"by an abuse of notation" vs "by abuse of notation"

[jaakov]

Dear all:

Which version is correct (and why):

- By an abuse of notation, we use the same symbol here as in Section 1.
- By abuse of notation, we use the same symbol here as in Section 1.

Thank you in advance,

Jaakov.

[Richard Tobin]

In article <mht9o7$cfr$2...@speranza.aioe.org>,

jaakov <jaakovR...@DELETEITro.ru> wrote:

>Which version is correct (and why):
>
>- By an abuse of notation, we use the same symbol here as in Section 1.
>- By abuse of notation, we use the same symbol here as in Section 1.

I don't know about correct, but the first seems more normal to.

-- Richard

[Athel Cornish-Bowden]

On 2015-04-30 15:12:41 +0200, jaakov <jaakovR...@DELETEITro.ru> said:

> Dear all:
>
> Which version is correct (and why):
>
> - By an abuse of notation, we use the same symbol here as in Section 1.
> - By abuse of notation, we use the same symbol here as in Section 1.

Impossible to answer without more information.



--
athel

[Richard Tobin]

In article <cqepvs...@mid.individual.net>,

Athel Cornish-Bowden <athe...@yahoo.co.uk> wrote:

>> Which version is correct (and why):
>>
>> - By an abuse of notation, we use the same symbol here as in Section 1.
>> - By abuse of notation, we use the same symbol here as in Section 1.

>Impossible to answer without more information.

It's a common phrase in maths. It refers to practices like
manipulating dy/dx as an ordinary fraction in contexts where it works.

-- Richard

[jaakov]

Here is the context.

In section 1, we defined 3-adic relation [right arrow]

x [right arrow]^a y :<=> ... .

In Section 2, we introduce another relation, a 4-adic one, which is
related to the previous 3-adic one:

x [right arrow]^w_i y :<=> ... .

Essentially, the 4-adic one is a relation on words w of length i, which
extends the 3-adic one, which is relation on symbols a (i.e. words of
length 1).

I wish to make the reader aware of the following facts in the most
compact and understandable way:
Logically speaking, these relations are different. But they are
intuitively similar, so that we decide to use the same symbol for them.

"By (an) abuse of (the) notation, we use here the same symbol as in
Section 1."

[Charles Bishop]

In article <mhtbsj$os7$2...@macpro.inf.ed.ac.uk>,

What happens if you do it where it doesn't work? Is it obvious which is
which from the equation(?) it's in?

--
Charles

[Jerry Friedman]

I'm not sure I've ever seen this phrase, but I'd be tempted to write
"In an abuse of notation". I wouldn't say either of your possibilities
is wrong, though.

--
Jerry Friedman

[jaakov]

Hm, I never saw your version before. An alternative would be to say
"Abusing (the) notation, we use ..."

[Iain Archer]

jaakov <jaakovR...@DELETEITro.ru> wrote on Thu, 30 Apr 2015 at
16:06:05:

Now that I know it's mathematico-speak, any of those alternatives looks
ok to me. When I search for use of the phrase by mathematicians, I find
also:

"With a slight abuse of notation, we will from now on use P to ..."

"and, by abuse of notation, its string encoding"

"As a slight abuse of notation we refer to both f and. sign(f) as
classifiers."
--
Iain Archer

[Peter Moylan]

On 01/05/15 00:06, jaakov wrote:
> Am 30.04.2015 um 15:49 schrieb Richard Tobin:
>> In article <cqepvs...@mid.individual.net>,
>> Athel Cornish-Bowden <athe...@yahoo.co.uk> wrote:
>>
>>>> Which version is correct (and why):
>>>>
>>>> - By an abuse of notation, we use the same symbol here as in Section 1.
>>>> - By abuse of notation, we use the same symbol here as in Section 1.
>>
>>> Impossible to answer without more information.
>>
>> It's a common phrase in maths. It refers to practices like
>> manipulating dy/dx as an ordinary fraction in contexts where it works.

As Iain has pointed out, "by a slight abuse of notation" seems to be
more common, and it's what I would have suggested before you gave the
context. (Have you noticed how many people change their answers once
they know the context?) We don't like to suggest that we're being really
abusive.

