Re: Corrective interpretation of real numbers

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 01/27/05 

Date: Thu, 27 Jan 2005 18:44:10 +0100

On 1/27/2005 5:31 PM, fishfry wrote:
> In article <41F8930F.6090309@et.uni-magdeburg.de>,
> Eckard Blumschein <blumschein@et.uni-magdeburg.de> wrote:
>
>> but rather for something fundamentally different from Peirce¹s
>> description: ³A continuum is precisely that every part of which has
>> parts².
>
> That characterization fails to distinguish between the reals and the
> rationals, for example.

Of course, it is as simple as the so-called Archimedes axiom: For any
natural number one can find a larger one.
Euclid characterized the point as having no parts.

What about the history of reals, I was pointed to Ramus who perhaps for
the first time commonly used the name numbers for anything that can be
used for calculations. He explicitly included the reals. In ancient
times, alogos stood for what we denote irrationals. The distinguished
people dealt with geometry/inequality while ordinary people dealt with
primitive counting/equality. In so far, the unifying use paved the way
for modern science.

However, it is the irrationals inside the reals that are still causing
trouble because they have to fulfil the impossible, e.g. the quadrature
of the circle. Without having tried to find or even provide a formal
evidence, I imagine the difference between rational (i.e. immediately
IN-related numbers) and the irrational numbers to gradually vanish
towards infinity and the continuum respectively. Why not interpret an
irrational number like a rational one with just an infinite period?
Infinity is not a particular limit but the opposite of that. I guess,
merely the very best brains could intuitively comprehend the essence of
the notions infinity and continuum. Cauchy was definitely among them. He
verbally described and applied what was formalized, decades later but
still within his lifetime, as Weierstrass's epsilon delta limit. Most
likely he shied back from the seemingly rigorous but actually imprudent
while perhaps inevitable exclusion of the case delta equal zero.

Eckard Blumschein

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