Re: Idempotence and reflexivity
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 14 Dec 2005 18:15:20 -0800
> "george" <greeneg@xxxxxxxxxx>
> > This is likely to be one of those things
> > where context matters.
Tron wrote,
> It is a danish single tome, "Politikens Filosofileksikon", Copenhagen 1983.
> Politiken, I believe, is the name of a publisher who is also behind
> the daily newspaper of the same name.
> It is edited by one Poul Luebcke. The individual entries are not signed,
> and since the list of attributed authors is very long, I'll save me the
> typing and y'all
> the guessing at philosophy professors from Denmark.
>
>
> The Entry
> Following a fairly long entry on reflexivity, there is a brief entry for
> "reflexive relation", which I'll give in full. The translation is mine, my
> comments in (/slashbrackets/).
>
> And I quote:
> "reflexive relation (from lat. re-, again, back, flectere (/no translation/)
> and relatio (/no translation/), properly (/the danish term "egentlig" ranges
> over "basically", "really", "essentially"/) that one item (/lit. "object"/)
> refers to or is connected to another).
> A relation, R, is a r.r. if for any arbitrary a, it is true (/lit.
> "holds"/), that aRa.
> This property of the relation is sometimes termed idempotence."
Well, yes, the context does help.
The operative word here is "sometimes".
Those times, in particular, would be,
"when R is a function".
When R is not a function, as Ken mentioned, it gets hard to
see how it could be idempotent, although I did give an example
that would apply to any relation that was composable with itself
(which, arguably, is any of them, in bare ZFC).
Relations that are not functions are just not even the right
type of thing for "idempotence".
There are many *different* types of things that could properly
be called idempotent, though. Any time there is any way in which
you are "operating" on something, the operator/operation could
be thought of as idempotent, but things that are NOT operators
can sometimes be thought of, metaphorically, as operators, or
used, instrumentally, as operators.
For example, you mentioned multiplication by zero, earlier.
If you think of that as an operation, IT is idempotent.
Starting with any number whatever, no matter how many times
you apply this operation to it (as long as it's a positive number of
times),
you get the same result. But there is also a usage when (given
that multiplication is the understood context) you could say that
1 is idempotent, because 1x1=1. What is the operation? Is it function
composition or is it multiplication? In any context where you can do
the latter, you can also do the former, so there are a lot of
different
ways in which this or that allegation could be right by accident.
Let me try again: juxtaposition can mean a lot of different things,
according to context. Idempotence means xx=x.
If x is a function then juxtaposition could mean function composition.
Or it could mean multiplication. Or it could mean a group product.
It needs to be something binary and associative, arguably, just
so you don't get into conflict about whether x(xy)xx could be different
from xx(yx)x.
.
- References:
- Idempotence and reflexivity
- From: Tron
- Re: Idempotence and reflexivity
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- Re: Idempotence and reflexivity
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