[Haskell-cafe] Category Theory woes

[Nick Rudnick]

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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