Re: Where's respect? was Re: Corrective interpretation of real numbers

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 01/28/05 

Date: Fri, 28 Jan 2005 13:22:27 +0100

On 1/28/2005 8:07 AM, Andreas Homrighausen wrote:

> Please accept: I've you and Prof. Mueckenheim want to
> proof that e.g. set theory fails, you have to proof
> this inside the theory

No. Why should we accept that? Theory of communism was also not
refutable from inside, neither is any religion.
I adore all those who found out very important things.
In my eyes there is something like mathematical beauty, the beauty of
convincing simplicity and absence of arbitrary choices.
On the other hand, I learned to judge myself, and I found set theory a
Pandora's box of paradoxes.
Meanwhile I am pretty sure to roughly understand what went wrong and
why. The problem does not reside within set theory itself.

not inside your interpretation
> of set theory. Please also accept that there is no
> axiom asking for clearness or physical realization.

What arrogance!

> Prof. Mueckenheim says there are only finite many
> natural numbers because of the finite number of atoms
> in universe.

I do not entirely share this argument. He is nonetheless correct that
the notion natural number depicts a method rather than something
tangible. So I see it a question of context. For my understanding, the
use of the expression "unendlich große Zahl"(= infinite number ?) by
Weierstrass and earlier by Poisson indicates lacking awareness of what
Gauss pointed to: "... so protestiere ich zuvörderst gegen den Gebrauch
einer unendlichen Größe als einer Vollendeten, welche in der Mathematik
niemals erlaubt ist. Das Unendliche ist nur eine Facon de perler, indem
man eingentlich von Grenzen spricht, denen gewisse verhältnisse so nahe
kommen als man will während anderen ohne Einschränkung zu wachsen
verstattet ist."

> But Peano axioms _define_ the infiniteness
> of the natural numbers. If you want to show failures
> in math you have to accept the definitions of math!

The essential part of Peano's axioms has been stolen from Archimedes.
Declaring zero a natural number is already a highly questionable, maybe
controversial (I am not a mathematician) additional issue. My parents
and me had to accept quite different political systems. So we got
skeptical against all arbitrariness. No. We do not have to accept all
axioms in order to show failures in mathematics because mathematics
includes a lot of knowledge that was created over the millennia, and
final acceptance of redefinitions would demand careful justification.

> Does Prof. Mueckenheim has reasonable arguments? ;-)

I would appreciate an understandable and correct clarification to what
extent he is possibly wrong and how serious are his objections.

> I know for sure that my statement about physicists is wrong,

Perhaps this does not matter.

> but Prof. Mueckenheim (and you?) really believes that millions of
> mathematicians errs and only he's able to see the truth.

Millions of mathematicians? What a treasure! Hopefully Cantor's issue
does not occupy their creative power for too long.
I guess, the majority of people in any discipline do not feel any reason
for questioning the very basics.

>> This would fail any double blind test.
>
> Sure! In the same manner it's easily shown that Prof. Mueckenheim
> errs in the world of math. Any undergraduate student can to this.

Reality is a more reliable touchstone than perhaps wooden tenets.

> Prof. Mueckenheim and your "ideas" have interesting aspects, but not
> in the world of math. Maybe in philosophy...

Do not equate Cantor's self-made "paradise" with mathematics.
What about philosophers, Wittgenstein was perhaps not the only one who
scathingly criticized Cantor. Mückenheim and me are no philosophers. We
are just experienced enough as to draw conclusions from obvious
inconsistencies. This includes revealing that Cantor himself associated
his demanding ideas with his own weird religious basics.

The question of actual infinity is definitely not a philosophical but a
practical one.

Greetings,
Eckard

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