[Haskell-cafe] Category Theory woes
Michael Matsko
msmatsko at comcast.net
Thu Feb 18 17:55:07 EST 2010
Nick,
Actually, clopen is a set that is both closed and open. Not one that is neither. Except in the case of half-open intervals, I can't remember talking much in topology about sets with a partial boundary.
Category theory-wise. No one seems to have mentioned MacLane's "Categories for the Working Mathematician." Although, I don't seem to recall instant enlightenment when I picked it up.
Mike
----- Original Message -----
From: "Nick Rudnick" <joerg.rudnick at t-online.de>
To: "Michael Matsko" <msmatsko at comcast.net>
Cc: haskell-cafe at haskell.org
Sent: Thursday, February 18, 2010 4:54:03 PM GMT -05:00 US/Canada Eastern
Subject: Re: [Haskell-cafe] Category Theory woes
Hi Mike,
of course... But in the same spirit, one could introduce a straightforward extension, «partially bordered», which would be as least as good as «clopen»... ;-)
I must admit we've come a little off the topic -- how to introduce to category theory. The intent was to present some examples that mathematical terminology culture is not that exemplary as one should expect, but to motivate an open discussion about how one might «rename refactor» category theory (of 2:48 PM).
I would be very interested in other people's proposals... :-)
Michael Matsko wrote:
Nick,
That is correct. An open set contains no point on its boundary.
A closed set contains its boundary, i.e. for a closed set c, Closure(c) = c.
Note that for a general set, which is neither closed or open (say the half closed interval (0,1]), may contain points on its boundary. Every set contains its interior, which is the part of the set without its boundary and is contained in its closure - for a given set x, Interior(x) is a subset of x is a subset of Closure(x).
Mike
----- Original Message -----
From: "Nick Rudnick" <joerg.rudnick at t-online.de>
To: "Michael Matsko" <msmatsko at comcast.net>
Cc: haskell-cafe at haskell.org
Sent: Thursday, February 18, 2010 3:15:49 PM GMT -05:00 US/Canada Eastern
Subject: Re: Fwd: [Haskell-cafe] Category Theory woes
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" <msmatsko at comcast.net>
To: "Nick Rudnick" <joerg.rudnick at t-online.de>
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" <joerg.rudnick at t-online.de>
To: "Gregg Reynolds" <dev at mobileink.com>
Cc: "Haskell Café List" <haskell-cafe at haskell.org>
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 < joerg.rudnick at 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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