Re: Database design

Mark Johnson wrote:
> "Marshall Spight" <marshall.spight_at_gmail.com> wrote:
>
> If something is said to be a set, the elements are unsorted.

Correct!

> They have no particular ranking with regard to any other element or member.

Correct!

> Fine. There can never be something known as an ordered or partially
> ordered set.

(Annoyingly, standard mathematical terminology includes both the terms "ordered set" and "partially ordered set" but neither one is a set! They are both ordered pairs of sets. An ordered pair is not a set.)

> The very idea of a proper order is meaningless when
> speaking of a set. Yes?

Yes.

> >> >Let's say you have some integers: 1 and 2.
>
> >> Let's say you have a roster of US Presidents.
>
> >we move on to your US Presidents example.
>
> Presidents is a great example.

It's certainly great at sidestepping the issues raised by my thought-experiment in dividing 1 by 2! The point of which was that values are immutable. We'll come back to that. Remember: values are immutable.

> >> So a set cannot be ordered because to place it in any order is to
> >> redefine it as non-set?
>
> >It is not "redefinition" at all.
>
> Then you deny its a set, at all.

It is the term "redefinition" I object to.

Let's say you have a set. Then you make a list of the members of the set, putting them in some specific order. Now you have a list. But you still have the original set! You can't redefine it, or make any least alteration to it in any way, because it's not a thing, it's a value. Values are immutable.

> >> To become a set, the most important attribute of that set must be
> >> discarded?
>
> >Sets cannot "become" things
>
> But things are becoming sets.

Nope. Sets cannot become things and things cannot become sets. No mathematical abstraction can become a real world object nor vice versa.

> To become a set, we're saying that the
> most important attribute, the most important bit of information, must
> simply be discarded?
>
> >We cannot "discard" an order attribute of a set, because
> >by definition no set ever had intrinsic order in the first place.
>
> So a list with an intrinsic ordering, which is basically most every
> list of every thing one might imagine,

Yes.

> is not a set,

Yes!

> and perhaps is not even properly an object for the RM, if you took it that far?

Well, that's not a very well-defined question. Set theory is foundational;
it can be used as the basis for substantially all of mathematics. Including
lists. But in set theory, instead of using lists as such, we instead have sets that represent the same information.

Consider the list of the first three presidents [Washington, Adams, Jefferson]

This structure is ordered. It is not a set. It could even be said to be intrinsically ordered, or to have a proper order, to use your terminology.

But we can construct an isomorphic set value using ordered pairs:

{ (1, Washington), (3, Jefferson), (2, Adams) }

This is a set of three members. The three members are unordered. This set contains exactly the same information as the previous list. Here is another way to write the *same* value:

{ (3, Jefferson), (1, Washington), (2, Adams) }

Here is a third way to write the *same* set:

2 Adams
1 Washington
3 Jefferson

That third way sometimes tricks people into thinking that relations are two-dimensional.

> >Thus it is not possible for order to be "the most important attribute."
>
> Unless it is. Sequence, rank and time. Sequence is not necessarily
> temporal but also spatial. But all can be termed - proper order.
>
> It's the most important.

I don't believe what you're describing has any mathematical validity. It is a psychological issue.

> >Now, you might have a *list* of things; that *would* have an
> >intrinsic order. You can make a set by taking the elements
> >of the list and removing the intrinsic order, and discarding
> >duplicates. The resulting set would not change the nature
> >of the original list.
>
> It would destroy it.

Nope. The list value would be unchanged, just like 1 remains unchanged when you divide 1 by 2. You get a result; the result is different than the original operands, but the operands themselves are unmodified.

If you used to programming in C or Java, you're probably trained to think in imperative terms--in terms of destructive updates. Math does not have destructive update. Set theory doesn't have destructive update. And actually there are programming languages that don't have destructive update. It's not a necessary idea (although it can make for efficient implementations.)

> If you remove the ordering, what you read,
> paragraph by paragraph, title by title, sidebar by sidebar, in the
> daily newspaper, or online, would become gibberish.

Well don't do that then.

> You'd have to try
> to piece it together, yourself, as a puzze. And on the other hand, if
> you necessarily retain the proper orde, then you have set of ordered
> entities and relations.

Not necessarily. You can capture the information contained in an order with yourself resorting to the use of order. So no information need be lost.

Marshall Received on Thu Feb 23 2006 - 08:44:10 CET