Re: Corrective interpretation of real numbers
From: Will Twentyman (wtwentyman_at_read.my.sig)
Date: 01/27/05
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Date: Thu, 27 Jan 2005 12:33:05 -0500
Eckard Blumschein wrote:
> 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”.
The above description is useless for doing mathematics. Did Peirce have
anything more precise to offer?
> 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.
Perhaps if you defined what a "cut" is it would help. If you don't know
which, if either, branch contains x, your definition is lacking.
> - 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.
Correctness demands logical consistency. Graphical representations may
be useful. Note: however, that 0 is neither positive nor negative.
> - 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.
Huh? You think there is no distinction between an element being or not
being in a set? What gave you that idea?
> 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.
Perhaps the above is supposed to be building on other posts you've made
that I haven't read, but it completely fails to stand on its own. Could
you fill in a few details?
-- Will Twentyman email: wtwentyman at copper dot net
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