[Haskell-cafe] Category Theory woes

[Daniel Fischer]

Am Freitag 19 Februar 2010 01:49:05 schrieb Nick Rudnick:
> Daniel Fischer wrote:
> > 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.
>
> «concept» is a word of many meanings; to become more specific: Its
> *definition* is...
>

It isn't. You can make it look simple (Given a topology T, a set V is 
called "open in T", if V is an element of T.) by moving all of the 
difficult parts to the other definitions, but the entire group of 
definitions contains a nontrivial amount of difficulties (I've seen fairly 
bright students take a couple of weeks to wrap their head well around it, 
even though they've been familiar with the stuff in the context of 
euclidean space.).

> >> 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).
>
> Even more workload to master... This strengthens the thesis that
> definition recognition requires a considerable amount of one's effort...
>

I don't know what "recognition" should mean here, but certainly, 
understanding a definition, its (near but not trivial) consequences and its 
purpose requires considerable effort. Especially if it's an abstract and 
very general definition.

> >> 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.
>
> So if you state that the used definitions are completely unpredictable

I don't. 

> so that they have to be studied completely

Many definitions contain details which you probably wouldn't think about 
before you've banged your head against a wall very hard several times 
because you didn't know such details even existed.
If you decide to ignore the hard work and experience that have gone into 
the carefully crafted definitions, you are bound to make the same mistakes, 
run up the same blind alleys as those who have shaped the definition to 
what it now is.

> -- which already ignores that human brain is an analogous «machine» --,

What is an "analogous machine", and why would such a machine not be 
suitable for studying definitions?

> you, by information theory,
> imply that these definitions are somewhat arbitrary, don't you?

In a sense, of course the definitions are completely arbitrary. You could 
go ahead and define whatever you wish. But of course, some definitions are 
more useful than others, so the definitions in use aren't very arbitrary, 
they're mostly the ones determined to be most useful.

The names given to the defined concepts are more arbitrary. You could call 
an open set a Pangalactic Gargleblaster and a closed set a Ravenous 
Bugblatter Beast of Traal. Mathematically, it would make no difference. It 
would just be harder to remember which was which.

A good name invokes enough imagery to remind the hearer/reader what the 
definition was [not the details, but the general idea], but not so much as 
to give false ideas about the consequences of the definition.

> This in
> my eyes would contradict the concept such definition systems have about
> themselves.
>
> To my best knowledge it is one of the best known characteristics of
> category theory that it revealed in how many cases maths is a repetition
> of certain patterns.

Backwards. Category Theory is a product of the realisation how often 
certain patterns appear in different guise in different parts of 
mathematics. Of course, once started, it revealed many more.

> Speaking categorically it is good practice to
> transfer knowledge from on domain to another, once the required
> isomorphisms could be established. This way, category theory itself has
> successfully torn down borders between several subdisciplines of maths
> and beyond.
>
> I just propose to expand the same to common sense matters...
>
> >> 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.
>
> Really? As «open == not closed» can similarly be implied,
> bordered/unbordered even in this concern remains at least equal...
>

I'm not saying current terminology is optimal - it isn't.
But to replace established terminology, the proposed replacement should be 
clearly superior. I don't think bordered/unbordered fits that criterion 
(especially, since the topologigal term is boundary, not border).

> >> 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.
>
> Why does the opposite work well for computing science?

Does it?

>
> All the best,
>
>     Nick

To you too,

Daniel