> Here is the context.
>
> In section 1, we defined 3-adic relation [right arrow]
>
> x [right arrow]^a y :<=> ... .
>
> In Section 2, we introduce another relation, a 4-adic one, which is
> related to the previous 3-adic one:
>
> x [right arrow]^w_i y :<=> ... .
>
> Essentially, the 4-adic one is a relation on words w of length i, which
> extends the 3-adic one, which is relation on symbols a (i.e. words of
> length 1).

Now that I know that, I would write "By an extension of the notation in
section 1, ...". It's not really an abuse; presumably you can tell which
meaning is intended by looking at what follows the right arrow. I assume
too that the two definitions agree when i=1. (If they didn't agree, THEN
it would be an abuse of the notation.)

Software people call this operator overloading, and many of them
consider that it's a Good Thing.

--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia

[jaakov]

Dear Iain and Peter,

I see, thanks a lot!!! "Extending the notation" sounds better.

Best,

Jaakov.

[Iain Archer]

Peter Moylan <pe...@pmoylan.org> wrote on Fri, 1 May 2015 at 11:22:25:
>Now that I know that, I would write "By an extension of the notation in
>section 1, ...". It's not really an abuse; presumably you can tell which
>meaning is intended by looking at what follows the right arrow. I assume
>too that the two definitions agree when i=1. (If they didn't agree, THEN
>it would be an abuse of the notation.)
>
>Software people call this operator overloading, and many of them
>consider that it's a Good Thing.

But pure mathematicians are the the least likely to welcome the silent
introduction of ambiguous operators. It seems they recognise the
notion of abuse of notation --
<http://en.wikipedia.org/wiki/Abuse_of_notation> -- and have their
mantras to frame its use. I wouldn't want to interfere with that.
--
Iain Archer

[Peter Moylan]

Let me quote from a book of mine. This is from Appendix A of
http://pmoylan.org/pages/research/DissBook.html
which happens to be a significantly mathematical section.

"For a row or column vector x, the condition x > 0 means that all
elements of x are real and positive, and x ≥ 0 means that all
elements of x are real and nonnegative. Similarly, x < 0 means
−x > 0, and x ≤ 0 means −x ≥ 0. For a square matrix, A > 0 instead means
that A is symmetric and positive definite."

Executive summary: the symbol '>' has one meaning for square matrices,
and an entirely different meaning for a column vector. This is, I
believe, consistent with a tradition that says we can re-use common
symbols if no ambiguity results; a tradition that is based on the
observation that we have a shortage of useful symbols. For my own
purposes, it helped that both meanings of '>' were in common use in the
literature.

I do agree, though, that "by a slight abuse of notation" is often seen.
The version without "slight" is less often seen.

[Iain Archer]

Peter Moylan <pe...@pmoylan.org> wrote on Fri, 1 May 2015 at 22:35:16:
>On 01/05/15 21:49, Iain Archer wrote:
>> Peter Moylan <pe...@pmoylan.org> wrote on Fri, 1 May 2015 at 11:22:25:
>>> Now that I know that, I would write "By an extension of the notation in
>>> section 1, ...". It's not really an abuse; presumably you can tell which
>>> meaning is intended by looking at what follows the right arrow. I assume
>>> too that the two definitions agree when i=1. (If they didn't agree, THEN
>>> it would be an abuse of the notation.)
>>>
>>> Software people call this operator overloading, and many of them
>>> consider that it's a Good Thing.
>>
>> But pure mathematicians are the the least likely to welcome the silent
>> introduction of ambiguous operators. It seems they recognise the
>> notion of abuse of notation --
>> <http://en.wikipedia.org/wiki/Abuse_of_notation> -- and have their
>> mantras to frame its use. I wouldn't want to interfere with that.
>
>Let me quote from a book of mine. This is from Appendix A of
> http://pmoylan.org/pages/research/DissBook.html
>which happens to be a significantly mathematical section.
>
>"For a row or column vector x, the condition x > 0 means that all

>elements of x are real and positive, and x 0 >of x are real and nonnegative. Similarly, x < 0 means 0 >means 0 >symmetric and positive definite."

>
>Executive summary: the symbol '>' has one meaning for square matrices,
>and an entirely different meaning for a column vector. This is, I
>believe, consistent with a tradition that says we can re-use common
>symbols if no ambiguity results; a tradition that is based on the
>observation that we have a shortage of useful symbols. For my own
>purposes, it helped that both meanings of '>' were in common use in the
>literature.

