On Thu, Feb 18, 2010 at 1:31 PM, Daniel Fischer <daniel.is.fischer@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?

 

> > 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.