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From: Deane Yang (deane_yang_at_[hidden])
Date: 2004-01-15 10:28:04

Andy Little wrote:
> "Deane Yang" <deane_yang_at_[hidden]> wrote
>>Way at the beginning,
>>I expressed some confusion about what a "dimension" was. I have yet
>>to see anyone articulate clearly and precisely what the definition
>>of a "dimension" is (please chime in if you have one!). Is it something
>>like this:
> You say you understand the mathematical concepts behind dimensional
> analysis. I am not qualified to tell if you do or you dont. However if you
> do, it would be extremely useful if you wrote a paper on the subject.

If you go back to the first posting whose subject contained "(long
reply)", I described all the essential details (just start reading
after "let's review"). There's really nothing more to dimensional
analysis than that.

Dimensional analysis is just a set of rules of what you can and
cannot do with measurements. There are different rules for
relative measurements (the difference between two different
measurements) and absolute measurements. I've listed all of
them in my posting.

When I say I "understand" the mathematical concepts, all I mean
is that I recognize the rules for relative measurements
as being exactly the same rules that dictate what you can and cannot do
with vectors in a 1-dimensional real vector space and
the rules for absolute measurements as being the same rules
for points in a 1-dimensional affine space.

Of course, you can do all of this in higher dimensions, too.
And it generalizes even further. All of this is just linear dimensional
analysis, where you always use linear units in your measurements.
If you use nonlinear measurements, then the mathematical foundations
of differential geometry apply, and a "dimension" becomes something
mathematicians call a "manifold".

But here is where your skepticism about mathematicians is well
justified. I love talking about all this fancy mathematical stuff,
but as far as programmers and physicists are concerned, it's just
all jargon and different terminology for things they already understand
perfectly well. I definitely do not think it is necessary to learn
about abstract vector and affine spaces. When they are described
as "dimensions", their properties (as described in my earlier posting)
all seem very sensible and logical. There's nothing more to learn.

So if you really want to understand what I am saying (I admit I probably
usually say too much), just read that one posting very carefully. Feel
free to ask me questions.

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