Gregg Reynolds wrote:
On Thu, Feb 18, 2010 at 1:31 PM, Daniel Fischer
<daniel.is.fisc...@web.de <mailto:daniel.is.fisc...@web.de>> wrote:
Am Donnerstag 18 Februar 2010 19:55:31 schrieb Nick Rudnick:
> Gregg Reynolds wrote:
> -- 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...
Doesn't work. You need a lot of training in abstraction to learn very
abstract concepts. Joe Sixpack's common sense isn't prepared for that.
True enough, but I also tend to think that with a little imagination
even many of the most abstract concepts can be illustrated with
intuitive, concrete examples, and it's a fun (to me) challenge to try
come up with them. For example, associativity can be nicely
illustrated in terms of donning socks and shoes - it's not hard to
imagine putting socks into shoes before putting feet into socks. A
little weird, but easily understandable. My guess is that with a
little effort one could find good concrete examples of at least
category, functor, and natural transformation. Hmm, how is a
cake-mixer like a cement-mixer? They're structurally and functionally
isomorphic. Objects in the category Mixer?
:-) This comes close to what I mean -- the beauty of category theory
does not end at the borders of mathematical subjects...
IMHO we are just beginning to discovery of the categorical world beyond
mathematics, and I think many findings original to computer science, but
less to maths may be of value then.
And I am definitely more optimistic on «Joe Sixpack's common sense»,
which still surpasses a good lot of things possible with AI -- no
categories at all there?? I can't believe...
> > Both have a border, just in different places.
>
> Which elements form the border of an open set??
The boundary of an open set is the boundary of its complement.
The boundary may be empty (happens if and only if the set is
simultaneously
open and closed, "clopen", as some say).
Right, that was what I meant; the point being that "boundary" (or
border, or periphery or whatever) is not sufficient to capture the
idea of closed v. open.
;-)) I did not claim «bordered» is the best choice, I just said
closed/open is NOT... IMHO this also does not affect what I understand
as a refactoring -- just imagine Coq had a refactoring browser; all
combinations of terms are possible as before, aren't they? But it was
not my aim to begin enumerating all variations of «bordered»,
«unbordered», «partially ordered» and STOP...
Should I come QUICKLY with a pendant to «clopen» now? This would be
«MATHS STYLE»...!
I neither say finding an appropriate word here is a quickshot, nor I
claim trying so is ridiculous, as it is impossible.
I think it is WORK, which is to be done in OPEN DISCUSSION -- and that,
at the long end, the result might be rewarding, similar as the effort
put into a rename refactoring will reveal rewarding. ;-))
Trying a refactored category theory (with a dictionary in the
appendix...) might open access to many interesting people and subjects
otherwise out of reach. And deeply contemplating terminology cannot
hurt, at the least...
All the best,
Nick
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