Re: [Phys-l] Equations

[John Denker <jsd@av8n.com>]

Quoting from Michael Edmiston (out of sequence, but hopefully not out of
context):

....
> if I can get students to discuss whether they are viewing
> physics equations as definitional or as cause/effect; and if
> cause/effect then which side is cause and which side is effect, then I
> think we have arrived at a higher level of appreciation of physics.
> Therefore this discussion of how equations are viewed is very
> worthwhile, especially if we focus on getting students to think about
> what equations might mean rather than nitpicking too much on the deeper
> philosophical issues of cause/effect.

1) I agree that questions of cause and effect are very important.

2) Not all deep issues are nitpicky.  Our working definitions ought to
 be *compatible* with a deep understanding of the issues, even if we
 don't expose the students to every nitpicky detail.

 To say the same thing in other words:  I don't object to incomplete
 coverage of the issues, but I object when the points that are covered
 are diametrically wrong.

> ....  I 
> always write it a = F/m and I tell students that writing it this way 
> ought to help them grasp it better than writing F = ma.

This is unhelpful for several reasons.

First, it needlessly blurs the distinction between equality and causality,
i.e. between equation and causation.
 -- Equality is (by definition!) an equivalence relation.  As such, it is
  reflexive, symmetric, and transitive.  Reasonable definitions and discussion
  can be found at
    http://en.wikipedia.org/wiki/Equivalence_relation
  In particular, "symmetric" means that if A = B, then B = A.
 -- Causality is (by any halfway reasonable definition) not symmetric.  A ==> B
  does *not* imply that B ==> A.

Equations is an important idea.  Cause-and-effect is an important idea.
These are not the same idea!  Absolutely, positively not the same.

It is a tremendous disservice to the students to teach them to confuse
causation using equation.  They will develop wrong ideas either about
causality or about equality.

As an application of this idea:  Saying that  "a equals F/m"  (for
nonzero m) has exactly the same meaning as saying "F equals ma".
If you want to say something about causality, go ahead, but call it
causality, not equality.

Say what you mean, and mean what you say.

In particular, it turns out that
    a is_caused_by F/m       (and not vice versa)
is false, even though
    a equals F/m
is true.  (We continue to assume nonzero m.)

Here's a useful illustration:  Consider a turntable, rotating uniformly.
A puck sits on the turntable.  I can measure the mass (m) of the puck,
and I can measure the force (F) required to hold the puck in position
relative to the turntable ... but it would be perverse to suggest that
the acceleration (a) is _caused_ by F/m.

Here's another bit of evidence supporting the main point:  Causes ought
to precede effects.  We know that /post hoc ergo propter hoc/ is a
fallacy, but the converse, /non post hoc ergo non proper hoc/ is reliable
rule.  Applying this to F=ma, we note that this equation is an equal-time
equation, i.e. F(t)=ma(t) for each and every time t.

Yes, there is a relationship between F and ma.  This relationship is
called equality ... not causality.  The vast majority of the formulas
encountered in first-year physics share this property:  they express
equality ... not causality.

> For example another way to view an equation is (idea or property) = 
> (definition).  I view work that way:  w = integral(F dot dr) is the 
> definition of work.

The _definition_ operator is not an equivalence relation.  In particular,
it is not symmetric.

I strongly recommend using the ":=" symbol to express definitions (instead
of the "=" symbol).  This is conventional in many circles.  The := symbol
looks non-symmetric, correctly corresponding to the non-symmetric nature
of the operation it symbolizes.

=====================

There is one more non-symmetric notion that must be discussed, namely
the "how-do-you-know" relationship, i.e. the "calculated-in-terms-of"
relationship.

This is closely related to the "assignment" operator in computer
languages.  Most regretably, many computer languages (including basic
and c) abuse the "=" symbol to represent assignment, as in
  x = x*x
which does _not_ represent a statement of equality;  instead it says
that x is a assigned a new value calculated by squaring the old value.
Note that some languages (notably the Algol family) use the ":="
symbol to represent assignment, as in
  x := x*x
which IMHO does a much better job of conveying the asymmetric nature
of the assignment operation.

Anyway, the important point here is that when people think they are
discussing causation of some physical result, very commonly they are
really just discussing how they _know_ the result.  For example, in the
case of the turntable, I know the F/m ratio at each point on the turntable,
_because_ I was told that the turntable is undergoing uniform rotational
motion.  I emphasize that the "because..." clause applies to the knowing,
not to the motion.

The turntable is a good example where F is easily calculated in terms of
ma.  There are of course innumerable examples (e.g. a slingshot) where
a is easily calculated in terms of F/m.

F=ma and a=F/m are equations.  For nonzero m, they express exactly the same
fact.  They express equality, not causality.  You can say  a  is associated
with  F/m, but you cannot reliably conclude that  a  is caused by  F/m.
If you want to make statements about causality, go ahead, but pleeeease
keep them separate from statements about equality.

===================

To summarize:  There are at least four distinct ideas in play here:
 -- _Equality_ is symmetric.
 -- _Causation_ is not symmetric.
 -- _Definining_ one thing in terms of another is not symmetric.
 -- _Calculating_ one thing in terms of another is not symmetric.

Each of these ideas is important, but they are definitely not the same
idea.  These are not picayune distinctions;  they are
  a) well within the grasp of ordinary high-school students, and
  b) essential to any real understanding of the topic.

For more on all this, see
  http://www.av8n.com/physics/causation.htm