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Re: [Axiom-developer] Unit package proposals and questions

From: C Y
Subject: Re: [Axiom-developer] Unit package proposals and questions
Date: Fri, 9 Sep 2005 07:30:37 -0700 (PDT)

--- William Sit <address@hidden> wrote:
> C Y wrote:
> >
> > Mass[r]*Length[r]/Time[r]^2
> No [IMHO]. This adds something unconventional and complicates things.

True, but reduced dimension itself is somewhat unconventional.  We are
dealing with a level of strictness virtually never used in the real
world (at least in my experience) so I'm not surprised normal usage
conventions are breaking down somewhat.  This is actually one of the
things making this so interesting for me :-).  I like getting down to
the roots of things and making it really correct.

Hopefully this could be done with some conditional outputform tweak
based on the Rep, so maybe a system setting to enable or disable this
would be a good idea, assuming I can convince you it's a good idea -
)set ShowReducedDimensions true  or some such.

The reduced dimension flag tells us that these dimensions are the
result of a substitution for some derived dimension, but otherwise we
allow things to proceed as normal - the reduced dimensions of work and
moment of force are the same, even though the derived dimensions
themselves are not.  The implication is that a reduced dimension
differs from a normal dimension in its history, but not in any
functional way, which also implies that its history is somehow
"inherent" in the Rep.  (one of the reasons I am in favor of preserving
the dimensional history of a reduced dimension in its Rep).  Just by
making the distinction, we are stating that there is something about a
reduced dimension compared to a regular dimension that MIGHT matter -
there may be some incompatibility not expressed in the dimensional
representation as it currently exists, although we don't know what. 
Certainly, if I divide work by moment of force, this is a dimensionless
number in terms of reduced dimensions but not in terms of derived
dimensions, and I would be happier if the "dimensionless" number
remembers something about this.  I'll have more details later - I'm
still chewing on this issue.  I might be thinking of reduced dimension
a bit differently than how it was originally defined - I'll have to
look - I'm viewing it as the dimension itself remembering its history,
rather than it being the name for a basic dimension definition of a
derived dimension.  

Hmm, I might be having too much fun thinking about this. Uh oh.

> There should be no difference in the output form. It should be quite
> obvious that a dimension or unit output is reduced or not. Plus, 
> there are dimensions whose reduced dimension is identical to the 
> dimension. The user can also explicitly ask for a reduced dimension 
> through functions such as reduce(identifier).

It is obvious in the initial substitution, but what if the resulting
quantity goes on to be used elsewhere, where its history is not
immediately apparent from the environment?

A user can't always ask for a reduced dimension - what would that mean
in the case of basic dimensions?  reduced from where?  from which
derived dimension?  reduced(Length) wouldn't mean anything, but the
Length which is part of the reduced dimension definition of Force is a
reduced dimension.

I'm not sure I agree there are dimensions whose reduced dimension is
identical to the dimension, depending on which definition of reduce we
are using.  I view a reduced dimension as a normal dimension which
calculates as normal, but preserves information about its origins.

I'm working on the interaction rules between reduced, derived, and
normal dimensions, although again I suspect I've gone a tad overboard
with the reduced idea.  Maybe this weekend I'll get some more written
on these thoughts, and realize they make no sense ;-).

> Reduced dimension can have reduced units (why not)?

Because this information is already in the dimension.  A unit might
have either a reduced dimension or a regular dimension associated with
it, depending on its history, but I see no reason to impart that
information to the unit as well since the unit MUST be associated with
a dimension.

Hmm.  Time to regroup and state my ideas on the definitions of reduced.


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