Re: Division by zero. Go ahead and laugh.
From: Abraham Buckingham (twizlewink_at_hotmail.com)
Date: 10/07/04
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Date: 7 Oct 2004 15:15:24 -0700
"Lefty" <Ye@h.Right> wrote in message news:<UP09d.80795$He1.209@attbi_s01>...
> No, I do not "divide by zero". But it did occur to me to mention it here as
> an illustration of mathematical culture.
>
> I have said before that mathematicians "do not like" paradoxes and
> inconsistencies. It's all quite natural that they wouldn't, and I don't
> blame them. But there are many who have voiced their disagreement with this
> opinion. Please let me qualify my belief.
>
> Division by zero creates an inconsistency in arithmetic, and any result can
> be derived if such a thing is allowed. Now then, instead of always
> stipulating for example, 1/x, x not 0, we simply state that 1/0 is
> "undefined", and avoid it completely. We avoid it like the plague. The only
> place I could find these things being treated seriously is in singularity
> theory. But in general, people do not even acknowledge that this is a
> singularity. They just state flatly that 1/0 "does not exist". And other
> indeterminate forms are treated pretty much the same.
>
> The fact that arithmetic falls apart when you let 1/0 is fascinating and
> amazing. There is no reason to relegate this operation to the trash can. It
> would be better to acknowledge the inconsistency, the fact that we don't
> really understand why it exists, and to simply work around it.
>
>
> The problem of division by zero is that it creates many paradoxes.
> Mathematicians hate this, and therefore they have defined division by zero
> to be "non-existent". Why didn't you guys just throw away zero altogether ?
> I'll tell you why. Because you want to keep zero around because it's a handy
> thing to have, but those damned paradoxes have got to go, so we'll just
> "define them away".
>
> It's the biggest cover up since Watergate.
>
> There are no paradoxes or inconsistencies in mathematics, unless you remove
> all the ad hoc stipulations and band aids which hold it all together.
>
> Confess.
Perhaps you should look up 'Zero Divisor' on google, I think you'll
find a lot of answers about the work done on such problems. Note that
1/0 in the usual sense is still undefined not because it's being
supressed, simply because doing so producing interesting and intuitive
results. Nothing prevents studying systems where division by zero
works yet those objects are recognized as being distinct from the
common length and area understanding of division.
Thinking in those terms, it's easy to see why division by 0 is
undefined in a geometric sense. What would 1/0 mean? It would answer
the question 'if the area is 1, and the length of one side is 0, what
is the length of the other side' and we realize there is no answer. We
see that for our rectangle to have a meaningful area it can't have a
side of length 0. Even if the other side was infinite in length the
area would still not be 1, and so we chose to leave it undefined and
in that specific context since one dimensional rectangles don't make
sense. Other contexts where a zero divisor might arise exist, yet
addition and division when seen in geometric terms clearly doesn't
allow 1/0 to be meaningful or useful. I hope that gives you some
insight.
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