Das Kalenderblatt 090621

21/06/2009 - 09:41 von WM | Report spam
Most of the debate on the internet about Cantor's Theory is junk. The
topic is a crank magnet. Most of the people who participate in the
debate, have no deep understanding of the issues. However, hidden
within all the noise, there does seem to be some signal.

While the pure mathematicians almost unanimously accept Cantor's
Theory (with the exception of a small group of constructivists), there
are lots of intelligent people who believe it to be an absurdity.
Typically, these people are non-experts in pure mathematics, but they
are people
who have who have found mathematics to be of great practical value in
science and technology, and who like to view mathematics itself as a
science.

These "anti-Cantorians" see an underlying reality to mathematics,
namely, computation. They tend to accept the idea that the computer
can be thought of as a microscope into the world of computation, and
mathematics is the science which studies the phenomena observed
through that microscope. They claim that that paradigm encompasses all
of the mathematics which has the potential to be applied to the task
of understanding phenomena in the real world (e.g. in science and
engineering).

Cantor's Theory, if taken seriously, would lead us to believe that
while the collection of all objects in the world of computation is a
countable set, and while the collection of all identifiable
abstractions derived from the world of computation is a countable set,
there nevertheless "exist" uncountable sets, implying (again,
according to Cantor's logic) the "existence" of a super-infinite
fantasy world having no connection to the underlying reality of
mathematics. The anti-Cantorians see such a belief as an absurdity (in
the sense of being disconnected from reality, rather than merely
counter-intuitive).

The mathematicians claim that they can "prove" the existence of
uncountable sets, and hence there's nothing to be debated. But that
merely calls into question the nature of "proof". Certainly infinite
sets and power sets exist as absractions. But, abstractions don't
necessarily obey exactly that same laws of logic as directly
observable objects. Assuming otherwise can turn abstractions into
fantasies, and proofs into absurdities, and that's the crux of the
anti-Cantorian's argument.

The pure mathematicians tend to view mathematics as an art form. They
seek to create beautiful theories, which may happen to be connected to
reality, but only by accident. Those who apply mathematics, tend to
view mathematics as a science which explores an objective reality (the
world of computation). In science, truth must have observable
implications, and such a "reality check" would reveal Cantor's Theory
to be a pseudoscience; many of the formal theorems in Cantor's Theory
have no observable implications. The artists see the requirement that
mathematical statements must have observable implications as a
restriction on their intellectual freedom.

The "anti-Cantorian" view has been around ever since Cantor introduced
his ideas. [...] In the contemporary mainstream mathematical
literature, there is almost no debate over the validity of Cantor's
Theory. [...] It was the advent of the internet which revealed just
how prevalent the anti-Cantorian view still is; there seems to be a
never-ending heated debate in the Usenet newsgroups sci.math and
sci.logic over the validity of Cantor's Theory. Typically, the anti-
Cantorians accuse the pure mathematicians of living in a dream world,
and the mathematicians respond by accusing the anti-Cantorians of
being imbeciles, idiots and crackpots.

It is plausible that in the future, mathematics will be split into two
disciplines - scientific mathematics (i.e. the science of phenomena
observable in the world of computation), and philosophical
mathematics, wherein Cantor's Theory is merely one of many possible
formal "theories" of the infinite.

[David Petry, sci.math, sci.logic, 20 Juli 2005]
http://groups.google.com/group/sci....%2C+Cantor
 

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#1 Christopher Creutzig
21/06/2009 - 14:10 | Warnen spam
WM wrote:

The "anti-Cantorian" view has been around ever since Cantor introduced
his ideas. [...] In the contemporary mainstream mathematical
literature, there is almost no debate over the validity of Cantor's
Theory.



Genau das Gleiche gilt (überigens auch für einen sehr großen Teil des
Restes) auch für die Relatvitàtstheorie, die Evolutionstheorie,
Quantenmechanik etc. Um mal nur Beispiele us den Naturwissenschaften zu
bringen, die ja deutlich dichter an der Mathematik sind als das große
Spektrum der Geisteswissenschaften.

Freiheit ist eine Zumutung. Aber sie ist zumutbar.

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