Mclean's Challenge accepted; with disclaimer

["Rauber, Joel Phys" <RAUBERJ@MG.SDSTATE.EDU>]

> ?  Jack (I think?) gave a fine description of how one might
> decide on different base dimensions, but I don't think we've seen a
> definition of 'dimension' itself.  Any takers?

I'll give it a shot, with the disclaimer that this is off the top of my head
on a friday afternoon; and the list will not doubt help to refine what I say
and correct its errors.

I assume we understand that there is a difference between "dimensions", what
I've called "physical dimensions" and "units".  BTW I say "physical
dimensions" to distinguish from "geometrical dimensionality", as in we live
in a Space-time of 4 dimensions (unless Kaluza-Klein and those type of folks
are correct).  I also assume that the distinction is somewhat slippery, or
why else are we having this discussion.

So having given the disclaimers, here goes:

In creating a system of units for measurements (in order to enhance and
enable communication with peers, i.e. what do the numbers in a measurement
actually mean, blah blah blah . . .)  one must decide on a base set of
units.  A base set is a set from which all other quantities will be defined.
 (all other quantities, means the units of all other quantities, these are
known as derived units).  The number of elements in the base set is
arbitrary.  For each element in the base set there is a corresponding
concept of a "physical dimension" associated with that element.  This
concept can be viewed as the core concept for which we must define the base
unit in our system of units.

I might add that the method of defining a unit is arbitrary as well; as long
as it is well defined; which presumably means reproducible in some coherent
fashion.

For example, the usual state of affairs for introductory mechanics:

We decide to have three "physical dimensions"; mass, length and time  (M,L
and T for short).  Hence, we need to define a corresponding base unit for
each of these concepts.  The SI system has the meter for the unit of length,
the kilogram for mass and the second for time.  All other units are derived,
meaning they are combinations of the base set.  E.g. The Newton, a unit of
force, which is Kg-m/s^2.

Note: the radian can be viewed as a derived unit (I don't think this is the
official US committee on standards position, if memory serves correctly they
view the radian as a supplementary unit; but I do not wish to go into that).
 It is a derivation of meters/meters in the SI system and hence is
dimensionless.

Another two examples:  Systems for mechanics and electromagnetism combined.
 The SI system chooses mass, length, time and current as the core "physical
dimension" concepts and uses the kilogram, meter, second and Amp as the
corresponding base units.  This is sometimes referred to as the MKSA system.

The system of units in the first two editions of Jackson (I usually refer to
it as cgs, but that might be inaccurate); furthermore it brought a tear to
my to see that his new 3rd edition has succumbed to the pressure of SI. :-(
    but I digress.

In that system one chooses only three "physical dimensions"  mass, length
and time for which the units are grams, centimeters and seconds.

Notice in the SI system Amps is a base unit and Coulombs (the units for
charge) are a derived unit. One can actually choose Charge as the core
phsyical dimension and treat Amps as a derived unit and you have the so
called MKSC system.

In what I referr to as the cgs system, both statcoulombs, and statamps; the
units of charge and current are derived units.  The choice of a different
number of physical dimensions in the two systems is what is responsible for
a lot of the different coefficients in Maxwell's equations as expressed by
the two systems and more importantly for the difference of dimensionality of
E and B in the SI system as oppossed to the cgs system where E and B have
the same dimensionality.  My opinion is that this is a powerful convenience
and arguement for that system.

Joel Rauber

All comments, corrections and criticisms, public or otherwise are welcome!