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

mike4...@xxxxxxxxx wrote:
Arturo Magidin wrote:
In article <1144285003.377445.114730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<mike4ty4@xxxxxxxxx> wrote:


<snip>


Yep. But *I* choose to call it, in a general sense, "the binary
operation"
and denote it by the UNAMBIGUOUS symbol "B.O."!


<snip>

How would it be so much harder to use something other than "*" and
"multiplication"? Why not "B.O." and "binary operation"? I'll admit,
you
don't have to, but I simply asking out of curiosity.


Just so you don't think that Arturo is winding you up...

Yes, it's true that your statement has unambiguous mathematical
meaning.

But good notation is more than just unambiguous. It also helps the
reader to follow what is actually being stated as part of a
mathematical argument/exposition.

Some of this is due to familiarity. You could, for example, create your
own notation for the standard symbol for a definite integral, say X!
dt f(t) min max. But if I have to read sheafs of equations in this
non-standard notation, I'm less likely to be able to quickly and easily
read it.

Similarly, when you use "+" in this setting, I assume that a + b = b +
a is an axiom for the operation. All of my intuitions about this
property can be brought in to play; because I instantly recognize the
abstract property "commutative", just by the notation. I also
understand that 3a = a + a + a.

But there is also the issue of whether your notation hides or exposes
fundamental underlying structure.

For example, in usual multiplicative notation for a group G with |G|
elements, we write a^n for a*a*a*... n times. Many theorems are easier
to write when we use this notation; for example, if n = |G|, then g^n =
e (the identity) for all g in G.

In your notation, a B.O. a B.O. a B.O. ... n times is written... how?

Your response brought up an intriguing Q: why are "multiplicative"
operations generally not considered commutative? Is it because of
matrix multiplication, quaternion multiplication, etc?


I would guess that it is associated with matrix multiplication, which
is usually the first example of we encounter of a non-commuting
operation. And non-abelian groups can be represented as the product of
matrices.

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!

But that confusion only exists if you insist that the symbol MUST
represent something which is "basically regular addition." And
->that<- is what you need to abandon.


Well, I may have abandoned it for *me*, but I used the notation
I used so that *others* would not get confused.


All I can add is that it apparantly didn't work :).

You agree that there would be no confusion if you go from

int(f(x)dx) to int(f(t)dt.

You know that whether you use an "x" or a "t" does not matter. There
are people for whom such a change might be confusing, but those are
people who have not really understood what the symbols mean and how to
operate with them; those are people who have trouble doing
integration, in some part for those hangups.


Yep.

Likewise, while certainly there are people who could be confused by
the use of the symbol "+" to mean somthing other than "adding
numbers", those are also people who will not (or should not) be
talking about general binary operations, groups, and so on. You need
to be able to step beyond that meaning if you want to become adept at
abstract algebra, just as you have to be able to step beyond any
queasiness you may feel at changing all the variables from "x" to "t"
if you want to become adept at integration.


So, then, the terms "addition" and "multiplication" are actually
*re-defined*
to mean something other than "the usuals", right, something a lot more
general?


Let's just take addition.

In the abstract, group addition has the properties "a + b = b + a" and
there is a special element which I will write as "0" such that "a + 0 =
a"; and for every a, there is an a' with a + a' = 0; in which case we
write a' as "-a", and we write "a + (-b)" as "a - b".

Now, this is true for adding in the integers; and it's also true for
adding nxn matrices with entries which are complex numbers; it's true
for addition modulo 5, and it's also true in the Klien group, etc. etc.

It's not so much that addition is "redefined" to mean something else;
instead it's a method of abstracting the operation "+" from the set
upon which it is operating.

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).


If you insist on getting stuck on the semantical meaning of the words
and symbols, rather than focus on their mathematical properties, you
simply will not get very far in doing mathematics.


I do focus on the math. properties. That's why I said, even if the
objects
"pigeon", "house", and "computer" do represent real objects, the
operation would still be valid (although it might not make any sense,
but
it nonetheless would still be a mathematically acceptable
relationship.).


"Even if"? In this domain of discourse, why would a mathematician
/ever/ talk about a bird, a house, or a computer in the way you
describe?

Mathematicians don't talk about concrete objects like that. A tree is
not where birds go to roost, a field is not a place where horses graze,
a group is not a collection of people, a ring is not round, etc., etc.

It just wouldn't be common sense to be called "addition",

I would argue that it would be a severe lack of common sense to think
that, in the context of a mathematical discussion, "pigeon + house =
computer" would have ANYTHING to do with birds, dwellings, electronic
appartusii, and the addition of numbers. People might not be clear on
what the symbols mean, but they would not be "confused" by thinking
that you might be talking about birds and dwellings


Actually, to make it clear, the items "pigeon", "house", and "computer"
represent just what anyone would think they do, a bird, a building, and
a machine.


I really can't imagine how such usage would possibly reduce confusion
in a mathematical context.

I can imagine something like "take one pigeon, and add another pigeon,
how many pigeons are there?" But this is again a case of abstraction; I
can just as easily imagine saying "take one floogle, add another
floogle, how many floogles are there?"

If the answer depends on what an /actual/ real world floogle is, and
knowledge of its real world properties, then I don't think it is
properly a mathematical question.

You are talking about terms of art, not about everyday language. You
are talking math, not speeches before the school assembly. If you
cannot differentiate, then you will also not get very far.


So what you are saying is, again and again, and even right at this
very point, that the "addition"/"multiplication" are just terms, and in
this
CONTEXT, they don't have "common-sense" meaning and since that
is ASSUMED, then there is NO confusion, even if you reference
addition of numbers and "addition" of pigeons and houses in the
same sentence: "Did you know that in the set Z of integers, 2 + 2 = 4,
and that in the set X = { pigeon, rabbit, computer, house, toad },
pigeon + house = computer?"


I guess I would then ask, is (X,+) isomorphic to Z_5 (addition mod 5)?
If so, you can just say "Z_5" - it has all the properties of (X,+), the
terminology is familiar, and there are many theorems that have been
proven regarding Z_5 which are then also true of (X,+).

<snip>

Actually, *I* can use the symbols "+" and "*". The point of the "B.O."
notations were to avoid confusing *others*, but I guess that with
100% certainty, anyone mathematically competent enough in this
subject would never be confused.


Actually, your use of "a B.O. b" was /more/ confusing than if you had
used a simple infix symbol like "a@b"; because the latter is what
people typically use, for various good reasons.

Cheers - Chas

.

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