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Re: Dimensions vs units



Hi David,

My view of a "dimension" as the specification of the KIND of a measurable
quantity does not preclude a "taxonomy and hierarchy of dimensions",
which may include expressing some dimensions as a compounding of other
dimensions, or identifying some dimensions as subsets of a "wider"
dimension. Indeed, the essence of analytical/experimental science might
be characterized as defining the most useful dimensions and establishing
the relations among them - ranging from the simple "velocity =
length/time" to the more esoteric "first law of thermodynamics", and
beyond.

Yes, "heat, work and torque" are different dimensions (concepts); and
establishing any valid relations among them is the business of physics.
The tangent or sine of an angle (at least for acute angles) could be
taken as dimensions (concepts) specifying angular openings, in place of
the radian concept (in the limit of small angles, the three even become
numerically equal). An angular velocity d(SinT)/dt might be used
instead of dT/dt (again, at least for acute angles T). (Instead of the
circle, we would be using the right triangle to specify acute angular
openings; I don't advocate any of this - I only illustrate the
principle.)

A decibel is a dimension (concept) specifying the relation of two
quantities in terms of the logarithm of their ratio. I would take the
"gigabyte" to be a unit (an amount or size) referring to a definite
amount of data (a dimension, concept).

Yes, in my view dimensions are names. "NAMING" a set of useful concepts
is the crucial beginning of any science, followed by establishing
measuring procedures, standards and units; for the purpose of
investigating quantitative relations among these concepts (dimensions).

-Bob

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor

-----Original Message-----
From: David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>
To: PHYS-L@LISTS.NAU.EDU <PHYS-L@LISTS.NAU.EDU>
Date: Saturday, December 12, 1998 9:44 AM
Subject: 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