Re: A new definition of natural numbers

In article <455B13A5.8090902@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/13/2006 8:17 PM, Virgil wrote:
In article <455892BB.7080009@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/6/2006 9:31 AM, Virgil wrote:
In article <454EE10F.7070506@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

Perhaps Archimedes was the first one who gave a still unrivalled
compelling description of natural numbers.

Not by modern standards.

Concerning the basics of mathematics: Do we need questionable modern
standards or a comprehensively correct and as plausible as possible
logic foundation?

Kooks like Eckie can question modern math, but have no logical
justification for rejecting it.

In the beginning I just rejected the possibility that one may decide at
will how to deal with the nil when splitting IR into IR+ and IR-.
Meanwhile I feel forced to not swallow the whole set theory. I see it
the reason for some benign practical imperfections, and I inceasingly
realized that any claimed fundamental of it proves unproven but based on
pure intuition and populist illusions.


Judge yourself: Methods by Euclid, Newton, Leibniz, Euler, Gauss etc.
were overly successful and will continue to do so.

If their methods were "overly" successful, then it is quite proper that
they be replaced. But in any case, none of them would support Eckie's
view that further discovery or development in mathematics is impossible.

I do not deny possible progress in mathematics. On the contrary.
Abandoning an Utopia will not create that huge heap of rubble which was
warned of by Fraenkel. I see it a new chance. Proficient mathematicians
will not be affected at all from critical evaluation.

Is there any need to use Cantor's transfinite set theory, any example of
useful application of aleph_2 or even more nonsensical phantasmagoria?

Measure theory, among other mathematical developments, requires more
than countable sets.

This might be a wrong interpretation. I will look at the matter. Can you
point me to an example?

Eckard Blumschein

http://en.wikipedia.org/wiki/Measure_theory
http://mathworld.wolfram.com/LebesgueMeasure.html
http://mathworld.wolfram.com/BorelMeasure.html
http://mathworld.wolfram.com/ProbabilityMeasure.html
.

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