Re: a few questions for you was Re: 150 years without Gauss

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 02/24/05 

Date: Thu, 24 Feb 2005 09:23:14 +0100

Those 'experts' who are ingenious in humiliating me tend to be less in
position to understand the essentials.

On 2/23/2005 7:53 PM, Dirk Van de moortel wrote:

> I think mr. Blumthing is one of those people who think that
> "the limit for n to infinity of a sum of n terms"
> really is
> "an infinite sum of terms"
> It is very difficult to talk to people like that.

Let's clarify who thinks "that the limit for n to infinity of a sum of n
terms really is an infinite sum of terms".

First of all, I reckon G. Cantor among them. His second diagonal
argument would otherwise fail to declare the non-algebraic irrationals
ueberabzaehlbar. Cantor explicitly excluded the possibility that the
coefficients a_n go into zero periodicity for very large values of n
because this would mutate his irrational reals into rationals.

Secondly, I include all those formalists who defend Cantor's actual
infinity and accordingly reject finitism.

Those who obey the identity between 0.99... and 1.0... might possibly
not be aware that this identity refers to a limit. In other words, it is
only correct for an actually infinite series of nines.

On the other hand, Mueckenheim denies the actual infinity. He did not
yet get much support for that.

What about me, I am ready to obey any reasonable rule. However, I found
out that mathematics includes contradictory points of view.

Initially, I compared the Anaxagoras/Peirce notion of a smooth
continuum with the atomistic one according to the Weierstrass's
epsilon-delta limit. I understood that the restriction to rationals is
reasonable because an actually infinite host of numbers would evade
practical use. I am suggesting to clearly distinguish between the realm
of numbers and the quite different world of the genuine continuum.

Just recently I dealt with Cantor's arguments proving something in
excess of bijection, and I found out that the border between the
practical atomistic continuum and the strictly speaking numerically
pointless smooth continuum has got its mathematical expression as the
border between rational and irrational numbers.

In so far, the notion of real numbers cannot be ubiquitously the same.
Why else did Robinso(h)n introduce his hyperreal numbers *IR?
Bachmann (1892) introduced the reals by means of intervals whose size
tends towards zero. This approach and all the other ones do not demand
intervals of exactly zero size. In other words, Cantor's reals are
fundamentally different from Bachmann's to which Robinso(h)n obviously
refers.

Cantor's reals satisfied two desires. They provided something that is in
ideal agreement with the Anaxagoras/Peirce notion, and they justified
Cantor's sensational claim that there are different infinities. The
community was split into scepticists and enthusiasts. While the latter
got the better, they were not in position to deliver convincing evidence
for benefits justifying their high-sounding expectations. Perhaps almost
nobody realized that Cantor's irrationals are impractical. Instead,
everybody calculated with rationals but used the more general expression
reals instead. To engineers like me in particular, the name reals
inspires confidence. I not just feel cheated. I beg for permission to
reveal disillusioning peculiarities of Cantor's reals, e.g to replace >=
by simply >.

Eckard Blumschein

Relevant Pages

  • Re: Does the Calculus rest on Euclid?
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  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: Cantor Confusion
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