Re: Fwd: [Haskell-cafe] Category Theory woes

18 Feb 2010

      Hi Mike,

so an open set does not contain elements constituting a border/boundary 
of it, does it?

But a closed set does, doesn't it?

Cheers,

    Nick

Michael Matsko wrote:
...
----- Forwarded Message -----
From: "Michael Matsko" 
To: "Nick Rudnick" 
Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada Eastern
Subject: Re: [Haskell-cafe] Category Theory woes
Gregg,
Topologically speaking, the border of an open set is called the 
boundary of the set.  The boundary is defined as the closure of the 
set minus the set itself.  As an example consider the open interval 
(0,1) on the real line.  The closure of the set is [0,1], the closed 
interval on 0, 1.  The boundary would be the points 0 and 1.
Mike Matsko
----- Original Message -----
From: "Nick Rudnick" 
To: "Gregg Reynolds" 
Cc: "Haskell Café List" 
Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada Eastern
Subject: Re: [Haskell-cafe] Category Theory woes
Gregg Reynolds wrote:
On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick
    mailto:joerg.rudnick@t-online.de> wrote:
IM(H??)O, a really introductive book on category theory still
        is to be written -- if category theory is really that
        fundamental (what I believe, due to its lifting of
        restrictions usually implicit at 'orthodox maths'), than it
        should find a reflection in our every day's common sense,
        shouldn't it?
Goldblatt works for me.
Accidentially, I have Goldblatt here, although I didn't read it before 
-- you agree with me it's far away from every day's common sense, even 
for a hobby coder?? I mean, this is not «Head first categories», is 
it? ;-)) With «every day's common sense» I did not mean «a 
mathematician's every day's common sense», but that of, e.g., a 
housewife or a child...
But I have became curious now for Goldblatt...
* the definition of open/closed sets in topology with the
        boundary elements of a closed set to considerable extent
        regardable as facing to an «outside» (so that reversing these
        terms could even appear more intuitive, or «bordered» instead
        of closed and «unbordered» instead of open),
Both have a border, just in different places.
Which elements form the border of an open set??
As an example, let's play a little:
Arrows: Arrows are more fundamental than objects, in fact,
        categories may be defined with arrows only. Although I like
        the term arrow (more than 'morphism'), I intuitively would
        find the term «reference» less contradictive with the actual
        intention, as this term
Arrows don't refer.
A *referrer* (object) refers to a *referee* (object) by a *reference* 
(arrow).
Categories: In every day's language, a category is a
        completely different thing, without the least
Not necesssarily (for Kantians, Aristoteleans?)
Are you sure...?? See 
http://en.wikipedia.org/wiki/Categories_(Aristotle) ...
If memory serves, MacLane says somewhere that he and Eilenberg
    picked the term "category" as an explicit play on the same term in
    philosophy.
In general I find mathematical terminology well-chosen and
    revealing, if one takes the trouble to do a little digging.  If
    you want to know what terminological chaos really looks like try
    linguistics.
;-) For linguistics, granted... In regard of «a little digging», don't 
you think terminology work takes a great share, especially at 
interdisciplinary efforts? Wouldn't it be great to be able to drop, 
say 20% or even more, of such efforts and be able to progress more 
fluidly ?
-g
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