Re: x*x-1=0

Vadim Tropashko wrote:
> In article <95bc3t$jd2$1_at_news.tue.nl>,
> hidders_at_win.tue.nl (Jan Hidders) wrote:
> >
> > You are falling in the "is" trap that I already warned you about in
> > previous postings. You are reading the "is defined as" as "is" but
> > that is wrong, wrong, wrong. The claim is not that a pair <a,b>
> > *is* {a,{b}} but that this constitutes a *model*. And it is hardly
> > surprising that there are several models to choose from. That is
> > always the case with models and it is not a bug but a feature.
>
> I disagree for 2 reasons. It is rare case in science when you have
> competitive models living at the same time period.

Are you kidding!? Do you have any real experience in science? Did you read any real scientific articles? Did you ever talk to researchers how are actually publishing and doing research? For starters; do you know how many different string theories there are?

> <a,b> = {a, {b}} vs. <a,b> = {{a}, b} is unresolved 100 years
> already.

That's because there is nothing to resolve. Both models make the same predictions, so it is a meaningless question.

> Second, what properties of pairs (or ordered sets, in general)
> could i derive from this model? The only one I can think of is
> assymetry.

No, not even that can be really derived because that was explicitly designed into it.

> Maybe, this is why i despise it.

I still think you misunderstand what the translation of ordered pairs into sets means. The only thing that is claimed is that they behave as a certain type of set. That is interesting because then you can reuse the same reasoning rules for these pairs that that you already found for sets. But this is not really a "deep" fact, just a matter of economy. And it allows you to say things like "if we have a complete set of acioms for set theory, then we can derive a complete set of acioms for a theory of sets + ordered pairs."

> > > > The expressive power becomes greater.
> > >
> > > For me this sounds pretty much like "If I rename variables in
> > > Maxwell equations, then they won't describe electromagnetic waves
> > > any more".
> >
> > The column names are *observables*; you can point to them and see
> > them. If you are going to ignore some observables in Maxwell's
> > theory then the theory becomes a different theory, if not simply
> > meaningless.
>
> IMHO, domains are observable, not column names. Again, the naming of a
> column (and relations, either;-) is so arbitrary! How could we
> eliminate this human factor (they always do that in science:-)?

Tables are a mental construction, made by humans. If you take away the human factor there is nothing left. It doesn't matter if you like the definition of tables or not, they are what they are.

> Besides, my relational algebra equations (not those abstract examples
> that were mentioned in the thread earler, but the ones I derived from
> realistic SQL) look messy with explicit rename operations in them.

Well, like I said, you can remove the rename operator but then other things get messy. So the messiness is an inherent property of the things that the theory describes, i.e., tables. Of course you can try to change the definition of tables but then you are no longer describing tables as they are defined in the relational model.

-- 
  Jan Hidders