Re: Is this a general property of an abelian group?

In article <XyUYf.4975$kg.1692@xxxxxxxxxxxxx>,
mike3 <mike4ty4@xxxxxxxxx> wrote:

[.snip.]


With notation that does not confuse an arbitrary operation with
good ol' "make-sense" addition & multiplication operations.

If you stay married to the notion that "+" can only mean "addition of
numbers", "*" can only mean "product/multiplication of numbers", and
that juxtaposition can only mean "composition of functions", you are
going to have a lot of trouble. It is important to be able to abstract
and to learn that the symbols don't have inherent meanings.

There is no such thing as "true addition" and "true multiplication",
and there is no reason for confusion.


"True addition/multiplication" means things that "make sense"
as addition/multiplication, etc.

The operation of any abelian group makes perfect sense as "addition";
all the usual identities are satisfied, so what is the problem? The
operation of any group makes perfect sense as the usual
composition/juxtaposition; all the usual identities are satisfied. So
what is the problem?

Note the pigeon-house-computer
example which makes no sense as such a thing,

Only if you insist that "+" can only mean "adding numbers." Your
argument is circular: "+" makes no sense because it can only mean
"adding numbers", and it can only mean "adding numbers" because it
does not make sense in any other sense.


Traditionally, we do NOT use infix notation with symbols that are
longer than a single character. You are trying to use infix notation
with a symbol with SIX characters.


Oh, you're including the periods? What's so bad about using multi-
character notation?

For infix notation? For starters, it is far more cumbersome, and it
pre-empts more symbols for use.

With your notation, we cannot use the letters I, B, O, or the period,
for any expression in the context of your discussion, unless as part
of your infix symbol notation; any other use invites confusion.


Well, I can fathom it, if the operation in question "makes sense" as
an addition or multiplication,

You are trapped in a circular argument.

Try to get out of it. Otherwise, you will be unable to learn anything
about groups, rings, vector spaces, matrices, and a plethora of
mathematics because you think the standard notation "makes no sense."
You will have to rewrite entire textbooks just so you can read them.

And ->that<- certainly makes no sense.

But, a much better suggestion is for you to try to get "unstuck" from
interpreting the symbols "+" and "*" exclusively as "addition of
numbers" and "multiplication of numbers".


Well, if it "makes sense" as addition or multiplication then it seems to
work

You are still repeating the same circular argument.

The symbol "+" is just a symbol. It has no inherent meaning as
"addition of numbers". The symbol "*" is just a symbol. It has no inherent
meaning as "multiplication of numbers." You say it makes no sense in
any other meaning because it only makes sense in that meaning. But why
does it only make sense in that meaning? Because you don't want it to
make sense in any other meaning.

(ie. pigeon + house = pigeon in house but pigeon + house =
computer doesn't, although it IS a perfectly OK operation provided
the set in question contains those elements, and therefore it would
make more sense to say pigeon B.O. house = computer.).

It makes no sense to get so stuck on notation that you cannot see the
forest because you are too busy saying that the only think that you
can call a "tree" is something that looks like the bush outside your
house.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

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