Corrective interpretation of real numbers
From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 01/27/05
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Date: Thu, 27 Jan 2005 08:06:55 +0100
Do not get me wrong. If I mention that Weierstrass’s notion of a limit
does never permit delta to be zero, I am fully aware of the fact that
this notion is the decisive basis not merely for a most rigorous picture
but rather for something fundamentally different from Peirce’s
description: “A continuum is precisely that every part of which has
parts”. So I do not share the widespread lack of understanding. I just
would like to suggest a corrective interpretation of real numbers. Let
me exemplary explain why.
As long as we neglect the potentiality of infinity, and we do so with
great success, we cannot avoid some unreasonable consequences.
- Let e.g. any number x cut IR. Then there is no consensus whether x
belongs to the smaller or the larger numbers. For x=0, both IR+ and IR-
need a neutral element of addition and should offer the option of
reunification.
- Buridan’s donkey is suffering starvation between two full mangers
because of lacking preference for the left or the right one.
- “Correctness” demands to graphically represent |sign(x)|=0 like a
singular point.
- Practice would appreciate to be released from obligation to always
carefully distinguish in IR between open and closed intervals just for
unspecified “mathematical” reasons even if such distinction does
obviously not make any sense.
Common sense provides the only reasonable elucidation and corrective
interpretation of real numbers if applied before or after calculating
with mathematics based on Weierstrass’s notion as usual:
Imagine delta equal to zero: Now, any single number x does not matter
any more. Infinitely many are required as to change a function f(x) by
addition or removal of numbers. Singularities only belong to
distributions. Equality of two irrational numbers tends to evade
numerical examination. When mathematicians like Stifel and Weyl used
profane terminology like fog or sauce as to express the essence of an
untamed continuum, then perhaps they did know why.
The alluded simple “external” reinterpretation can be used without any
scruples but with much ease and success as compared to hyperreal or
surreal numbers in order to correct for unreasonable consequences of the
standard analysis.
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