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Hey Jamie,
I've got a few more thoughts on this whole situation; lemme know
what you think...
I wonder if it would be accurate to say that an
infinite series is convergent as long as it only has a certain similarity
of process in it, like; only addition and only
positive numbers, for example? In other words-- if it is always a
process of progressing from smaller to larger, with no reversals in the
process...
Because it seems to me that the reversals in the process are what
create the divergency-- adding then subtracting (or adding negative numbers to
positive numbers which is the same thing)... one step forward, one step back. A
rhythm is created, with the time signature specified by the parentheses. As soon
as time is involved, "when" those reversals happen is capable of
having a major impact on the whole thing.
I see this as a bit of yin and yang action: if it's all yin or all
yang, it's convergent, but as soon as there's a mix of both, then the way the
two are mixed is going to matter and timing/sequencing becomes a critical issue.
The fact that the universe is a balance of yin and yang means that there are not
likely to be many convergent situations, it seems to me. Would there even be
ANY?
Judith
PS: A friend pointed out a third result from this exercise: it
could equal -1. Antimatter! We'd have to shuffle the order of the numbers as
well as the parentheses, though: I'm not sure what that means, mathematically.
Any feedback on that?
----- Original Message -----
Sent: Monday, August 15, 2005 10:16
AM
Subject: Re: [ROSEN] Judith's
mathematical insight...
You have hit upon the crucial factor, Judith, which,
though
in and of itself is difficult to specific - since we live
inside
it and can't readily pare it down (through standard
reductionist logic as
your father did in your example here) -
the harder challenge comes
next.
The step which you just depicted essentially corresponds to
this:
- within a natural environment there are multiple
simultaneous
frames of refence (habitats), which, though they use the same
mathematics, ordinate them differently and therefore coordinate
in
different functional ways ... both independently -and-
inter-
involvedly. They are functionaly independent -and- bound at
the
same time.
The challenge is specifying their coordinated
behaviors, when
one uses base=0 and the other uses
base=1.
:-)
Jamie
> Judith Rosen
wrote:
>
> I actually had a mathematical insight over the
weekend while reading one of the
> self-published books I created
from an unpublished partial manuscript of my
> father's ("The Limits of
the Limits of Science"). I thought I might as well post
> it for the
list:
>
> In this particular passage, Robert Rosen described the
nature of the problem in
> mathematics in dealing with infinite sums and
infinite products. The problem
> is that judicious placing of the
parentheses can create entirely different
> results, in spite of the
fact that the numbers haven't changed at all. In the
> example he used:
1+ (-1) + 1+ (-1) +1 +(-1)...=? If the parentheses are placed
> like
this: (1 + (-1)) + (1 + (-1)) +..... the answer will be "zero". If the
>
parentheses are placed like this, however:
> 1 + ((-1) + 1) + ((-1) + 1)
+..... the answer will be "one"
>
> He than goes on to talk about
Cauchy, a mathematician, who introduced the notion
> of "convergence",
specifically to solve this problem. Cauchy's solution
> formulated a set
of criteria for discriminating between cases where the "value"
> (limit)
of an infinite sum or product was dependent on how it was evaluated
and
> when it was INdependent of that. RR's conclusion: Cauchy showed
that the
> generalization of the behavior of mathematics-- and therefore
the
> assumed applicability of approach-- from finite to infinite (at
least with
> regards to the behaviors and properties of sums and
products)-- wasn't
> appropriate: To preserve the independence of a
value or limit from how it is
> evaluated, an independence which is
automatic in the finite realm, we need to
> cut the "infinite"
world back down again-- by imposing convergence conditions.
>
>
Here's my little insight from the weekend: What's being illustrated by
this
> whole situation is the impact of the relational effect. The
relational aspect of
> causality, which is represented here by the
parentheses, is magnified in
> infinite processes, whereas in finite
processes it can be safely ignored. The
> reason has to do with the
nature of complexity: The parentheses represent the
> temporal aspects
of sequencing... It's common knowledge in non-mathematical
> realms that
the timing of "when" something happens is sometimes more important
>
than the fact that it happens at all.
>
>
> Furthermore:
the transition from the way mathematics predictably behaves in the
>
process of generating sums and products in the FINITE realm to the behavior
of
> mathematics engaged in the exact same processes-- but in the
INfinite realm-- is
> a transition that creates the potential for
relational aspects to have a far
> greater influence than they have in
the finite realm. Therefore, it is
> inappropriate to generalize from
finite to infinite. This exactly mirrors the
> case between the impact
of relations in systems with simple organizations and
> systems with
complex organizations. As such, it's not a paradox at all: it's
>
telling us something about the potential entailments of relations on
>
causality in an infinite realm. It's also an object lesson for science on
what
> not to do in making a similar transition from studying simple
systems to
> studying complex systems. Generalizing from finite to
infinite is a dangerous
> thing to do.
>
> Judith
Rosen
>
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