Re: Notation (was Re: Is this a general property of an abelian group?)


Arturo Magidin wrote:
In article <1144271326.669001.236310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<mike4ty4@xxxxxxxxx> wrote:

[.snip.]

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.


Oh, because what if you had an equation like

B BO O = Pigeon?

You wouldn't know if "BO" is a product, or the binary operation, right?

Basically; it invites confusion. Infix notation is, ->traditionally<-
(as I wrote above) used only for single-symbol operands.

Since you seem so fixed on the idea that we should not throw the
traditional meanings of things out the window, is strikes me as rather
ironic that you are perfectly willing to toss that particular
tradition out without a second thought.


So, what if I just used an "@" sign? a @ b?


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.


Ohhh... so you have to know by CONTEXT whether we are talking about
addition or some other operation, yep. But I was using what I wanted
because I thought some people could get confused.

And yet, if you stop thinking that it is better to think of "+"
exclusively as addition-of-numbers-related, you would have seen much
more clearly what your conditions entailed and what your conditions
required. Instead of saying

"a I.B.O. b = b I.B.O. a"

you would have obtained, as I did, "a + a = b + b" or "aa=bb", or
even "a#a = b#b". And this would have told you "if a#a is always the
same"...

Leibnitz used to say: if you have good notation, half your problem is
already solved. We ->use<- + to denote commutative operations in
abelian groups ->because it works<- and ->because it is good
notation<-. We use juxtaposition for possibly non-commutative
operations in arbitrary groups and rings for the same reason.


Actually, I saw that it reduced to a "+" a = b "+" b (did you see at
the end where it said b B.O. b = a B.O. a (which is the same as
a B.O. a = b B.O. b, symmetry property of equality!). I just didn't
know what to make of that, and writing it as "+", "*", "#", or "@"
instead of "B.O." wouldn't have been much more helpful.


(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.


Actually, I can see the ideas just fine. I just had a little objection
to the
notation, because it didn't seem to make sense why it would be used
if those symbols are already taken (and it might create confusion).

That's the point. They aren't "taken" in a pre-emptive sense. And you
should not think of them as taken.

If you insist on doing that, you will only complicate your life
unnecessarily. You do not need to invent a new symbol for every new
idea; matrix multiplication is very different from number
multiplication, but it is ->extremely useful<- to use the same symbol
for it.


Yes, but nonetheless it is called "multiplication".

Are you under the like impression that we should only use "x" to
denote independent variables, "f" to denote one-variable functions,
"y" to denote dependent variables?


No. I could use h as an independent, or s() as a function, t as
dependent if I wanted, etc. but the whole point I'm trying to make is:
TRYING TO ELIMINATE CONFUSION!

Is
it
because perhaps there are so many different operations on different
things called "addition" and "multiplication" that one might as well
just toss sense to the wind

We are not "tossing sense to the wind." That's the point. And that's
where your circular argument lies.

ALL the things that are called "addition" share the same basic
properties; we have ->abstracted<- what is important about the
operation we usually call "addition", and ignore all the itinerant
things that are completely irrelevant; then ANYTHING that shares those
same basic properties is called "addition" because then all the things
that are true about one of them will be true about all of them,
because they share those important basic, essential notions (rather
than itinerant things like whether they have to have numbers sticking
to either side of them). That's the whole point of abstraction.

The same with multiplication, composition, and juxtaposition. You are
->abstracting<-. And it seems clear that, despite your protestations,
you have a hard time seeing just what ->is<- and what ->is not<-
important about the operations.

and call anything satisfying the axioms they do
"addition" and "multiplication" even if they don't "feel like" a form
of addition/multiplication (like in the phc example I gave)?

The example you gave makes perfect sense, UNLESS you insist that
"pigeon" HAS to mean a kind of bird, that "house" HAS to mean a
certain kind of building, and that "computer" HAS to mean a specific
item, and that "+" HAS to mean 'adding things like numbers'. That's
where your discomfort and "does not feel like" comes from.


But they could all mean what they do, a bird, building, and machine,
and the operation would still work (remember, sets can consist of
ANYTHING). It just wouldn't be common sense to be called "addition",
or "multiplication", although I suppose you could (you can call
anything
ANYTHING you want, I could call a cardboard box, a PC, and a CRT
television a "box" because they all (roughly) have a box shape just
like
how the "addition"/"multiplication" "works like"
addition/multiplication in
terms of properties, there's no problem with it but it could confuse),
but it might create confusion.

But those are ->precisely<- the things you need to leave behind if you
want to abstract. Those are SEMANTIC meanings. The point in
abstracting is to stop relying on the semantics, and consider only
those things which follow merely from the basic properties, not their
linguistic meanings.


So what you're saying is that "addition"/"multiplication" are general
terms,
just like how anything shaped like (roughly) a rectangular
parallelepiped can
be called a "box".

But the whole point is this -- I want to avoid confusion. So are you
saying
that there is no confusion to eliminate in the first place, because of
how
general the terms have become? What if you reference number addition
with "+" and pigeon-house-computer abeliangroup B.O.ing in the same
sentence, and use "+" for the latter as well? Will this create
confusion?

Final question: Will there be virtually no confusion or ambiguity even
if
I use "+" and "*"?

--
======================================================================
"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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