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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