[Haskell-cafe] Category Theory woes
Hans Aberg
haberg at math.su.se
Thu Feb 18 15:54:24 EST 2010
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.
Hans
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