Newsgroups: bit.listserv.words-l
Date:         Tue, 7 Apr 1992 22:14:59 CDT
 <WORDS-L%UGA.bitnet@VTVM2.CC.VT.EDU>
From:         Tushar Samant <tushar%ZAPHOD.UCHICAGO.EDU@VTVM2.CC.VT.EDU>
Subject:      econometrics (long)

>when you're right, you're right.
>
>Thanks for the explanation.
>
>JAG
>^bwp


You must understand that all this outpouring of jargon is a joke.
You are expected to say "My, what nerds," or something like that.

Second, despite what Steve says, I am not competing for shortest proof.
I want the most understandable proof, and one which uses only the
information which is really necessary.

The problem we were talking about is very general (which is why even I
could answer it, without knowing what "input requirement set" really
means.)

To see how simple the statement of the problem is, here is a special case
(which is not Steve's). You are given a chunk of three-space and a point
outside it. You want to know whether there is a point in the chunk which
nearest the outside point.

The answer seems to be an obvious "yes". But you must consider all kinds
"chunks" (subsets). For instance, take 1-space (the number line). Take
the chunk to be all points greater than zero. Take the outside point to
be zero. Now is there a point greater than zero which is nearest zero ?
It's not 0.1, nor 0.01, nor 0.001, ... in short there is NO such point.

Going back to 3-space, it is more or less obvious that the point in the
subset nearest to an outside point has to be on the boundary (surface).
Now if the boundary were peeled of from the chunk, there would be no
nearest point. This particular property of having a boundary can be
formalised, and it's called "closed"ness. The property of having the
boundary "peeled off" is made rigorous and called "closed"ness.

And now, we are just trying to prove that if the set is closed, it does
in fact have a nearest point.

Why go through all the nonsense of proving if it is obvious ? Because
once the definitions are made, "open" and "closed" mightn't just mean
what we vaguely think they mean. Besides, the definitions are much more
general, and so the proof got by formalising mental pictures will carry
over to all kinds of other situations. For instance, showing that a
minimum cost exists.

"Compactness" is a property got by distilling the familiar notion of
compactness. A set in 3-space which is closed ("has boundary") and
has a bounded extent is compact. So a set which has a fuzzy boundary
(i.e. no boundary, like only the inside of a three dimensional ball,
without the surface) or a set which extends to infinity is not compact.

In my opinion, it's the property of compactness which is crucial in this
problem, although not in the most obvious way (i.e. it is not as if the
input requirement set is itself compact.)

This is the best I can do. You will have to believe that all this can be
made completely rigorous, and in fact my argument and Steve's were totally
rigorous. But to make it rigorous, I am afraid SOME notation will have to
be used.

tushar

PS The reason why so much "math" is used in "economics" is that there is
no such thing as math vs. economics. If economists find that the methods
used by traditional mathematicians are useful to them, they use them.
Might as well ask why physicists learn so much math. We are human beings.
We can only think of math.
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