[Haskell-cafe] Category Theory woes

[Nick Rudnick]

Hans Aberg wrote:
> On 18 Feb 2010, at 19:19, Nick Rudnick wrote:
>
>> 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?
>>
>> * that's (for a very simple concept) 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). 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.»
>>
>> 418 bytes in my file system... how many in my brain...? Is it 
>> efficient, inevitable?
>
> Yes, it is efficient conceptually. The idea of closed sets let to 
> topology, and in combination with abstractions of differential 
> geometry led to cohomology theory which needed category theory solving 
> problems in number theory, used in a computer language called Haskell 
> using a feature called Currying, named after a logician and 
> mathematician, though only one person.
It is SUCCESSFUL, NO MATTER... :-)

But I spoke about efficiency, in the Pareto sense 
(http://en.wikipedia.org/wiki/Pareto_efficiency)... Can we say that the 
way in which things are done now cannot be improved??

Hans, you were the most specific response to my actual intention -- 
could I clear up the reference thing for you?

All the best,

    Nick


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>   Hans
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