Re: .999... ?= 1

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 06/11/04 

Date: Fri, 11 Jun 2004 09:14:56 +0100

Robin Chapman wrote:
> Johan Kullstam wrote:

>>Not to be a smart ass, but what is a "number"? Is there some sort of
>>generally agreed upon definition?
>>
> As I mentioned in another post: what systems are counted as "number systems"
> is largely a historical accident. We generally regard the complexes
> as "numbers" but generally not matrices or quaternions. In short
> we know what rational numbers, real numbers etc. are, but "numbers" per
> se are not defined. That doesn't stop Blumsheit from seeing some
> essence of "number" such that "infinity" is one :-(

I feel mistaken.

Perhaps nobody will be able to seriously object against my objection
that it does not make any sense in any practical application of a continuum
to exclude a particular value as does mathematics for instance
when it distinguishes between open and closed intervals.

So what is wrong? People like David Rosenthal and Lothar Brendel who
trust more in tenets they learned than in their own shallow reasoning
feel great as an inquisitor: An outsider must be wrong.
However, this does not resolve the problem. Do we really know what rational
numbers are if numbers per se are not defined?

Where might be a basic fallacy? As long as we are not able to agree on
the meaning of a number it might be premature to exclude the possibility
  that a continuum like Q and the natural numbers behave differently
although they formally coincide in part.

I am suggesting a simple way out. Q and R include the natural numbers in
a manner that kills their discriminability. They are hidden somewhere in
what Cantor called an abyss. This must not worry us. For instance zero
can stll be exactly symbolized by various combinations of other numbers.
Theoretically nothing has changed. We have merely to check whether or
not it makes sense to include or exclude zero after we got aware that it
behaves like infinity outside of any possibility to attribute adjacent
smaller and larger values, respectively.

I reiterate it quite clearly: Any rational number requires infinitely
much of decimals (or alternatively an overbar even in case of omitted
zeros) as to describe a single position. While theoretically exact, this
       number is absolutely impractical. In so far, the natural number 1
and the coinciding real number 0.9 overbar = 1.0 overbar are formally
identical but of different use. This has to do with recognition of objects.

Now we are at the roots of mathematics. I faced protest when ascribing a
purpose to numbers. Do not believe numbers given by god. They are
manmade tools though very reasonable ones in the sense they were
discovered rather than invented. When I got aware that the primary
fields of applicaton for mathematics were counting and geometry I
perhaps correctly concluded that this might be a basic contradiction
which has been nicely expressed in Buridan's donkey.

Now I am still facing an ongoing outcray: "Do not destroy Cantor's
paradise! You are simply too stupid as to admire its greatness!"
Being a layman, I was surprized that already Abraham Robinson
refurbished the standard analysis. I am not yet sure but I got the
impression his hyperreals are rather artifical, not something that
ingeneers have to study thoroughly. Perhaps it would be better if the
physicists were told that as common sense dictates, the distinction
between open and closed intervals is absolutely necessary when one deals
with natural numbers but nonsense in case of well understood rational
and real numbers.

Eckard Blumschein

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