[Axiom-developer] Units and dimensions

[Ralf Hemmecke]

Hi Cliff,

At the end of section 4.4.2 of
http://portal.axiom-developer.org/Members/starseeker/axiomunit_diagramtest.pdf/download
where you have discussed the problem of curvature, you say that the SI 
system is not a perfect one and that it is missing a dimension of 
<CircularCurvature>.

At the same time you say:

  Unfortunately, Axiom must work with systems as they exist and are used
  rather than than their ideal implementations, we must somehow allow
  the current fundamental disodance to exist in Axiom's SI definitions
  and function non the less.

I understand your wish, but to my taste that is a non-mathematical 
approach. As you have described, expressed in the SI base dimensions, 
<Work> and <Momentum of Force> are indistinguishable. So why would you 
want to implement an imperfect system? If there is nothing in SI that 
helps you to distinguish <Work> and <Momentum of Force> then you cannot 
express it. You are basically saying that the SI system is 
7-dimensional, but the world is 8-dimensional (or even 
higher-dimensional), but you want to describe your world with 7 
dimensions. That with fewer dimensions you don't see everything is 
clear---draw a 4-dimensional hypercube on a sheet of (two-dimensional) 
paper and try to figure out from the latter, what properties the 4-dim 
object has. Of course, impossible.

So if I were to implement dimensions and units, I would design a perfect 
system and allow the user to add dimensions and work with them. 
Depending on which system the user chooses, he/she can work with 
<CircularCurvature> or leave it out. Such an open system would even 
enable one to experiment whether <CircularCurvature> is the right concept.

In the SI-system plus <CircularCurvature> you could distinguish <Work> 
and <Momentum of Force> in the pure SI system you cannot.

All I want to say, forget about the nature and all those systems like SI 
and cgs or whatever. In Axiom you should build a system that is flexible 
enough to accomodate for any system you like. If, for example, you want 
to replace the "fundamental dimension" <Length> by <Force>, then that is 
possible mathematically and should be allowed by Axiom. <Length> is then 
simply a "derived dimension" that depends on the "fundamental 
dimensions" <Force>, <Time>, and <Mass>. Where would be the problem? As 
a mathematician I can choose any axioms (fundamental things) I like as 
long as they are non-contradicting. Maybe a physicist thinks differently 
here?

By the way, if I look at section 4.2.2, then that very much suggests 
that the *type* of a unit is a *dimension*.
To my taste, that suggests to implement 7 (or 8 or how many dimensions 
you like) dimension domains:

Length: Dimension == add {
  Rep == -- I don't yet know.
  meter: % == ...
  inch: % == ...
  ...
}

or you don't hardcode meter, inch etc., but make those units dynamic.
Similar domains for other dimensions.

I could even imagine something like

Length(u: Unit): Dimension == add {
   Rep == Unit;
   unit: Unit == ...
}

where out of the many units that are possible for lengths units, u would 
be the reference unit for this dimension. (So you could have the SI 
system and cgs system easily by using the Length constructor with a 
different unit.

I don't yet have a good idea for the domain Unit, though. :-(
Please don't take all this too serious, I am just thinking aloud.

Once you have these dimension domains, you could form an algebra with 
them, ehm, a abelian group, of course. No, no, there are still no 
coefficients. So with that "abelian group" you cannot express 3m or 6s. 
All you could do is to express, what you probably called "derived 
dimensions". So Length, Length*Time^(-1) etc. are examples for such 
monomials. As you see, each "fundamental dimension" is then also a 
"derived dimension", but no conversions would be possible, since you 
don't have coefficients.

Now, having the monomials, you can think about quantities (= units (as 
elements of dimensions) together with numbers).

I don't have a well developed idea of how I would implement what I am 
writing about here, but it reminds me a bit of what I am currently doing 
in aldor-combinat. There we define combinatorial species (which are 
endo-functors from the category of finite sets and isomorphisms to 
itself. They are implemented as domain constructors in Aldor, but as the 
theory goes, one can form a semi-ring where species would be the 
elements. This is in so far similar to what I was writing about above as 
I like dimensions to be domains. But later on you should be able to 
multiply together such domains.

Another remark concerns dimensional analysis... If in Axiom dimensions 
are implemented as types, couldn't then the dimensional analysis be done 
by the compiler? Types on the right and left hand side must match. What 
about thinking along these lines for implementing dimensions in Axiom?

Maybe some day I'll find time to code my ideas into Aldor which would 
then be open for criticism.

Ralf