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Re: Dimensions vs units



Regarding where Bob Sciamanda wrote the following about 'dimension':
It seems to me that the "dimension" of a measurable quantity answers the
question "What are we measuring?"; a question of KIND or CONCEPT - often
based on some conceptual model of the phenomenon under consideration and
its interaction with the measuring instrument.
...
I would classify length, mass, time, angular opening, charge, etc as
"dimensions"; and meters, kilograms, seconds, radians, and coulombs as
"units". The radian is not dimensionless anymore than the centimeter is.
They are each the unit of a dimension (angular opening and linear
extent). The fact that the size of an angular opening can be
measured/defined (in radians) by dividing two lengths is neither
bothersome nor unique.

Bob, I'm not sure I follow your argument. If it is the case that "the
'dimension' of a measurable quantity answers the question 'What are we
measuring?'; a question of KIND or CONCEPT" would you also consider
'heat', 'work', & 'torque' to have different dimensions considering they
seem to be different kinds or concepts? What kind of dimension would you
give to a decibel or a gigabyte? What about the dimension of the slope of
a line plotted as y vs. x? In the latter case (once we have established
our coordinate system) we know that the displacement of an object along
the 'y' axis direction is a physically different concept than the
displacement of that object along the 'x' axis direction (because the
object ends up in different places under these two different actions).
Just how similar or how different do two concepts have to be to have
distinguishing dimensions? Also, if the dimension answers the question
of conceptual identity, isn't the use of dimensions then redundant with
just the use of the names of the quantities? Under such circumstances I
fail to see any point in even having the idea of dimensions at all.

Unless I'm reading Bob very wrongly here (& I very well may be) I think I
like Joel's way of defining things better. I prefer the idea of
dimensions as a place holding means of 'power counting' when defining
composite concepts in terms of conceptually simpler ones--the simplest of
which are taken as the physical quantities which serve as the defining
the quantities for the base units for one's measurement system of units.
Under such a scheme angle is dimensionless and represents a ratio of
lengths (or a fraction of a period of a phenomenon or function which is
periodic over an appropriate infinite domain or a fraction of the
circumference of some abstract closed geometric space).

I could be wrong here, and if so would ask John Mallinckrodt to correct
me, but I somehow got the idea that John's earlier suggestion for giving
'angle' an irreducible dimension was essentially equivalent to a call to
consider the 'radian' as a base unit (rather than its current status as an
officially supplementary, but in practice derived one). Giving angle a
dimension would have the effect of putting powers of angle in the
dimensions of quantities (and corresponding powers of radians in the
units of) defined in terms of rotations or their generators such as
torque, moment of inertia etc.. This complication would have the
beneficial pedagogical effect of helping distinguish these rotation-based
quantities from other ones (such as work and mass quadrupole moment) that
are not defined with reference to rotations.

David Bowman
dbowman@georgetowncollege.edu