Re: definitions +- equations

[John Denker <jsd@MONMOUTH.COM>]

At 07:26 AM 10/15/00 -0400, Robert A Cohen wrote:

> we need to teach students to interpret equations appropriately.

Right.

>  Sometimes order is important, as in definitions.  We write
> that v = dx/dt as a definition of v but dx=vdt is not a definition of dx.

OK.

Some suggestions:

1) Do not expect (nor teach students to expect) that
        v = dx/dt
will be interpreted as a definition.


2) When defining things, be explicit about what is being defined.  It is
better to write explicitly
        "v = dx/dt by definition of v"
than the nonexplicit
        "v = dx/dt by definition"
or the even worse
        "v - dx/dt = 0 by definition"


3) There are many good reasons to prefer the ":=" symbol
        "v := dx/dt"
were we use an asymmetric symbol ":=" to denote an asymmetric operation
(definition).

Note that the definition
        v := dx/dt
implies the equation
        v = dx/dt
but the converse does not hold;  the equation does not imply the definition.


4) It is important to pay attention to the quantifiers that are attached to
equations.  For instance the universal quantifier in the predicate
        for all x, sin^2(x) + cos^2(x) - 1 = 0
is very different from the existential quantifier in the predicate
        for some x, sin^2(x) - 0.5 = 0
which is different from the instruction
        find x such that sin^2(x) - 0.5 = 0



> We write F=ma and a=F/m and say both are equivalent.  We need to emphasize
> that an empirical relationship just identifies the relationship - it does
> not identify cause/effect.

Right!