[Haskell-cafe] Category Theory woes
Daniel Fischer
daniel.is.fischer at web.de
Thu Feb 18 14:20:49 EST 2010
Am Donnerstag 18 Februar 2010 19:19:36 schrieb Nick Rudnick:
> Hi Hans,
>
> agreed, but, in my eyes, you directly point to the problem:
>
> * doesn't this just delegate the problem to the topic of limit
> operations, i.e., in how far is the term «closed» here more perspicuous?
It's fairly natural in German, abgeschlossen: closed, finished, complete;
offen: open, ongoing.
>
> * that's (for a very simple concept)
That concept (open and closed sets, topology more generally) is *not* very
simple. It has many surprising aspects.
> the way that maths prescribes:
> + historical background: «I take "closed" as coming from being closed
> under limit operations - the origin from analysis.»
> + definition backtracking: «A closure operation c is defined by the
> property c(c(x)) = c(x).
Actually, that's incomplete, missing are
- c(x) contains x
- c(x) is minimal among the sets containing x with y = c(y).
> If one takes c(X) = the set of limit points of
Not limit points, "Berührpunkte" (touching points).
> X, then it is the smallest closed set under this operation. The closed
> sets X are those that satisfy c(X) = X. Naming the complements of the
> closed sets open might have been introduced as an opposite of closed.»
>
> 418 bytes in my file system... how many in my brain...? Is it efficient,
> inevitable? The most fundamentalist justification I heard in this regard
> is: «It keeps people off from thinking the could go without the
> definition...» Meanwhile, we backtrack definition trees filling books,
> no, even more... In my eyes, this comes equal to claiming: «You have
> nothing to understand this beyond the provided authoritative definitions
> -- your understanding is done by strictly following these.»
But you can't understand it except by familiarising yourself with the
definitions and investigating their consequences.
The name of a concept can only help you remembering what the definition
was. Choosing "obvious" names tends to be misleading, because there usually
are things satisfying the definition which do not behave like the "obvious"
name implies.
>
> Back to the case of open/closed, given we have an idea about sets -- we
> in most cases are able to derive the concept of two disjunct sets facing
> each other ourselves, don't we? The only lore missing is just a Bool:
> Which term fits which idea? With a reliable terminology using
> «bordered/unbordered», there is no ambiguity, and we can pass on
> reading, without any additional effort.
And we'd be very wrong. There are sets which are simultaneously open and
closed. It is bad enough with the terminology as is, throwing in the
boundary (which is an even more difficult concept than open/closed) would
only make things worse.
>
> Picking such an opportunity thus may save a lot of time and even error
> -- allowing you to utilize your individual knowledge and experience. I
When learning a formal theory, individual knowledge and experience (except
coming from similar enough disciplines) tend to be misleading more than
helpful.
> have hope that this approach would be of great help in learning category
> theory.
>
> All the best,
>
> Nick
>
> Hans Aberg wrote:
> > On 18 Feb 2010, at 14:48, Nick Rudnick wrote:
> >> * 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),
> >
> > I take "closed" as coming from being closed under limit operations -
> > the origin from analysis. A closure operation c is defined by the
> > property c(c(x)) = c(x). If one takes c(X) = the set of limit points
> > of X, then it is the smallest closed set under this operation. The
> > closed sets X are those that satisfy c(X) = X. Naming the complements
> > of the closed sets open might have been introduced as an opposite of
> > closed.
> >
> > Hans
>
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