Re: The horse race

Mark Johnson wrote:
> "Marshall Spight" <marshall.spight_at_gmail.com> wrote:
>
> Then the example of a simple relation ought to be discussed. I've
> proposed many, many examples, which you seemed to suggest
> confused you no end in another message.

The wide variety of examples doesn't help clarify the situation; it obscures it. The ability to master a topic depends utterly on the ability to understand the basics. I suggest you put your energy in that direction, rather than trying to come up with more examples. Just think how much better equipped you'll be to find the order-related flaws in set theory once you know what it says on the topic!

> >Anyway, set theory doesn't discount order
>
> A simple relation is said to be unordered. If it represents ordered
> data - then what? It's not a relation? Or it must be divided into
> n-relations? And then what comes from that?

I have explained how this works in detail on two previous occasions; I am disinclined to type the info in a third time. I refer you to either of my previous posts explaining order relations.

Or perhaps you like wikipedia? Try this:

http://en.wikipedia.org/wiki/Order_relation

> >> Perhaps for other purposes that set may be unordered.
>
> >That's right.
>
> You're saying it always is, by definition.

Correct.

> But, here, let's see:
>
> >> If I give you page one, with story A in column left and story B
> >> beginning over in column right, with A paragraph one following the
> >> headlines and all falling below the banner, and A paragraph three
> >> following two and preceding four, one might say the set of all
> >> paragraphs is an unordered set. But there is an intrinsic order.
>
> >Yes. But that order is not part of the set of paragraphs S.
>
> They would be meaningless without it. It is intrinsic to both the
> semantics and the syntax of that group of paragraphs. They must appear
> in - proper order.

The amount of information contained in the order of the four paragraphs is tiny: less than 5 bits of data; not enough for a single ascii character.

You can't think of a single useful thing you could do with the paragraphs out of order? There are various unix utilities whose output will be the same regardless of the order that you fed them the paragraphs: spell, sort, wc, etc.

Perhaps it is not the case that the paragraphs *must* appear in proper order after all. Perhaps the value of the order is dependent on the application? Perhaps it is even the case that an order on a set could be abstracted away from the set. Even more outrageously, perhaps we could represent that order as a set--that is, represent order with an unordered data structure!

> >may capture that order in an order relation R.
>
> An ordered relation?

Nope. An order relation. Try reading the wikipedia article.

> >The pair (S, R) is called an ordered set.
>
> And set that is ordered.

I couldn't parse that.

> But in the example, elements of R are
> inextricably part of the elements of S,

Since I extracated them, I don't see how you can call them inextricable.

> and furthermore would be
> ordered by S-order, which supposedly can't be true in either case? It
> has to be one or the other.

I couldn't parse that.

> >The set S is not called ordered;
>
> But it is, nonetheless.

Incorrect by definition.

> >R and S are sets; they are unordered by definition.
>
> Which seems to be the problem, once again.

The problem here being that you can't seem to accept the idea that implicit order could be abstracted away--that it's not *required* for anything.

> I don't know if I'm supposed to be Abbott or Costello in this.

The one that kept asking "who's on first" was Costello.

> And information is not unordered.

*Some* information is unordered. The earlier example of the three physical dimensions was pretty good. If I have a box, 3' by 2' by 1', which of those dimensions comes first? Or are you going to claim that my 3x2x1 box is a different size than your 1x2x3 box?

And, as it turns out, all information can be captured with the vehicle of unordered information, including ordering information.

> It must hold to its proper order. It means nothing without it.

The wc example refutes this.

> >But R specifies a (not "the") order for set S.
>
> There is only one order for that group of paragraphs, titles,
> footnotes, etc.

There are exactly 24 paragraph orders to your four paragraph story. You are fixated on the one the author produced, but there are others, include longest to shortest, alphabetical, etc. No one of the 24 is distinguished in absolute terms; they are all equally valid although some may be more applicable to a particular task that others.

And again, there are lots of useful things you can do even without the order.

> Anything else would be characterized as computer
> error, a potential virus attack, somebody drunk on the job. The
> structure must be respected and retained with the text, links, markup
> and what have you. Without it, you have nothing but a 1000 unnumbered
> punch cards spilled onto the floor.

What if the task at hand was to identify how many cards were in the given deck? If I dropped the deck and it went everywhere, and then I picked it up again, it would have no impact on my ability to answer that question.

Marshall Received on Thu Feb 23 2006 - 05:54:01 CET