Re: .999... ?= 1

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 06/11/04 

Date: Fri, 11 Jun 2004 10:05:07 +0100

Eckard Blumschein wrote:

> 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.

There are many practical uses in mathematics for the distinction
between open and closed intervals. For instance [a,b] is compact,
while (a,b) is not. Compactness is a very useful property.

> 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.

OOOOoooooh! A victim complex.

> However, this does not resolve the problem. Do we really know what
> rational numbers are if numbers per se are not defined?

Yes.

> 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.

There are many differences between Q and N --- these number systems
have very different properties.

> I am suggesting a simple way out. Q and R include the natural numbers in
> a manner that kills their discriminability.

Private language time: "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.

Incoherent.

> I reiterate it quite clearly:

You never do.

> Any rational number requires infinitely
> much of decimals

bollocks

> (or alternatively an overbar even in case of omitted
> zeros)

bollocks

> as to describe a single position. While theoretically exact, this
> number is absolutely impractical.

what's "impractical" about 3/4 say?

> Now we are at the roots of mathematics. I faced protest when ascribing a
> purpose to numbers.

No, you faced protest for posting your incoherent ravings on sci.math.

> 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.

Again, your donkey is irrelevant to mathematics.

> Now I am still facing an ongoing outcray: "Do not destroy Cantor's
> paradise! You are simply too stupid as to admire its greatness!"

No one has said that to you. You have no power to "destroy
Cantor's paradise", no matter how stupid you be.

> 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.

No, as I have pointed out to you many times, any "nonsense"
in this thread is entirely contained in your babblings. 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

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