Re: "equals" versus "is"

[John Denker <jsd@MONMOUTH.COM>]

At 07:48 PM 1/25/01 -0700, Jim Green wrote:

> Does E equal mc^2 or is it that E _is_ mc^2 .

1) A few months ago, at 01:06 PM 10/10/00 -0400, I wrote:
> > The expression F = ma uses the "=" sign which represents the "equality"
> > operator.  Formally, equality is member of the class of _equivalence
> > relations_, because equality is reflexive, symmetric, and
> > transitive.  Reference:
> >    http://www.ms.uky.edu/~carl/ma502/html/green1.html

And I stand by that.  The meaning of the symbols is clear:  E equals mc^2.
There is no other way to read the symbols.

2a) In contrast, the English word "is" generally does not express an
equivalence relationship.  More usually, it expresses "set membership", or
the very similar notion of "implication".  Examples include
        Every Mustang is a Ford.                (set membership)
        Playing with matches is dangerous.      (implication)
These relationships are reflexive and transitive, but they are not symmetric.

2b) OTOH sometimes "is the" (unlike "is a") expresses equivalence.  For example
        The president is the commander in chief.        (equivalence)
        The commander in chief is the president.        (equivalence)

> F=ma means that if I calculate/measure F and calculate m x a they will be
> equal.

Fine.

> F=mg means that the force _is_ m x g.

No, the symbols F=mg clearly state that F _equals_ mg.
There is no other way to read the symbols.

There is one slightly subtle point here.  The expression F=mg is a
shorthand that hides some important physics.  It hides the fact that mg is
often not the total force.  So it would be better to write
        F_grav = mg
to leave open the possibility of writing
        F = F_tot = F_grav + other forces


> Again, mg is a force, ma is not.

I disagree.  I would say
        mg is a force           (set membership)
        ma is a force           (set membership)

Note that mg may or may not be "the" force (i.e. the total force).

Probably I would say that anything that has dimensions of force "is a" force.