Re: What is this thing known as "=" ?


huangxienchen@xxxxxxxxx wrote:
Chip Eastham wrote:
huangxienchen@xxxxxxxxx wrote:
T.H. Ray wrote:
T.H. Ray wrote:

> that someone else wrote:

There are different kinds of "equal". Which kind
are
you talking about?

Nonsense. Equivalence, in a mathematical equation,
has only one meaning. I have recited it for you.
Whatever else you want to say about equivalence may
have some personal meaning to you, but it has
nothing to
to do with mathematics.

Ah, you think that identity is trivial?

http://hdebruijn.soo.dto.tudelft.nl/fototjes/gezocht.h
tm

Han de Bruijn


Do you think that every equation is an identity
equation?

Tom


Tom, I still think that I'm miscommunicating what I'm going after in
this.

Look at this proof again,
http://en.wikipedia.org/wiki/Zero_element

--------------------------------------------------------
Proving the uniqueness of a zero element is equivalent to proving the
uniqueness of an additive identity. Assuming there are two, 0 and 0',
we have that 0 = 0 + 0' = 0', so that 0 must be unique. Thus we can
speak of the zero element in a system.
--------------------------------------------------------

OK - they are saying that since 0 = 0', hence 0 must be unique because
0 _is_ 0'.

Simply showing that zero is quantitatively equal to zero' does not
prove uniqueness in my opinion. Unless you are suggesting that quantity
is a property which is inherently endowed with uniqueness - maybe
that's the case. Then, quantitative equivalence would imply that 0 "is"
0' and the proof would establish uniqueness, but that entails a pretty
substantial assumption that you have already proved the uniqueness of
the reals, in which case uniqueness of zero is implied anyway.

[huangxienchen@xxxxxxxxx wrote:]
I dont like this usage of " = ".

Fine. Use or interpret it however you wish. You are scarcely the
first to consider alternative interpretations of "=" as some sort of
equivalence _other than_ identity. But a lot is known about this
"generalization" and it doesn't really lead to anything interesting.

If it suits your tastes, though, then by all means go for it. It's a
wonderful thing to have the freedom in mathematics to reinterpret
the basic concepts within a common framework of operational
rules.

regards, chip


Well, I never really gave it too much thought until now. And I'm not
trying to pick a fight over it - but seriously - I really think that
this whole disagreement boils down to a problem of linguistics and how
people read math.

Many people will see something like y = t^2. They might read this as "y
is t^2". But what it really means is not that y "is" t^2, but that y is
quantitatively equal to t^2.

y is a quantity, and t^2 is another quantity, and = simply equates
them.

This is quite different from y actually "being" t^2, and that's what
the proof in question is attempting to do.

[huangxienchen@xxxxxxxxx wrote:]
Sure, sodium "is" sodium, potassium "is" potassium, and 50gNa "=" 50gK
. But Na =/= K.

The proof says that 0 = 0'. That 0 "is" 0'. I disagree.

There are many contexts in which one introduces equivalence
relations and then, by "abuse of notation", refers to equality
in place of a strict observance of notation that reflects the
underlying choice of representatives of equivalence classes.
Esp. in a text medium such as this, where mathematical
notation is "cramped" by a limited typography, one needs
to exercise a bit of good faith effort to follow what others
are trying to say.

Your example of "equal" masses of non-identical substances
is a nice one. Chemists will even speak of "mass equivalents"
for clarity in cases like this.

Of course _sodium "is" sodium_ is also "equally"
open to attack. Are there not distinct isotopes of sodium?
Is not the gram of sodium locked up in my shaker of table
salt physically distinct from the gram of metallic sodium
kept in oil to prevent oxidation? One of the Buddhist
critiques is that material objects are like a river flowing,
never the same in any two viewings, so that "identity" is
always a conventional rather than absolute "truth".

Yet in math we have no need to appeal to any absolute
reality beyond the conventional one that is described by
axioms, so apart from unsettling a personal attachment
to "numbers" or "geometry" as having a Platonic reality,
this critique passes harmlessly over mathematicians.

"Modulo" an equivalence relationship imposed, then
equality means, in mathematics, the true identity of the
corresponding equivalence classes. Numerical value 0
may arguably be different in the real numbers than it is
in the complex numbers, and certainly than in the ring
of integers modulo 7 or the zero vector in another context.

But if you stick to one of these contexts, or in some
cases contrive to combine a pair of them, eg. viewing
the real numbers as a subring of the complex numbers,
then objections vanish.

It really is important to understand that just naming
two things "zero" is not enough to establish "identity".
Hopefully, however, the use of same names is apt and
expresses some "identicalness" that is relevant, and
not used simply to mislead the reader/listener.

An argument can go wrong when a characteristic is
evoked that, rather than being common to all and
thus independent of choices of representatives in
equivalence classes, actually depends on the
particular representative chosen. Failures of this
kind earn the sobriquet "not well-defined" as the
notion "well-defined" refers to establishing that a
characteristic is independent of choice of the
equivalence class representatives.

Making these interpretations appropriately, as the
context requires, is part of what is conventionally
called "mathematical maturity". I applaud the
independence of spirit that leads you to question
this authority of "names". If you think, after careful
study, that an author is "abusing notation" to draw
incorrect conclusions, by all means point it out in
a polite way. I've certainly been misled by my
own shorthand notations at various times. But
let's try not to "split hairs", okay?

regards, chip

.

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