Re: theory and practice: ying and yang

Alexandr Savinov wrote:
> Tony Andrews schrieb:
> > How can you possibly say that sets "do not exist in reality"?
>
> I mean that we cannot *represent* sets in such a way that they remain
> sets.

You're talking about implementation below the level of whatever language we're discussing. At the end of the day, it's all bits, right? But that doesn't mean we need program only in machine code. If a language supports only set primitives, then sets DO exist, with respect to that language.

> We can think of a number of things as a set but we are not able to
> store or pass them. We always need some underlying *representation*
> mechanism like arrays.

?

If you define a set, add items to it (discarding duplicates), allow intersection and union and such, and probably allow some sort of unordered traversal, then you have a set. Whether it's implemented as an array is irrelevant - its representation can still be a set.

> Sets do not exist in reality because elements cannot exist in vacuum
> like in set theory. Elements of a set *must* have some coordinates
> (offsets, positions etc.) in order to be qualified as (separate)
> elements.

Many languages distinguish object identity (truly separate) from equality (equivalence of "separate values").

> Thus when we say we have a set we normally mean we have some
> representation of them (but we do not care how concretely it is
> organized, particularly, we do not care its order).

I don't understand the importance of this point. Arrays don't exist either, nor do lists, nor does anything in a processor or disk save bits.

• Eric