My inexactitudinous mistake then. For "ambiguous operators" I should
perhaps have said "operators which appear ambiguous in a particular
context". Even then, I'm not sure I'd escape the possibility of
objection. Common ground, I'd suggest, is that ambiguity, actual or
potential, is (with some exceptions) deprecated and avoided within all
language systems. Sort of, innit.

>
>I do agree, though, that "by a slight abuse of notation" is often seen.
>The version without "slight" is less often seen.

It's been new to me. I dare say there may even be some informal
regularity and convention in the accepted types and descriptions of the
said abuse, but I don't have the domain knowledge to know or tell.
>
--
Iain Archer

[jaakov]

The only problem I see is that avoiding ambiguity is costly (space to
write, time to read, and, under circumstances, time to understand).

>>
>> I do agree, though, that "by a slight abuse of notation" is often seen.
>> The version without "slight" is less often seen.
>

To me, "slight" is new in the context.

> It's been new to me. I dare say there may even be some informal
> regularity and convention in the accepted types and descriptions of the
> said abuse, but I don't have the domain knowledge to know or tell.
>>

Anyway, thanks you both!

[Charles Bishop]

In article <mi04n2$up7$1...@speranza.aioe.org>,

What is the cost if there is ambiguity? I propose that it must be more
costly, with the costs you enumerate as well as mistakes that may come
from the ambiguity.

[snip]

-- c

[Jerry Friedman]

On Thursday, April 30, 2015 at 3:16:06 PM UTC-6, jaakov wrote:
> Am 30.04.2015 um 21:29 schrieb Jerry Friedman:
> > On Thursday, April 30, 2015 at 7:12:43 AM UTC-6, jaakov wrote:
> >> Dear all:
> >>
> >> Which version is correct (and why):
> >>
> >> - By an abuse of notation, we use the same symbol here as in Section 1.
> >> - By abuse of notation, we use the same symbol here as in Section 1.
> >>
> >> Thank you in advance,
> >
> > I'm not sure I've ever seen this phrase, but I'd be tempted to write
> > "In an abuse of notation". I wouldn't say either of your possibilities
> > is wrong, though.
> >
> Hm, I never saw your version before.

It does exist, according to Google. I didn't try to figure out how
common it is among native-English-speaking mathematicians. Anyway, you
might as well go with what you're familiar with.

> An alternative would be to say
> "Abusing (the) notation, we use ..."

I think that could work. I'd prefer the "the". It's also more pleasant,
in my opinion, not to put the verbs "abuse" and "use" so close together
unless they have corresponding meanings, so "employ" might be better than
"use". (Putting the noun "abuse" next to the verb "use" isn't so bad in
my opinion, maybe because of the difference in pronunciation.)

--
Jerry Friedman

[Jaakov]

So,
"Abusing the notation, we employ here the same symbol as in Section 1"
?

[Mark Brader]

"Jaakov" asks about:

> > - By an abuse of notation, we use the same symbol here as in Section 1.
> > - By abuse of notation, we use the same symbol here as in Section 1.

Jerry Friedman:

> I'm not sure I've ever seen this phrase, but I'd be tempted to write
> "In an abuse of notation".

I agree. The "by" version suggests that you are describing the *result*
of abusing the notation, not stating the fact you are about to abuse it.
"By an abuse of notation like this, we can 'prove' that 19/94 = 1/4."

> I wouldn't say either of your possibilities is wrong, though.

All three are understandable.
--
Mark Brader, Toronto "'Run me,' Alice?"
m...@vex.net -- Tom Neff

My text in this article is in the public domain.

[Peter Moylan]

I don't believe that there is any ambiguity in the notation jaakov is
using. You can tell which meaning is meant by looking at the operands.

OK, there is technically an ambiguity in the special case of a string of
length 1, but I assume that in that case the two definitions collapse to
being identical with each other.

[Jaakov]

> I don't believe that there is any ambiguity in the notation jaakov is
> using. You can tell which meaning is meant by looking at the operands.
>
> OK, there is technically an ambiguity in the special case of a string of
> length 1, but I assume that in that case the two definitions collapse to
> being identical with each other.
>

Right.