* Kragen Javier Sitaker <kragen at canonical.org> [2008-07-21 09:40]:
> now expressed in kg*s/s, or merely kg.  So our two
> relativity-based equivalences (mass as energy, and time as
> distance) turned out to be merely two facets of the same
> equivalence.

This is how Einstein stumbled into E = mc², of course! In a
letter to his friend Conrad Habicht, he wrote:

    Eine Konsequenz der elektrodynamischen Arbeit ist mir noch in
    den Sinn gekommen. Das Relativitätsprinzip in Zusammenhang
    mit den Maxwellschen Grundgleichungen verlangt nämlich, daß
    die Masse direkt ein Maß für die im Körper enthaltene Energie
    ist; das Licht überträgt Masse. Eine merkliche Abnahme der
    Masse müßte beim Radium erfolgen. Die Überlegung ist lustig
    und bestechend; aber ob der liebe Herrgott nicht darüber
    lacht und mich an der Nase herumgeführt hat, das kann ich
    nicht wissen.

Or in English:

    Another consequence of the work on electrodynamics has also
    occured to me. Namely, the principle of relativity in
    connection with Maxwell’s equations requires that its mass
    be a direct measure of the energy contained in a body; light
    transfers mass. A noticeable decrease of mass should occur in
    the case of radium. The thought is amusing and intriguing;
    but whether the good Lord isn’t laughing about it and hasn’t
    led me down the garden path, I cannot know.

Regards,
-- 
Aristotle Pagaltzis // <http://plasmasturm.org/>

I was reading Raphael Finkel's book "Advanced Programming Language
Design", and he mentions a language somebody named Finkel invented in
the 1970s called AL.  AL had dimensional analysis built in, with four
base dimensions: 'time', 'distance', 'angle', and 'mass'.

(As Finkel says, this is the kind of "type safety" that's really
needed in everyday calculations.  His language did it statically, but
you can do it dynamically too, as units(1) does.)

He points out that 'angle' really shouldn't be included since radians
are really dimensionless, which leaves us with 'time', 'distance', and
'mass'.  But no 'speed', 'force' or 'energy'.

However, they can be derived from the base units; speed is a ratio
distance/time --- e.g. m/s.  Acceleration is speed/time, or m/s², and
force can be thought of as mass * acceleration, or kg*m/s².  Finally,
energy is force * distance, so you can express it in units of
kg*m²/s².  This is in fact how units(1) represents it.

But do we really need all three of 'time', 'distance', and 'mass'?
The speed of light provides a natural conversion factor between time
and distance, and mass can be equivalently measured as energy,
according to the well-known formula E = mc².  So speed is really just
a dimensionless quantity, and acceleration is the reciprocal of a time
interval, namely the time to reach the speed of light at that constant
acceleration; so its units are really 1/s.  So force can really be
measured merely with kg/s.

Unfortunately this doesn't really help us get rid of mass: energy is
now expressed in kg*s/s, or merely kg.  So our two relativity-based
equivalences (mass as energy, and time as distance) turned out to be
merely two facets of the same equivalence.

But I suspect that in everyday calculations, the equivalence of mass
and energy is rarely useful; it's far more likely to hide errors than
to reduce the amount of work necessary.  With time, distance, and
mass, the only incommensurable quantities I commonly run into with
commensurable units are:

- torque and energy;
- various dimensionless quantities;
- perhaps pressure and stress;
- stress and young's modulus.

Collapsing mass with energy and time with distance creates many more
"units collisions".