[Haskell-cafe] Category Theory woes
Nick Rudnick
joerg.rudnick at t-online.de
Thu Feb 18 17:43:11 EST 2010
Gregg Reynolds wrote:
> On Thu, Feb 18, 2010 at 1:31 PM, Daniel Fischer
> <daniel.is.fischer at web.de <mailto:daniel.is.fischer at 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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