Re: equality versus identity

["John S. Denker" <jsd@MONMOUTH.COM>]

At 05:02 PM 10/14/01 -0400, Larry Cartwright wrote:
> *identical* or merely *equal*?

> Jim Green wrote:
> > I am not sure why you seem to denigrate the noble quality of "equal" with
> > "merely".

At 11:31 PM 10/14/01 -0400, Larry Cartwright wrote:

> Just a personal perspective.  It seems rarely do we identify two things
> that initially appear to be different but turn out to be identical, and
> I view such identity as far more profound than the uncountable physical
> examples of things that are just mathematically equal.

I agree that equality and identity are different, but I would describe the
difference differently:

Consider the following contrast:
  *) F=ma expresses equality, plain old equality.
  *) ma=ma expresses a tautology, a tautological equality, an identity.

Identity is the stronger statement, because every identity implies
equality, but not every equality is an identity.

However, the plain old equality is the more interesting statement.  It
expresses a fact about nature that we might not otherwise have known.  In
contrast, a tautology cannot tell us anything about nature.

To determine whether F=ma is true, we need to know what the symbols mean (F
means force, et cetera).  We can prove that ma=ma without assigning any
physical meaning to the symbols.