Re: "equals" versus "is"

[Jack Uretsky <jlu@HEP.ANL.GOV>]

        While I don't disagree with anything that John wrote, I think
he misses an important point that is troublesome to students.  The
dichotomy is not between "equals" and "is", but between "equals" and
"is the same as".  This is a distinction that mathematicians often make
by use of the identity symbol ("=" topped by a "~").  In physics, and
elementary math courses, we seldom make that distinction, with the result
that "A=B" may have one of two meanings.
        If, for example, I have independent ways to define the quantities
force, mass and acceleration (not a trivial task), then the equation
F=ma is a statement about the result of three different measurents, as
John stated.  When, on the other hand, I write: 6=3x2 I am stating that
the number 6 <is the same as> the product of the numbers 3 and 2.  These
are, conceptually, two different uses of the equal sign.
        Students often confuse these two different uses.  If you require
your students to write out and explain their steps in solving problems,
you will often find students making statements such as "since F is the
same as mass times acceleration...".  Such a statement is conceptually
wrong, but seems to follow naturally from the student's prior experience
with the use of "=".  I think that teachers should be alert for this
misconception and begin to correct it from its earliest appearance.
                                Regards,
                                        Jack



On Thu, 25 Jan 2001, John Denker wrote:

> 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.

--
While [Jane] Austen's majestic use of language is surely diminished in its
translation to English, it is hoped that the following translation conveys
at least a sense of her exquisite command of her native tongue.
                        Greg Nagan from "Sense and Sensibility" in
                     <The 5-MINUTE ILIAD and Other Classics>