[Haskell-cafe] Category Theory woes

[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...
>   
>> 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...
>> 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 
so that they have to be studied completely -- which already ignores that 
human brain is an analogous «machine» --, you, by information theory, 
imply that these definitions are somewhat arbitrary, don't you? 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. 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...
>> 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?

All the best,

    Nick

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