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Re: "equals" versus "is"



At 02:13 PM 1/26/01 -0600, Jack Uretsky wrote:
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.

We are getting into pretty deep waters here.

I agree that we have independent definitions of force, mass, and
acceleration. Therefore F=ma is not a tautology; it is something that
must be proved, to some appropriate degree of accuracy. We do not define F
in terms of m times a, so the equation is not written F := ma.

Meanwhile, we also have independent definitions of 2, 3, and 6. The
concept of 6 is not usually defined in terms of 2x3; the assertion that
6=2x3 must be proved. The proof is a couple of pages long; it is not a
trivial task. The result is exact.

So I would disagree with Jack. I would say that 6 = 2x3 in essentially the
same sense that F equals ma. One is deduced using the axioms of number
theory, while the other is deduced from theoretical mechanics principles
and/or induced from innumerable experiments. One result is absolutely
exact, while the other holds only in the nonrelativistic limit.

There is no special symbol that denotes "equality in the nonrelativistic
limit" or "equality for all practical purposes". Physicists just use the
"=" sign for this, the same sign as is used for exact arithmetical equality.

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

Jack also mentioned the notion of "identically equal" which is usually
typeset using three parallel horizontal lines, and often represented in
email as "==". This is another tricky concept. Some people use
f(x) == 0
as a shorthand for the universal quantifier, i.e.
for all x, f(x) is equal to zero
in contrast to
f(x) = 0
which suggests the existential quantifier, i.e.
there exists some x such that f(x) is equal to zero.

=========

I do not recommend using "==" to represent the notion of "defined in terms
of" since the latter is not an equivalence relation; it does not have the
symmetry property required of equivalence relations. It should be
represented by an asymmetric symbol such as ":=".