In particular that only numbers and
> quantity values are used within:
> 1) <http://physics.nist.gov/Pubs/SP811/sec07.html> clause 7.11; and
> 2) the definition of ITS90. (08)
I was trying for a degree of generality which would not require us to
go back and re-do the axioms for cases like vector scales. So
'measure' is simply a way of saying 'number' without committing to
decimals for everything. Also, even when numerals are used to indicate
measures, these are not always what they seem, eg the hardness scale
points are not truly numbers, since it makes no sense to subtract them. (09)
>
> I am aware that there may be something I have missed, and I am happy
> to wait
> and see how things work out as we go forward.
>
> Best regards,
> David
>
> p.s. Your statement "Well, a function *is* a set of pairs, in all
> cases" is
> nearly but not quite true. Wikipedia gets it right - see
> http://en.wikipedia.org/wiki/Function_%28mathematics%29#Definition .
> Trivially there is a difference between a total function and a partial
> function - how can this be if a function is just a set of pairs? (010)
One's view on this is hostage to whether you take a set-theoretic or
category-theoretic approach to foundational questions in mathematics.
I was raised in set theory, myself :-) In my education, a function can
be total (or not) on a domain, but the function still *is* a set of
pairs. So, the function from an integer to its largest prime factor is
total on the natural numbers and partial on the reals, but its the
same function in both cases. Category theory would have it be two
distinct mappings related by a forgetful functor between the
categories. I dislike the idea of incorporating the domain and
(especially) the codomain into the actual definition as it leads to a
needless excess of pointless distinctions, eg the successor function
with domain N and codomain N is distinct from the successor function
with domain N and codomain R. BTW, in CL (as in most formal logics),
all functions are considered by the logic to be total on the entire
universe of discourse, so the issue is somewhat moot. (011)
Pat (012)
PS. Any feedback on this? (013)
>>
>> PPS. On terminology. Is there a widely accepted classification of
>> scales into linear, affine, etc..? I simply made up some terminology,
>> but it would be more satisfactory to use a standard if one is
>> available. (014)
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