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Registered:
12/4/12
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Re: Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 6:17 AM
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On 3/14/2013 4:47 AM, david petry wrote:
>
> In a previous aritcle a while back, "marcus_b" wrote:
>
> ** start quote **
> In math, the great pons asinorum now is Cantor's diagonal proof. There seem to be scads of people out there who just cannot quite get it and who yearn to achieve their rightful 15 minutes of fame by trying to shoot it down, thus thrusting themselves ahead of Cantor in the pantheon of mathematical geniuses. Can you shed some light on the motivation here?
> ** end quote **
>
>
> The "cranks" who insist on pointing out the absurdity of Cantor's argument are puzzled that mathematicians "cannot quite get it".
>
> Cantor's argument relies on accepting the notion of an actual infinite, and that's problematic.
If you are going to paraphrase Cantor's argument, please do
so correctly.
Cantor's "argument" is an argument scheme.
It is an argument for constructing a counter-example
for some particular claim.
A particular claim must be made before Cantor's argument
can even apply.
That particular claim is that there is a single infinity
that is referred to in language as an object.
The referential notion is important because that
underlies the fact that a completed infinity is being
assumed in the claim.
Given that, the notion of infinity that will apply
is the notion of the natural numbers since Euclid
proved that there is no greatest prime number and
every natural number has a prime factorization.
So, the person who will apply Cantor's argument
can do nothing until the person referring to
infinity accepts an enumeration of real numbers
using some sort of infinitary naming scheme
(such as any expansion based on long division).
Once an enumeration claiming to list the infinite
collection of real numbers in an ordinal sequence
is presented and acceptable to both parties as
a starting point, Cantor's argument may be
applied to show that the enumeration cannot
be total.
One need not even believe in an actual infinite
in order to apply Cantor's argument.
For example, Wittgenstein understood perfectly
well how to apply Cantor's argument and he
certainly is not thought of as believing in
a completed infinity.
However, he also did not attack the mathematicians
who conducted investigations along those lines.
He recognized that he had a responsibility to
pursue answers to accommodate mathematical
practice. He also recognized that the ultimate
difficulty and ground for criticism has to do
with the nature of "all" and its interpretation.
Such considerations have led to fields
of mathematics based on constructive ideas.
Unfortunately, constructive mathematics has
yet to be able to explain the successes of the
differential calculus. In fact, it has yet to
be able to prove the fundamental theorem of
algebra.
It is all of the "other mathematics," including
the difficulty with infinitesimals, that leads
mathematicians to accept completed infinities
as more reasonable than other alternatives.
In other words, there are responsible ways to address
this problem and other ways which are less responsible.
You are making the latter choice.
But since you are, let me invite you to explain
a resolution to Zeno's paradox without completed
infinities. We can turn back the clock. We would
just like you to run it back forward.
That is not so easy now, is it?
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